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Understanding multiplication

Commutative property

Here you will learn about the commutative property, including what it is, and how to use it to solve problems.

Students will first learn about the commutative property as part of operations and algebraic thinking in 3rd grade.

What is the commutative property?

The commutative property says that when you add or multiply numbers, you can change the order of the numbers and the answer will still be the same.

For example,

This is also true when multiplying numbers.

The commutative property can be used to create friendly numbers when solving.

Friendly numbers are numbers that are easy to add or multiply mentally – like multiples of 10.

The commutative property lets us change the order and create friendlier numbers.

10 + 25 is easier to solve mentally than 3 + 25 + 7 = 28 + 7.

The commutative property lets us regroup and create friendlier numbers.

10 \times 8 is easier to solve mentally than 2 \times 8 \times 5=16 \times 5.

The commutative property can also be referred to as the commutative property of addition and the commutative property of multiplication, or more generally as the commutative law.

Common Core State Standards

How does this relate to 3rd grade math?

  • Grade 3 – Operations and Algebraic Thinking (3.OA.B.5) Apply properties of operations as strategies to multiply and divide. Examples: If 6 \times 4 = 24 is known, then 4 \times 6 = 24 is also known. (Commutative property of multiplication.) 3 \times 5 \times 2 can be found by 3 \times 5 = 15, then 15 \times 2 = 30, or by 5 \times 2 = 10, then 3 \times 10 = 30. (Associative property of multiplication.) Knowing that 8 \times 5 = 40 and 8 \times 2 = 16, one can find 8 \times 7 as 8 \times (5 + 2) = (8 \times 5) + (8 \times 2) = 40 + 16 = 56. (Distributive property.)

[FREE] Properties of Equality Check for Understanding Quiz (Grade 3 to 6)

Use this quiz to check your grade 3 to 6 students’ understanding of properties of equality. 10+ questions with answers covering a range of 3rd, 5th and 6th grade properties of equality topics to identify areas of strength and support!

How to use the commutative property

In order to use the commutative property:

Check to see that the operation is addition or multiplication.

Change the order of the numbers and solve.

Commutative property examples

Example 1: simple commutative property with addition.

Give an example of the commutative property using 4 + 9.

All the numbers are being added, so the commutative property can be used.

2 Change the order of the numbers and solve.

Changing the order in the equation does not change the sum.

Example 2: simple commutative property with multiplication

Give an example of the commutative property using 10 \times 6.

All the numbers are being multiplied, so the commutative property can be used.

Changing the order in the equation does not change the product.

Example 3: commutative property – addition with friendly numbers

Use the commutative property to create a friendly number and solve 6 + 32 + 14.

Example 4: commutative property – multiplication with friendly numbers

Use the commutative property to create a friendly number and solve 3 \times 8 \times 3.

Notice that when multiplying, friendly numbers can also be single digit numbers. If you know your basic facts, it is easier to solve 9 \times 8 than solving 3 \times 8 \times 3=24 \times 3.

Example 5: commutative property – addition with friendly numbers

Use the commutative property to create a friendly number and solve 41 + 17 + 9.

Example 6: commutative property – multiplication with friendly numbers

Use the commutative property to create a friendly number and solve 3 \times 5 \times 4.

Notice that when multiplying, friendly numbers can also be numbers that are basic facts. If you have memorized the basic multiplication facts from 1-12, it is easier to solve 12 \times 5 than solving 3 \times 5 \times 4=15 \times 4.

Teaching tips for the commutative property

  • Be intentional about choosing problems where the commutative property makes solving easier, since it is not always useful or necessary in all solving situations.
  • Instead of just telling students the commutative property definition, draw attention to examples of the commutative property when they naturally occur  in daily math activities. Record the different examples you see in the classroom on an anchor chart. Over time, students will start recognizing and using the property on their own. Then, after there are sufficient examples, you can introduce students to the property name and definition by using their own examples.
  • Include plenty of student discourse around this property so that students understand changing the order of numbers when adding or multiplying does not change the final result. This could include students sharing their thinking or critiquing the thinking of others.

Easy mistakes to make

  • Using the commutative property for subtraction or division The commutative property only works when changing the order of the numbers doesn’t change the answer. This is not true for subtraction or division and they are considered non-commutative arithmetic operations. For example, 11-5 = 6 \; AND \; 5-11 = -6 Changing the order of the numbers, changes the answer.
  • Thinking there is only one way to use the commutative property to solve with friendly numbers Sometimes there is more than one way to use the commutative property when solving. For example, \begin{aligned} & 6 \times 4 \times 5 \hspace{2.1cm} 6 \times 4 \times 5 \\ & =5 \times 4 \times 6 \hspace{1.7cm} =6 \times 5 \times 4 \\ & =20 \times 6 \hspace{2.05cm} =30 \times 4 \\ & =120 \hspace{2.4cm} =120 \end{aligned}
  • Confusing the order of operations Equations are always solved moving from left to right. It is not necessary to formally introduce students to the order of operations, but they need to understand and read equations in this way. Otherwise the commutative property may not mean anything to them.

Related properties of equality lessons

This commutative property topic guide is part of our series on properties of equality. You may find it helpful to start with the main properties of equality topic guide for a summary of what to expect or use the step-by-step guides below for further detail on individual topics. Other topic guides in this series include:

  • Properties of equality
  • Order of operations
  • Associative property
  • Distributive property

Practice commutative property questions

1. Which of the following equations shows the commutative property?

The commutative property says that changing the order in the equation does not change the product.

2. Which of the following equations shows the commutative property?

The commutative property says that changing the order in the equation does not change the sum.

3. Which of the following equations shows how to solve 2 \times 9 \times 5 using the commutative property?

4. Which of the following equations shows how to solve 37 + 28 + 23 using the commutative property?

5. Which of the following equations shows how to solve 8 \times 4 \times 5 using the commutative property to create a friendly number?

The commutative property says that changing the order in the equation does not change the product. Friendly numbers are numbers that are easy to multiply mentally – like multiples of 10.

6. Which of the following equations shows how to solve 16+18+22 using the commutative property to create a friendly number?

The commutative property says that changing the order in the equation does not change the sum. Friendly numbers are numbers that are easy to multiply mentally – like multiples of 10.

Commutative property FAQs

No, you can solve the numbers as they appear in the equation, without changing the order. The commutative property just gives you flexibility to add or multiply in a different order.

Yes, the commutative property can be used with integers, rational numbers and any real number, as long as they are all being added or multiplied.

The associative property of addition states that you can change the grouping of numbers when adding (using parentheses) and the sum will still be the same. The order of operations changes, but not the written order of the numbers in the equation. The commutative property of addition says you can change the written order of the numbers when adding and the sum will still be the same.

It is one of the properties of numbers for mathematical operations. This property states that any number added to 0 will still result in the same number (0 + a = a) or any number multiplied by 1 will still result in the same number (1 \times a=a).

The next lessons are

  • Addition and subtraction
  • Multiplication and division
  • Types of numbers

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solved problems in commutative algebra

Commutative Algebra

Can commutative algebra solve real-world problems.

solved problems in commutative algebra

Srikanth Iyengar

“When we first study advanced math, we learn to solve linear and quadratic equations, generally a single equation and in one variable,” said Srikanth Iyengar , Professor of Mathematics at the U. “But most real-world problems aren’t quite so easy—they often involve multiple equations in multiple variables.”

Finding explicit solutions to such equations is generally not feasible nor useful—it’s much more helpful to look for overall structure in the collection of all possible solutions. These solution sets are called algebraic varieties. The word algebraic indicates their origin is from polynomial equations, as opposed to equations involving things like trigonometric and exponential functions. Over the centuries, mathematicians have developed various tools to study these objects. One of them is to study functions on the space of solutions, and algebra is a good way to begin. These functions form a mathematical structure called a commutative ring. Commutative algebra is the study of commutative rings and modules, or algebraic structures over such rings.

Iyengar’s research focuses on understanding these structures, which have links to different areas of mathematics, particularly topology and representation theory.

Iyengar joined the Mathematics Department in 2014. He grew up in Hyderabad, India, and received a master’s degree and Ph.D. from Purdue University . Before joining the U, he taught at the University of Nebraska-Lincoln.

The foundation of commutative algebra lies in the work of 20th century German mathematician David Hilbert , whose work on invariant theory was motivated by questions in physics.

solved problems in commutative algebra

Srikanth Iyengar, Professor of Mathematics at the University of Utah

As a subject on its own, commutative algebra began under the name “ideal theory” with the work of mathematician Richard Dedekind , a giant of the late 19th and early 20th centuries. In turn, Dedekind’s work relied on the earlier work of Ernst Kummer and Leopold Kronecker . The mathematician responsible for the modern study of commutative algebra was Wolfgang Krull , who introduced concepts that are now central to the study of the subject, as well as Oscar Zariski, who made commutative algebra a foundation for the study of algebraic varieties.

“One of the things I enjoy about my research is how commutative algebra has so many connections to other things,” said Iyengar. “It makes for rich and lively research. Commutative algebra is continually reinvigorated by problems and perspectives from other fields.” Funding for Iyengar’s research is from the National Science Foundation . The Humboldt Foundation and the Simons Foundation have also provided support.

Commutative rings arise in diverse contexts in mathematics, physics, and computer science, among other fields. Within mathematics, besides functions on algebraic varieties, examples of commutative rings include rings of algebraic integers—the stuff of number theory. Commutative rings also arise, in myriad ways, in the study of symmetries of objects—algebraic topology, graph theory, and combinatorics, among others. One of the areas of physics where commutative algebra is useful is with string theory.

In recent years, ideas and   techniques from commutative algebra have begun to play an increasingly prominent role in coding theory, in reconstructions, and biology with neural networks.   While not everything Iyengar does day-to-day (or perhaps even in the span of a few years) has a direct impact in the field, mathematicians have a way of impacting other areas far from their original source, often decades later. There are many striking examples of this phenomenon. The “unreasonable effectiveness of mathematics” is well known. The phrase is part of a title of an article published in 1960 by Eugene Wigner , a Hungarian-American mathematician and theoretical physicist.

“I work by thinking about a piece of mathematics—perhaps it’s a research paper or a problem I run into somewhere in a textbook or a talk,” said Iyengar. “This sometimes leads to interesting research projects; at other times, it ends in a dead end. My perspective on research is that it’s more like a garden (or many interconnected gardens) waiting to be explored, rather than peaks to be climbed. Sure, there are landmarks but there’s rarely a point when I can say, Well, this is it—there’s nothing more to be achieved.’’

 - by Michele Swaner   First Published in Discover Magazine, Fall 2019

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solved problems in commutative algebra

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Applications of DG Homological Algebra in Commutative Algebra

Homological algebra comes from algebraic topology and provides valuable techniques in studying commutative ring theory. Auslander-Buchsbaum and Serre’s solution to the localization conjecture of regular local rings firmly established homological algebra as a powerful tool to solve problems in commutative algebra. Solving this conjecture by homological methods was a revolution that opened a new horizon to commutative algebraists and encouraged them to develop more homological techniques to study the properties of commutative rings. Another tool from algebraic topology (specifically, rational homotopy theory) that has sparked interest among commutative algebraists is differential graded (DG) homological algebra. The use of techniques from DG homological algebra was established by Avramov, Buchsbaum, Eisenbud, Foxby, Halperin, Kustin, Miller, and Weyman in commutative algebra, for instance, via DG algebra structures on Koszul complexes and free resolutions. It has been shown recently that these techniques can be applied to solve non-trivial problems in commutative algebra.

In this talk, we will discuss the following major problems:

Auslander-Reiten Conjecture (1975) . If M is a finitely generated module over a local ring $R$ with Ext i R (M, M ⊕ R) = 0 for all i >> 0, then pd R (M) < ∞.

Vasconcelos’ Conjecture (1974) . There are only finitely many semidualizing modules, up to isomorphism, over a local ring.

The Auslander-Reiten Conjecture originates from representation theory of Artin algebras. In a part of this talk, which is based on joint works with Luchezar Avramov, Srikanth Iyengar, and Sean Sather-Wagstaff, we present results about vanishing of homology over trivial extensions of DG algebras and use them to introduce new classes of commutative local rings that satisfy this conjecture. We then sketch a possible approach that involves techniques from differential equations which may result in a solution to this conjecture in full generality; this is based on an in-progress joint work with Maiko Ono and Yuji Yoshino. Vasconcelos’ Conjecture, is about an important class of finitely generated modules that are called semidualizing modules. Examples of these modules include the dualizing modules in the sense of Grothendieck. These modules were introduced by Foxby and rediscovered independently by several authors for different applications. Another part of the talk, which is based on a joint work with Sean Sather-Wagstaff, is devoted to sketch a complete solution to Vasconcelos’ Conjecture using geometric aspects of representation theory for DG algebras.

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9.3.1: Associative, Commutative, and Distributive Properties

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Learning Objectives

  • Identify and use the commutative properties for addition and multiplication.
  • Identify and use the associative properties for addition and multiplication.
  • Identify and use the distributive property.

Introduction

There are many times in algebra when you need to simplify an expression. The properties of real numbers provide tools to help you take a complicated expression and simplify it.

The associative, commutative, and distributive properties of algebra are the properties most often used to simplify algebraic expressions. You will want to have a good understanding of these properties to make the problems in algebra easier to solve.

The Commutative Properties of Addition and Multiplication

You may encounter daily routines in which the order of tasks can be switched without changing the outcome. For example, think of pouring a cup of coffee in the morning. You would end up with the same tasty cup of coffee whether you added the ingredients in either of the following ways:

  • Pour 12 ounces of coffee into mug, then add splash of milk.
  • Add a splash of milk to mug, then add 12 ounces of coffee.

The order that you add ingredients does not matter. In the same way, it does not matter whether you put on your left shoe or right shoe first before heading out to work. As long as you are wearing both shoes when you leave your house, you are on the right track!

In mathematics, we say that these situations are commutative—the outcome will be the same (the coffee is prepared to your liking; you leave the house with both shoes on) no matter the order in which the tasks are done.

Likewise, the commutative property of addition states that when two numbers are being added, their order can be changed without affecting the sum. For example, \(\ 30+25\) has the same sum as \(\ 25+30\).

\(\ 30+25=55\)

\(\ 25+30=55\)

Multiplication behaves in a similar way. The commutative property of multiplication states that when two numbers are being multiplied, their order can be changed without affecting the product. For example, \(\ 7 \cdot 12\) has the same product as \(\ 12 \cdot 7\).

\(\ 7 \cdot 12=84\)

\(\ 12 \cdot 7=84\)

These properties apply to all real numbers. Let’s take a look at a few addition examples.

Commutative Property of Addition

For any real numbers \(\ a\) and \(\ b\), \(\ a+b=b+a\).

Subtraction is not commutative. For example, \(\ 4-7\) does not have the same difference as \(\ 7-4\). The \(\ -\) sign here means subtraction.

However, recall that \(\ 4-7\) can be rewritten as \(\ 4+(-7)\), since subtracting a number is the same as adding its opposite. Applying the commutative property for addition here, you can say that \(\ 4+(-7)\) is the same as \(\ (-7)+4\). Notice how this expression is very different than \(\ 7-4\).

Now look at some multiplication examples.

Commutative Property of Multiplication

For any real numbers \(\ a\) and \(\ b\), \(\ a \cdot b=b \cdot a\).

Order does not matter as long as the two quantities are being multiplied together. This property works for real numbers and for variables that represent real numbers.

Just as subtraction is not commutative, neither is division commutative. \(\ 4 \div 2\) does not have the same quotient as \(\ 2 \div 4\).

Write the expression \(\ (-15.5)+35.5\) in a different way, using the commutative property of addition, and show that both expressions result in the same answer.

\(\ (-15.5)+35.5=20\) and \(\ 35.5+(-15.5)=20\)

Rewrite \(\ 52 \cdot y\) in a different way, using the commutative property of multiplication. Note that \(\ y\) represents a real number.

  • \(\ 5 y \cdot 2\)
  • \(\ 26 \cdot 2 \cdot y\)
  • \(\ y \cdot 52\)
  • Incorrect. You cannot switch one digit from 52 and attach it to the variable \(\ y\). The correct answer is \(\ y \cdot 52\).
  • Incorrect. This is another way to rewrite \(\ 52 \cdot y\), but the commutative property has not been used. The correct answer is \(\ y \cdot 52\).
  • Incorrect. You do not need to factor 52 into \(\ 26 \cdot 2\). The correct answer is \(\ y \cdot 52\).
  • Correct. The order of factors is reversed.

The Associative Properties of Addition and Multiplication

The associative property of addition states that numbers in an addition expression can be grouped in different ways without changing the sum. You can remember the meaning of the associative property by remembering that when you associate with family members, friends, and co-workers, you end up forming groups with them.

Below are two ways of simplifying the same addition problem. In the first example, 4 is grouped with 5, and \(\ 4+5=9\).

\(\ 4+5+6=9+6=15\)

Here, the same problem is worked by grouping 5 and 6 first, \(\ 5+6=11\).

\(\ 4+5+6=4+11=15\)

In both cases, the sum is the same. This illustrates that changing the grouping of numbers when adding yields the same sum.

Mathematicians often use parentheses to indicate which operation should be done first in an algebraic equation. The addition problems from above are rewritten here, this time using parentheses to indicate the associative grouping.

\(\ (4+5)+6=9+6=15\)

\(\ 4+(5+6)=4+11=15\)

It is clear that the parentheses do not affect the sum; the sum is the same regardless of where the parentheses are placed.

Associative Property of Addition

For any real numbers \(\ a\), \(\ b\), and \(\ c\),

\(\ (a+b)+c=a+(b+c)\).

The example below shows how the associative property can be used to simplify expressions with real numbers.

Rewrite \(\ 7+2+8.5-3.5\) in two different ways using the associative property of addition. Show that the expressions yield the same answer.

\(\ (7+2)+8.5-3.5=14\) and \(\ 7+2+(8.5+(-3.5))=14\)

Multiplication has an associative property that works exactly the same as the one for addition. The associative property of multiplication states that numbers in a multiplication expression can be regrouped using parentheses. For example, the expression below can be rewritten in two different ways using the associative property.

Original expression: \(\ -\frac{5}{2} \cdot 6 \cdot 4\)

Expression 1: \(\ \left(-\frac{5}{2} \cdot 6\right) \cdot 4=\left(-\frac{30}{2}\right) \cdot 4=-15 \cdot 4=-60\)

Expression 2: \(\ -\frac{5}{2} \cdot(6 \cdot 4)=-\frac{5}{2} \cdot 24=-\frac{120}{2}=-60\)

The parentheses do not affect the product. The product is the same regardless of where the parentheses are.

Associative Property of Multiplication

For any real numbers \(\ a\), \(\ b\), and \(\ c\), \(\ (a \cdot b) \cdot c=a \cdot(b \cdot c)\).

Rewrite \(\ \frac{1}{2} \cdot\left(\frac{5}{6} \cdot 6\right)\) using only the associative property.

  • \(\ \left(\frac{1}{2} \cdot \frac{5}{6}\right) \cdot 6\)
  • \(\ \left(\frac{5}{6} \cdot 6\right) \cdot \frac{1}{2}\)
  • \(\ 6 \cdot\left(\frac{5}{6} \cdot \frac{1}{2}\right)\)
  • \(\ \frac{1}{2} \cdot 5\)
  • Correct. Here, the numbers are regrouped. Now \(\ \frac{1}{2}\) and \(\ \frac{5}{6}\) are grouped in parentheses instead of \(\ \frac{5}{6}\) and \(\ 6\).
  • Incorrect. The order of numbers is not changed when you are rewriting the expression using the associative property of multiplication. How they are grouped should change. The correct answer is \(\ \left(\frac{1}{2} \cdot \frac{5}{6}\right) \cdot 6\).
  • Incorrect. The order of numbers is not changed when you are rewriting the expression using the associative property of multiplication. Only how they are grouped should change. The correct answer is \(\ \left(\frac{1}{2} \cdot \frac{5}{6}\right) \cdot 6\).
  • Incorrect. Multiplying within the parentheses is not an application of the property. The correct answer is \(\ \left(\frac{1}{2} \cdot \frac{5}{6}\right) \cdot 6\).

Using the Associative and Commutative Properties

You will find that the associative and commutative properties are helpful tools in algebra, especially when you evaluate expressions. Using the commutative and associative properties, you can reorder terms in an expression so that compatible numbers are next to each other and grouped together. Compatible numbers are numbers that are easy for you to compute, such as \(\ 5+5\), or \(\ 3 \cdot 10\), or \(\ 12-2\), or \(\ 100 \div 20\). (The main criteria for compatible numbers is that they “work well” together.) The two examples below show how this is done.

Evaluate the expression \(\ 4 \cdot(x \cdot 27)\) when \(\ x=-\frac{3}{4}\).

\(\ 4 \cdot(x \cdot 27)=-81\) when \(\ x=\left(-\frac{3}{4}\right)\)

Simplify: \(\ 4+12+3+4-8\)

\(\ 4+12+3+4-8=15\)

Simplify the expression: \(\ -5+25-15+2+8\)

  • Incorrect. When you use the commutative property to rearrange the addends, make sure that negative addends carry their negative signs. The correct answer is 15.
  • Correct. Use the commutative property to rearrange the expression so that compatible numbers are next to each other, and then use the associative property to group them.
  • Incorrect. Check your addition and subtraction, and think about the order in which you are adding these numbers. Use the commutative property to rearrange the addends so that compatible numbers are next to each other. The correct answer is 15.
  • Incorrect. It looks like you ignored the negative signs here. When you use the commutative property to rearrange the addends, make sure that negative addends carry their negative signs. The correct answer is 15.

The Distributive Property

The distributive property of multiplication is a very useful property that lets you rewrite expressions in which you are multiplying a number by a sum or difference. The property states that the product of a sum or difference, such as \(\ 6(5-2)\), is equal to the sum or difference of products, in this case, \(\ 6(5)-6(2)\).

\(\ \begin{array}{l} 6(5-2)=6(3)=18 \\ 6(5)-6(2)=30-12=18 \end{array}\)

The distributive property of multiplication can be used when you multiply a number by a sum. For example, suppose you want to multiply 3 by the sum of \(\ 10+2\).

\(\ 3(10+2)=?\)

According to this property, you can add the numbers 10 and 2 first and then multiply by 3, as shown here: \(\ 3(10+2)=3(12)=36\). Alternatively, you can first multiply each addend by the 3 (this is called distributing the 3), and then you can add the products. This process is shown here.

Screen Shot 2021-05-24 at 11.30.11 PM.png

\(\ \begin{array}{l} 3(10+2)=3(12)=36 \\ 3(10)+3(2)=30+6=36 \end{array}\)

The products are the same.

Since multiplication is commutative, you can use the distributive property regardless of the order of the factors.

Screen Shot 2021-05-24 at 11.49.14 PM.png

The Distributive Properties

For any real numbers \(\ a\), \(\ b\), and \(\ c\):

Multiplication distributes over addition:

\(\ a(b+c)=a b+a c\)

Multiplication distributes over subtraction:

\(\ a(b-c)=a b-a c\)

Rewrite the expression \(\ 10(9-6)\) using the distributive property.

  • \(\ 10(6)-10(9)\)
  • \(\ 10(3)\)
  • \(\ 10(6-9)\)
  • \(\ 10(9)-10(6)\)
  • Incorrect. Since subtraction isn’t commutative, you can’t change the order. The correct answer is \(\ 10(9)-10(6)\).
  • Incorrect. This is a correct way to find the answer. But the question asked you to rewrite the problem using the distributive property. The correct answer is \(\ 10(9)-10(6)\)
  • Incorrect. You changed the order of the 6 and the 9. Note that subtraction is not commutative and you did not use the distributive property. The correct answer is \(\ 10(9)-10(6)\).
  • Correct. The 10 is correctly distributed so that it is used to multiply the 9 and the 6 separately.

Distributing with Variables

As long as variables represent real numbers, the distributive property can be used with variables. The distributive property is important in algebra, and you will often see expressions like this: \(\ 3(x-5)\). If you are asked to expand this expression, you can apply the distributive property just as you would if you were working with integers.

Screen Shot 2021-05-25 at 12.04.35 AM.png

Remember, when you multiply a number and a variable, you can just write them side by side to express the multiplied quantity. So, the expression “three times the variable \(\ x\)” can be written in a number of ways: \(\ 3 x\), \(\ 3(x)\), or \(\ 3 \cdot x\).

Use the distributive property to expand the expression \(\ 9(4+x)\).

\(\ 9(4+x)=36+9 x\)

Use the distributive property to evaluate the expression \(\ 5(2 x-3)\) when \(\ x=2\).

When \(\ x=2,5(2 x-3)=5\).

In the example above, what do you think would happen if you substituted \(\ x=2\) before distributing the 5? Would you get the same answer of 5? The example below shows what would happen.

Combining Like Terms

The distributive property can also help you understand a fundamental idea in algebra: that quantities such as \(\ 3x\) and \(\ 12x\) can be added and subtracted in the same way as the numbers 3 and 12. Let’s look at one example and see how it can be done.

Add: \(\ 3 x+12 x\)

\(\ 3 x+12 x=15 x\)

Do you see what happened? By thinking of the \(\ x\) as a distributed quantity, you can see that \(\ 3x+12x=15x\). (If you’re not sure about this, try substituting any number for in this expression…you will find that it holds true!)

Groups of terms that consist of a coefficient multiplied by the same variable are called “like terms”. The table below shows some different groups of like terms:

Whenever you see like terms in an algebraic expression or equation, you can add or subtract them just like you would add or subtract real numbers. So, for example,

\(\ 10 y+12 y=22 y\), and \(\ 8 x-3 x-2 x=3 x\).

Be careful not to combine terms that do not have the same variable: \(\ 4 x+2 y\) is not \(\ 6 x y\)!

Simplify: \(\ 10 y+5 y+9 x-6 x-x\).

\(\ 10 y+5 y+9 x-6 x-x=15 y+2 x\)

Simplify: \(\ 12 x-x+2 x-8 x\).

  • Incorrect. It looks like you added all of the terms. Notice that \(\ -x\) and \(\ -8 x\) are negative. The correct answer is \(\ 5 x\).
  • Incorrect. You combined the integers correctly, but remember to include the variable too! The correct answer is \(\ 5x\).
  • Correct. When you combine these like terms, you end up with a sum of \(\ 5x\)
  • Incorrect. It looks like you subtracted all of the terms from \(\ 12x\). Notice that \(\ -x\) and \(\ -8 x\) are negative, but that \(\ 2 x\) is positive. The correct answer is \(\ 5 x\).

The commutative, associative, and distributive properties help you rewrite a complicated algebraic expression into one that is easier to deal with. When you rewrite an expression by a commutative property, you change the order of the numbers being added or multiplied. When you rewrite an expression using an associative property, you group a different pair of numbers together using parentheses. You can use the commutative and associative properties to regroup and reorder any number in an expression as long as the expression is made up entirely of addends or factors (and not a combination of them). The distributive property can be used to rewrite expressions for a variety of purposes. When you are multiplying a number by a sum, you can add and then multiply. You can also multiply each addend first and then add the products together. The same principle applies if you are multiplying a number by a difference.

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Applications of commutative algebra

Hi. I'm preparing a thesis in commutative algebra, and when I say this to my friends they always ask me what are the applications to "real-world", and I don't know what to answer. This let me think that I'm studying something useless. I'm studying on the Matsumura and on the Herzog-Bruns. Any of you know some applications of this abstract algebra to the real-world?

  • applications
  • applied-mathematics
  • ac.commutative-algebra
  • 4 $\begingroup$ I personally do not know of any uses (however, I am positive I will find out about some of them in the answers to this question). But why is it so terrible to be studying something simply because you find it fascinating? Something doesn't have to save lives or build bridges to be worthwhile. $\endgroup$ –  Zev Chonoles Apr 22, 2011 at 23:10
  • 2 $\begingroup$ The two generic answers are "algebraic number theory" and "algebraic geometry". $\endgroup$ –  Mark Apr 23, 2011 at 0:19
  • 10 $\begingroup$ I'd much rather hear about your thesis. $\endgroup$ –  Theo Johnson-Freyd Apr 23, 2011 at 0:31

6 Answers 6

The book "Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra" by Cox, Little & O'Shea, contains some "real world" applications, specifically chapter 6 (of the 3rd edition) is titled "Robotics and Automatic Geometric Theorem Proving".

Commutative Algebra and Algebraic Geometry are of relevance to Statistics and in recent years there was quite a lot of activity on this.

See e.g. http://en.wikipedia.org/wiki/Algebraic_statistics (and scroll down, the beginning is perhaps also interesting for your purpose, but what I mean is rather at the end of the page).

For example there is this book L. Pachter and B. Sturmfels. Algebraic Statistics for Computational Biology from 2005.

And there is a fairly recent (I believe) Activity Group of SIAM (Society for Applied and Industrial Mathematics) for Algebraic Geometry (which perhaps is close enough CA), about to hold its first conference http://www.siam.org/meetings/ag11/ (looking up the planery speakers should yield further details; there is a considerable intersection with names I. Rivin gives).

Another topic at the borderline of commutative algebra and number theory is Elliptic Curve Cryptography see http://en.wikipedia.org/wiki/Elliptic_curve_cryptography and also other cryptographic problems, but in part they feeel perhaps too number theoretic for you.

Finally, not really your question, but apparently the motivation: to convince your friends, depending on the background of your friends, I suggest to explain them the (simple) congruence arithmetic behind the final digit of the ISBN numbers. This was the only thing that I found that I felt had some real impact on the opinion of some of my friends on the usefulnes of pure mathematics.

The answer that I am going to give is implicitly contained in a few answers already given, but it is a bit too implicit, to my taste, so let me give it out and loud: Gröbner bases . When you solve a system of linear equations, you use Gaussian elimination, when you solve a system of polynomial equations of higher degrees, you use Gröbner bases, and it is very clear that solving systems of polynomial equations is something that people have to do for all sorts of applications.

That "very clear" is not just a belief held by a pure mathematician: on a few occasions that I talked about something mathematical to people doing research in some real world questions of statistics, biology, engineering, Gröbner bases would be the only aspect of somewhat advanced algebra, not just commutative algebra, that they would have ever heard of. You can see some relevant bits of software solving applied problems in various areas here: http://www.risc-software.at/en/ .

I can't resist from also saying that in some areas of pure maths, for a long time, saying the words "Gröbner bases" was a bit of faux pas , something that a true pure mathematician should rather leave as a discussion topic to people concerned with applications, something as silly and naive and so not worth mentioning as using a calculator to multiply two numbers. However, besides being a useful tool for computations, Gröbner bases and their generalisations also give methods to construct resolutions (starting from work of Anick in 1980s), and in particular to prove that a certain algebra (or an operad) is Koszul etc. So it certainly is something worth being aware of, really.

Google "Gunnar Carlsson" and "Rob Ghrist" and "Bernd Sturmfels" and "John Canny", and...

  • $\begingroup$ +1 although I'm not sure how the first two apply commutative algebra (beyond coefficient rings for homology)? $\endgroup$ –  Mark Grant Apr 23, 2011 at 0:36
  • $\begingroup$ @unknown (google): a fair point, but I choose to be chicken (or is it egg?) @Mark Grant: philistine that I am, I view any subject beset by many projective resolution as part of (or at least almost a part of) commutative algebra. $\endgroup$ –  Igor Rivin Apr 23, 2011 at 1:18
  • $\begingroup$ @Igor Rivin, thanks for the answer; of course, this is a legitimate point of view. $\endgroup$ –  user9072 Apr 23, 2011 at 1:43

Counting (partially) magic squares (and in fact combinatorics and commutative algebra have had really fruitful interactions).

One should also look at this question .

For applications in physics (string theory) see http://link.springer.com/chapter/10.1007%2F978-1-4614-5292-8_2 (Some Applications of Commutative Algebra to String Theory, by P.S. Aspinwall) and http://arxiv.org/abs/hep-th/0703279 (Topological D-Branes and Commutative Algebra, by the same author).

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Commutative Property – Definition, Examples, FAQs

Commutative property.

  • Commutative Property of Addition
  • Commutative Property of Multiplication

Solved Examples on Commutative Property

Practice problems on commutative property, frequently asked questions on commutative property.

The commutative property states that the numbers on which we operate can be moved or swapped from their position without making any difference to the answer. The property holds for Addition and Multiplication, but not for subtraction and division. Let’s see.

Commutative property for addition, Subtraction, multiplication and division

The above examples clearly show that the commutative property holds true for addition and multiplication but not for subtraction and division . So, if we swap the position of numbers in subtraction or division statements, it changes the entire problem. 

So, mathematically commutative property for addition and multiplication looks like this:

Add Numbers Using Column Addition Game

Commutative Property of Addition:

a + b = b + a; where a and b are any 2 whole numbers

Related Worksheets

1 and 2 more within 10: Horizontal Addition Worksheet

Commutative Property of Multiplication:

a × b = b × a; where a and b are any 2 non zero whole numbers

Use Cases of Commutative Property

  • Myra has 6 apples and 2 peaches. Kim has 2 apples and 6 peaches. Who has more fruits?

Even if both have different numbers of apples and peaches, they have an equal number of fruits, because 2 + 6 = 6 + 2.

  • Sara buys 3 packs of buns. Each pack has 4 buns. Mila buys 4 packs of buns and each pack has 3 buns. Who bought more buns?

Even if both have different numbers of bun packs with each having a different number of buns in them, they both bought an equal number of buns, because 3 × 4 = 4 × 3.

Example 1: Fill in the missing numbers using the commutative property.

  • _________ + 27 = 27 + 11
  • 45 + 89 = 89 + _________
  • 84 × ______ = 77 × 84
  • 118 × 36 = ________ × 118
  • 11; by commutative property of addition
  • 45; by commutative property of addition
  • 77; by commutative property of multiplication
  • 36; by commutative property of multiplication

Example 2: Use 14 × 15 = 210, to find 15 × 14.

Solution: 

As per commutative property of multiplication, 15 × 14 = 14 × 15. 

Since, 14 × 15 = 210, so, 15 × 14 also equals 210.

Example 3: Use 827 + 389 = 1,216 to find 389 + 827. 

As per commutative property of addition , 827 + 389 = 389 + 827. 

Since, 827 + 389 = 1,216, so, 389 + 827 also equals 1,216.

Example 4: Use the commutative property of addition to write the equation, 3 + 5 + 9 = 17, in a different sequence of the addends.

3 + 9 + 5 = 17 (because 5 + 9 = 9 + 5)

5 + 3 + 9 = 17 (because 3 + 5 = 5 + 3)

5 + 9 + 3 = 17 (because 3 + 9 = 9 + 3)

Similarly, we can rearrange the addends and write:

9 + 3 + 5 = 17

9 + 5 + 3 = 17

Example 4: Ben bought 3 packets of 6 pens each. Mia bought 6 packets of 3 pens each. Did they buy an equal number of pens or not?

Ben bought 3 packets of 6 pens each.

So, the total number of pens that Ben bought = 3 × 6

Mia bought 6 packets of 3 pens each.

So, the total number of pens that Ben bought = 6 × 3

By the commutative property of multiplication, 3 × 6 = 6 × 3. 

So, both Ben and Mia bought an equal number of pens.

Example 5: Lisa has 78 red and 6 blue marbles. Beth has 6 packets of 78 marbles each. Do they have an equal number of marbles?

Since Lisa has 78 red and 6 blue marbles.

So, the total number of marbles with Lisa = 78 + 6

Beth has 6 packets of 78 marbles each.

So, the total number of marbles with Beth = 6 × 78

Clearly, adding and multiplying two numbers gives different results. (Except 2 + 2 and 2 × 2.

That is, 78 + 6 ≠ 6 × 78

So, Lisa and Beth don’t have an equal number of marbles.

Attend this Quiz & Test your knowledge.

Which of the following represents the commutative property of addition?

Which of the following represents the commutative property of multiplication, which of the following expressions will follow the commutative property, choose the set of numbers to make the statement true. 5 + _____ = 4 + ______.

Can you apply the commutative property of addition/multiplication to 3 numbers?

Yes. By definition, commutative property is applied on 2 numbers, but the result remains the same for 3 numbers as well. This is because we can apply this property on two numbers out of 3 in various combinations.

Which operations do not follow commutative property?

Commutative property cannot be applied to subtraction and division.

What is the associative property of addition (or multiplication)?

This property states that when three or more numbers are added (or multiplied), the sum (or the product) is the same regardless of the grouping of the addends (or the multiplicands). That is, 

(a + b) + c = a + (b + c) (a × b) × c = a × (b × c) where a, b, and c are whole numbers.

For which all operations does the associative property hold true?

The Associative property holds true for addition and multiplication.

What is the distributive property of multiplication?

By the distributive property of multiplication over addition, we mean that multiplying the sum of two or more addends by a number will give the same result as multiplying each addend individually by the number and then adding the products together. That is,

a × (b + c) = (a × b) + (a × c) where a, b, and c are whole numbers.

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  • Commutative Property

The commutative property is one of several properties in math that allow us to evaluate expressions or compute mental math in a quicker, easier way. This is a well known number property that is used very often in math.

This property was first given it's name by a Frenchman named Francois Servois in 1814. He used the french word " commutatif ", which means "switchable".

Why the word "switchable"? The commutative property can be used with addition or multiplication and it says that you can add or multiply numbers in any order and you will still end up with the same answer.

Let's first take a closer look at the commutative property for addition.

Commutative Property of Addition

You can add numbers in any order and still have the same sum.

Algebraic Definition: a + b = b + a

5 + 3 = 8   and   3 + 5 = 8

6 + 8 + 4 = 18  and   8 + 4 + 6 = 18

So, not too bad! Just switch the numbers around in your addition problem and you have used this property.

Now let's take a quick look at the Commutative Property for Multiplication. It is utilized in exactly the same way.

Commutative Property of Multiplication

You can multiply numbers (factors) in any order and still end up with the same product.

Algebraic Definition: ab = ba

5(3) = 15    and      3(5) = 15

2(9)(5) = 90   and     5(2)(9) = 90

Same concept - just switch the numbers around in your multiplication problem and you will end up with the same product.

One thing we don't want to forget.....

The commutative property only works for addition and multiplication.  It does NOT work for subtraction or division.

For example:

5 - 3 = 2      but     3 - 5 = -2

18/2 = 9    but     2/18 = 1/9

So, Why is this Property so Helpful?

Is your mental math strong? If not, let's see how this property can make you a mental math whiz! Take a look at the following example.

Mental Math Strategies Using the Commutative Property

Let's say that you are given the following problem to compute mentally. Try to compute this problem mentally, in the order that it is presented!

20 + 3 + 55 + 7

Yeah, this is tough because you have to think about the sum of 23 + 55 which is 78 and then add another 7 to get 85.

Now, let's experiment.

Same numbers, different order. Let's compute this mentally and see if we can work any faster.

20 + 55 + 3 + 7

Did it take a shorter amount of time? This is easier because you know that                20 + 55 is 75 and 3 + 7 is 10, so 75 + 10 = 85.

We used the commutative property to rewrite these numbers in a different order. I ended up with the same answer, but it was easier to add 20 +55 and then 7+3.

The commutative property helps to make your mental math much faster!

Work smarter not harder!! Make sure that you make yourself more efficient with mental math.

The commutative property is an extremely helpful property used in math.

The associative property and distributive property are the two other number properties that are also used very often in math.

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  • Math Article

Commutative Property

In mathematics, commutative property or commutative law explains that order of terms doesn’t matter while performing arithmetic operations . 

Commutative property is applicable only for addition and multiplication processes. Thus, it means we can change the position or swap the numbers when adding or multiplying any two numbers. This is one of the major properties of integers .

For example: 1+2 = 2+1 and 2 x 3 = 3 x 2. 

What is Commutative Property?

As we already discussed in the introduction, as per the commutative property or commutative law, when two numbers are added or multiplied together, then a change in their positions does not change the result. 

Commutative property

  • 2+3 = 3+2 = 5
  • 2 x 3 = 3 x 2 = 6
  • 5 + 10 = 10 + 5 = 15
  • 5 x 10 = 10 x 5 = 50

So, there can be two categories of operations that obeys commutative property:

  • Commutative property of addition
  • Commutative property of multiplication

Although the official use of commutative property began at the end of the 18th century, it was known even in the ancient era.

The word, Commutative, originated from the French word ‘commute or commuter’ means to switch or move around, combined with the suffix ‘-ative’ means ‘tend to’. Therefore, the literal meaning of the word is tending to switch or move around. It states that if we swipe the positions of the integers, the result will remain the same.

Commutative Property of Addition

According to the commutative property of addition, when we add two integers, the answer will remain unchanged even if the position of the numbers are changed.

Let A and B be the two integers, then;

Examples of Commutative Property of Addition

  • 1 + 2 = 2 + 1 = 3
  • 3 + 8 = 8 + 3 = 11
  • 12 + 5 = 5 + 12 = 17

Commutative Property of Multiplication

As per the commutative property of multiplication, when we multiply two integers, the answer we get after multiplication will remain the same, even if the position of the integers are interchanged.

Examples of Commutative Property of Multiplication

  • 1 × 2 = 2 × 1 = 2
  • 3 × 8 = 8 × 3 = 24
  • 12 × 5 = 5 × 12 = 60

Important Facts Of Commutative Property

  • Commutative property is only applicable for two arithmetic operations: Addition and Multiplication
  • Changing the order of operands, does not change the result 
  • Commutative property of addition: A + B = B + A
  • Commutative property of multiplication: A.B = B.A

Other Properties

The other major properties of addition and multiplication are:

  • Associative Property
  • Distributive Property

Now, observe the other properties as well here:

Associative Property of Addition and Multiplication

According to the associative law, regardless of how the numbers are grouped, you can add or multiply them together, the answer will be the same. In other words, the placement of parentheses does not matter when it comes to adding or multiplying.

  • 1 + (2+3) = (1+2) + 3 → 6
  • 3 x (4 x 2) = (3 x 4) x 2 → 24

Distributive Property of Multiplication

The distributive property of Multiplication states that multiplying a sum by a number is the same as multiplying each addend by the value and adding the products then.

According to the Distributive Property, if a, b, c are real numbers, then:

a x (b + c) = (a x b) + (a x c)

  • 2 x (5 + 8) = (2 x 5) + (2 x 8)
  • 2 x (13) = 10 + 16

There are certain other properties such as Identity property, closure property which are introduced for integers.

Non-Commutative Property

Some operations are non-commutative. By non-commutative, we mean the switching of the order will give different results. The mathematical operations, subtraction and division are the two non-commutative operations. Unlike addition, in subtraction switching of orders of terms results in different answers.

Example: 4 – 3 = 1 but 3 – 4 = -1  which are two different integers.

Also, the division does not follow the commutative law. That is,

2 ÷ 6  = 1/3

Hence, 6 ÷ 2 ≠  2 ÷ 6

Solved Examples on Commutative Property

Example 1: Which of the following obeys commutative law?

  • 36 – 6

Solution: Options 1, 2 and 5 follow the commutative law

Explanation:

  • 3 × 12 = 36 and

       12 x 3 = 36

=> 3 x 12 = 12 x 3 (commutative)

  • 4 + 20 = 24 and

     20 + 4 = 24

     => 4 + 20 = 20 + 4 (commutative)

  • 36 ÷ 6 = 6 and

     6 ÷ 36 = 0.167

=> 36 ÷ 6 ≠ 6 ÷ 36  (non commutative)

  • 36 − 6 = 30 and 

      6 – 36 = – 30

=> 36 – 6 ≠ 6 – 36  (non commutative)

  • −3 × 4 = -12 and

       4 x -3 = -12

=> −3 × 4 = 4 x -3  (commutative)

Q.2: Prove that a+ b = b+a if a = 10 and b = 9.

Sol: Given that, a = 10 and b = 9 

LHS = a+b = 10 + 9 = 19   ……(1)

RHS = b + a = 9 + 10 = 19 ……(2)

By equation 1 and 2, as per commutative property of addition, we get;

Hence, proved.

Q.3: Prove that A.B = B.A, if A = 4 and B = 3.

Sol: Given, A = 4 and B = 3.

A.B = 4.3 = 12  ….. (1)

B.A = 3.4 = 12  …..(2)

By eq.(1) and (2), as per the commutative property of multiplication, we get;

Practice Questions

Find which of the following is the commutative property of addition and multiplication.

  • 3 + 4 = 4 + 3
  • 10 x 7 = 7 x 10
  • 8 x 9 = 9 x 8
  • 6 + 4 = 4 + 6

Frequently Asked Questions – FAQs

What is commutative property give examples., what is commutative property of addition, what is the commutative property of multiplication, what are the major four properties in maths, what is the difference between commutative and associative property, leave a comment cancel reply.

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Commutative Property – Definition, Examples, FAQs

Created: January 4, 2024

Last updated: January 10, 2024

Welcome to Brighterly , the ultimate destination for children to discover the fascinating world of math! At Brighterly, we believe that learning math can be both enjoyable and educational. Our mission is to make math concepts accessible and engaging for children of all ages.

Today, we’ll unravel the mystery behind the Commutative Property – a fundamental concept that lies at the heart of arithmetic operations. This simple yet powerful rule has the potential to transform the way your child approaches math problems, making the learning process more exciting and rewarding. Are you ready to embark on this mathematical adventure? Let’s get started!

What is Commutative Property?

The Commutative Property is a basic principle in mathematics that states that the order of numbers doesn’t matter when you’re performing certain arithmetic operations. In other words, you’ll get the same result regardless of the order in which you arrange the numbers. This property applies to two primary operations: addition and multiplication. The Commutative Property does not apply to subtraction and division, as the order of the numbers does matter in these cases.

By understanding the Commutative Property, children can develop their mental math skills and improve their ability to solve problems more efficiently. This property is crucial for building a strong foundation in arithmetic and algebra.

Commutative Property of Addition:

The Commutative Property of Addition states that changing the order of numbers in an addition equation does not affect the sum. In other words, if you’re adding two numbers, it doesn’t matter which one you put first – the result will be the same. Mathematically, this can be expressed as:

a + b = b + a

For example:

  • 3 + 5 = 5 + 3
  • 8 + 2 = 2 + 8

This property can be extended to more than two numbers as well:

  • 3 + 5 + 7 = 5 + 3 + 7 = 7 + 5 + 3

Commutative Property of Multiplication:

The Commutative Property of Multiplication works similarly to the property of addition. It states that changing the order of numbers in a multiplication equation does not affect the product. Mathematically, this can be expressed as:

a × b = b × a

  • 4 × 6 = 6 × 4
  • 9 × 3 = 3 × 9

This property also extends to more than two numbers:

  • 4 × 6 × 2 = 6 × 4 × 2 = 2 × 6 × 4

Commutative Property vs Associative Property

While the Commutative Property focuses on the order of numbers, the Associative Property deals with the grouping of numbers. The Associative Property states that changing the grouping of numbers does not affect the result, as long as the operation remains the same. It is important to note that both the Commutative and Associative Properties apply to addition and multiplication, but not to subtraction and division.

For example, the Associative Property of Addition can be expressed as:

(a + b) + c = a + (b + c)

And the Associative Property of Multiplication can be expressed as:

(a × b) × c = a × (b × c)

Solved Examples on Commutative Property

Let’s look at some examples that demonstrate the Commutative Property in action:

  • Addition: 5 + 9 = 9 + 5 In this case, 5 + 9 = 14, and 9 + 5 = 14. Both sums are equal, so the Commutative Property of Addition holds true.
  • Multiplication: 7 × 4 = 4 × 7 In this case, 7 × 4 = 28, and 4 × 7 = 28. Both products are equal, so the Commutative Property of Multiplication holds true.

Practice Problems On Commutative Property

Now that you understand the Commutative Property, it’s time to put your knowledge to the test! Try solving these practice problems:

Addition: a) 6 + 11 = ? + 6 b) 15 + 7 = ? + 15

Multiplication: a) 8 × 5 = ? × 8 b) 3 × 10 = ? × 3

Feel free to use the Commutative Property to help you solve these problems!

In conclusion, the Commutative Property is a pivotal concept in mathematics that plays a significant role in simplifying addition and multiplication problems. By mastering this property, children can unlock their full potential in mental math and enhance their problem-solving abilities.

At Brighterly, we’re committed to nurturing your child’s mathematical prowess and cultivating a lifelong love for learning. Always remember, the Commutative Property is your trusty ally when it comes to addition and multiplication, but not subtraction and division. Keep exploring the amazing world of math with Brighterly, and watch your child’s confidence and skills soar to new heights!

Frequently Asked Questions On Commutative Property

Does the commutative property apply to subtraction.

No, the Commutative Property does not apply to subtraction. The order of numbers in a subtraction problem does matter. For example, 7 – 5 ≠ 5 – 7.

Does the Commutative Property apply to division?

No, the Commutative Property does not apply to division. The order of numbers in a division problem does matter. For example, 12 ÷ 4 ≠ 4 ÷ 12.

How does the Commutative Property help in solving problems?

The Commutative Property helps by allowing you to rearrange numbers when adding or multiplying, making calculations easier and more efficient. This can be especially helpful in mental math and when solving more complex problems.

Are there any other properties related to the Commutative Property?

Yes, the Associative Property is another important property related to the Commutative Property. While the Commutative Property deals with the order of numbers, the Associative Property deals with the grouping of numbers.

To learn more about the Commutative Property and other math concepts, check out these reputable sources:

  • National Council of Teachers of Mathematics (NCTM) – Commutative Property
  • OpenStax – Commutative Property
  • BBC Bitesize – Commutative Property

Happy learning, and have fun exploring the world of math with Brighterly!

I am a seasoned math tutor with over seven years of experience in the field. Holding a Master’s Degree in Education, I take great joy in nurturing young math enthusiasts, regardless of their age, grade, and skill level. Beyond teaching, I am passionate about spending time with my family, reading, and watching movies. My background also includes knowledge in child psychology, which aids in delivering personalized and effective teaching strategies.

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The number 43000 is spelled as “forty-three thousand”. It is forty-three sets of one thousand each. For example, if you have forty-three thousand coins, you have forty-three thousand coins and then extra coins to make forty-three thousand. Thousands Hundreds Tens Ones 43 0 0 0 How to Write 43000 in Words? Writing the number 43000 […]

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In the grand, enchanting world of mathematics, a sound understanding of simple yet profound concepts like halves can pave the way to a brighter future. At Brighterly, we believe in fostering a deeper understanding of these core concepts, breaking them down in a way that children not only understand but also enjoy. This immersive journey […]

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  • Commutative Algebra

solved problems in commutative algebra

Commutative Algebra Explained: In-Depth Examples, Equations, and Expert Insights

mugenziwzn6

The permutations and combinations of a b c d taken 3 at a time are respectively.

Ghillardi4Pi

Show that the image of P n × P m under the Segre embedding ψ is actually irreducible.

musicbachv7

Is function composition commutative?

Lillianna Andersen

If B and C are A -algebras with ring morphisms f : A → B and g : A → C , and D = B ⊗ A C is an A -algebra with morphism a ↦ f ( a ) ⊗ g ( a ) , then u f = v g , where u : B → D is u ( b ) = b ⊗ 1 .The map v : C → D is not defined in the text, but my guess is it's v ( c ) = 1 ⊗ c .I don't understand why the diagram is commutative though. That would imply f ( a ) ⊗ 1 = 1 ⊗ g ( a ) for all a ∈ A . Is that true, or is v something else?Added: On second thought, does this follow since f ( a ) ⊗ 1 = a ⋅ ( 1 ⊗ 1 ) and 1 ⊗ g ( a ) = a ⋅ ( 1 ⊗ 1 ) where ⋅ is the A -module structure on D ? A → f B g ↓ # ↓ u C → v D

slijmigrd

Let A ⊆ B be rings, B integral over A ; let q , q ′ be prime ideals of B such that q ⊆ q ′ and q c = q ′ c = p say. Then q = q ′ . Question 1. Why n c = n ′ c = m ? My attempt: Since p ⊆ q , we have m = S − 1 p ⊆ S − 1 q = n ⊆ B p . But m is maximal in A p , which is not necessarily maximal in B p . I can't get m = n by this. Question 2. When we use that notation A p , which means the localization S − 1 A of A at the prime ideal p of A . But in this corollary, p doesn't necessarily be a prime ideal of B . Why can he write B p ? Should we write S − 1 B rigorously?

Kyle Sutton

Proof of commutative property in Boolean algebra a ∨ b = b ∨ a a ∧ b = b ∧ a

orlovskihmw

The theorem is stated in the context of commutative Banach (unitary) algebras, but the proof seems to show that it is valid for any commutative algebra defined as a linear space where a commutative, associative and distributive (with respect to the addition) multiplication is defined such that ∀ α ∈ K α ( x y ) = ( α x ) y = x ( α y ) . In any case, whether it concerns only commutative Banach unitary algebras or commutative algebras as defined above, I think we must intend contained as properly contained. Am I right?

Lorena Beard

Let H be a Hilbert space and A a commutative norm-closed unital ∗ -subalgebra of B ( H ) . Let M be the weak operator closure of A . Question: For given a projection P ∈ M , is the following true? P = inf { A ∈ A : P ≤ A ≤ 1 } It seems that the infimum must exist and is a projection, but I am not able to show that the resulting projection cannot be strictly bigger than P . Also, if the above is true, what happens if A is non-commutative?

Kaeden Hoffman

Just a simple question. What does Eisenbud mean by ( x : y ) where x , y ∈ R a ring. An example on this is in the section 17 discussing the homology of the koszul complex. I assume it's something along the lines of { r ∣ a x = r y } for some a ∈ R .

Brock Byrd

Let R be a commutative k -algebra, where k is a field of characteristic zero. Could one please give an example of such R which is also: (i) Not affine (= infinitely generated as a k -algebra). and (ii) Not an integral domain (= has zero divisors).My first thought was k [ x 1 , x 2 , … ] , the polynomial ring over k in infinitely many variables, but unfortunately, it satisfies condition (i) only. It is not difficult to see that it is an integral domain: If f g = 0 for some f , g ∈ k [ x 1 , x 2 , … ] , then there exists M ∈ N such that f , g ∈ k [ x 1 , … , x M ] , so if we think of f g = 0 in k [ x 1 , … , x M ] , we get that f = 0 or g = 0 , and we are done.

rmd1228887e

Let A be a commutative Banach algebra. Let χ 1 and χ 2 be characters of A . I am having some difficulty seeing why the following statement is true: If ker ⁡ χ 1 = ker ⁡ χ 2 , then since χ 1 ( 1 ) = χ 2 ( 1 ) = 1 , we have that χ 1 = χ 2 .

Wisniewool

Need to find a functor T : Set → Set such that Alg(T) is concretely isomorphic to the category of commutative binary algebras. The first idea is that the functor is likely to map object X ∈ O b ( to the X × X because then we have to get a binary algebra, i.e., the operation X × X → X , which have to be commutative. So the question (if these thoughts are right) is: how to map X to X × X to get later a commutative binary algebra?

therightwomanwf

I would like to know, under what condition on the group G (abelian, compact or localement compact ...), the algebra L 1 ( G ) is commutative?

Maliyah Robles

Let A be a C ∗ -algebra. (i) Let φ be a state on a u ∈ A -algebra A . Suppose that | φ ( u ) | = 1 for all unitary elements u∈A. Show that φ is a pure state. [Hint: span ⁡ U ( A ) = A ] (ii) Let φ be a multiplicative functional on a C ∗ -algebra A . Show that φ is a pure state on A . (iii) Show that the pure and multiplicative states coincide for commutative A . I managed to work out the first two problems but I have no idea about the last one. How to see from being an extreme element in the state space of a commutative A that the extreme element is multiplicative?

Yesenia Obrien

I am at a very initial stage of commutative algebra. I want to know whether the power of a prime ideal in a commutative ring is prime ideal or not?

Callum Dudley

A ⊂ B is a ring extension. Let y , z ∈ B elements which satisfy quadratic integral dependance y 2 + a y + b = 0 and z 2 + c z + d = 0 over A . Find explicit integral dependance relations for y + z and y z .

Sam Hardin

Content ( f g ) ⊂ Content ( f ) Content ( g ) ⊂ rad ( Content ( f g ) ) to deduce that if Content ( f ) contains a nonzerodivisor of R , then f is nonzerodivisor of S = R [ x 1 , . . . , x r ] .

ntaraxq

Let g be a complex linear Lie algebra. Assume that the center z of g is trivial Let r be the radical of g . If r is abelian, then g is semisimple? What if g is the Lie algebra of an algebraic complex linear group?

Aganippe76

Is it true that every commutative Hopf algebra is related to a Group in such a way that the co-multiplication is originated from the multiplication of the group, the antipode from the inverse? Making it more explicitly, can all commutative and co-comutative Hopf algebra H be written in this form: H = C [ G ] , with the usual group algebra structure η : 1 → e , m : g ⊗ h ↦ g h the coalgebra structure ε : g ↦ 1 , Δ : g ↦ g ⊗ g where all of the maps above are defined on the basis of group elements?

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solved problems in commutative algebra

How To : Solve math problems with the commutative law of addition

Solve math problems with the commutative law of addition

From Ramanujan to calculus co-creator Gottfried Leibniz, many of the world's best and brightest mathematical minds have belonged to autodidacts. And, thanks to the Internet, it's easier than ever to follow in their footsteps (or just finish your homework or study for that next big test). With this installment from Internet pedagogical superstar Salman Khan's series of free math tutorials, you'll learn how to unpack and solve word problems requiring use of the commutative law of addition.

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Google's new math app solves nearly any problem with AI: Here's how to use it

artie

Stuck on a tricky math problem? Google's newest app will use AI to help you solve it. 

Two years ago, Google announced the purchase of a math problem-solving app called Photomath. And earlier this week, that app was officially brought under the company's app umbrella . 

Also: How ChatGPT (and other AI chatbots) can help you write an essay

The app itself isn't new, having debuted back in 2014 and picking up over 100 million downloads since. But it is now officially a Google app. It works on a wide range of math, from basic elementary school problems like division and multiplication to advanced math like trigonometry and calculus. 

Once a problem is scanned with the app, the AI starts working. After a few moments, an answer is displayed, along with step-by-step details of how the problem was solved. The latter part is likely the most useful part of this app. Not only is a solution given, but a student can also learn how to get that same answer on their own. 

While several other apps do the same thing, Photomath is often regarded by users as not only the most accurate but also the fastest. 

The app's camera can recognize both printed and handwritten problems and even shows multiple methods for problems that can be solved in different ways. And it works without needing a data or Wi-Fi connection, meaning parents can let their kids use it without worrying about them wandering off online.

Also: The best AI chatbots

If you're thinking that an app like this is nothing but a way for a student to rip through homework in record time, that's not all it's used for. While that certainly does happen, it has plenty of other uses -- a parent checking their kid's homework, a student practicing before a test or catching up on a missed class, or simply a 24/7 tutor. 

Since Google Lens already has a homework filter that's designed to solve problems, what's the need for Photomath? As AI becomes more and more present  in the classroom, Google is likely just making sure it stays ahead of its competition. While nothing has been announced, there's a good chance Photomath will be integrated into Google Lens and even traditional Google search, making those features even more reliable.

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Microsoft Research Blog

Orca-math: demonstrating the potential of slms with model specialization.

Published March 5, 2024

By Arindam Mitra , Senior Researcher Hamed Khanpour , Senior Program Manager Corby Rosset , Senior Researcher Ahmed Awadallah , Partner Research Manager

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Our work on Orca and Orca 2 demonstrated the power of improved training signals and methods to enhance the reasoning abilities of smaller language models, getting closer to the levels found in much larger language models. Orca-Math is another step in this direction, where we explore the capabilities of small language models (SLMs) when specialized in a certain area, in this case solving grade school math problems, which has long been recognized as a complex task for SLMs.

Orca-Math is a 7 billion parameters model created by fine-tuning the Mistral 7B model. Orca-Math achieves 86.81% on GSM8k pass@1, exceeding the performance of much bigger models including general models (e.g. LLAMA-2-70, Gemini Pro and GPT-3.5) and math-specific models (e.g. MetaMath-70B and WizardMa8th-70B). Note that the base model (Mistral-7B) achieves 37.83% on GSM8K.

Alt Text: Bar graph comparing GSM8K score of different models with an upward trend in quality. The models are LLAMA-2-70, GPT-3.5, Gemini Pro,  WizardMath-70B, MetaMath-70B and Orca-Math-7B. The graph shows that the Orca-Math-7B model outperforms other bigger models on GSM8K.

The state-of-the-art (SOTA) performance of the Orca-Math model can be attributed to two key insights:

  • Training on high-quality synthetic data with 200,000 math problems, created using multi-agents ( AutoGen ). This is smaller than other math datasets, which could have millions of problems. The smaller model and smaller dataset mean faster and cheaper training.
  • In addition to traditional supervised fine-tuning, the model was trained using an iterative learning process, where the model is allowed to practice solving problems and continues to improve based on feedback from a teacher.

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Peter Lee, head of Microsoft Research, and Ashley Llorens, AI scientist and engineer, discuss the future of AI research and the potential for GPT-4 as a medical copilot.

Our findings show that smaller models are valuable in specialized settings, where they can match the performance of much larger models while also highlighting the potential of continual learning and using feedback to improve language models. We are making the dataset (opens in new tab) publicly available, along with a report (opens in new tab) describing the training procedure to encourage research on the improvement and specialization of smaller language models.

Teaching SLMs math

Solving mathematical word problems has long been recognized as a complex task for SLMs. Models that achieve over 80% accuracy on the GSM8K benchmark (GSM8K, which stands for Grade School Math 8K, is a dataset of 8,500 high-quality grade school mathematical word problems that require multi-step reasoning) typically exceed 30 billion parameters.

To reach higher levels of performance with smaller models, researchers often train SLMs to generate code, or use calculators to help avoid calculation errors. Additionally, they employ a technique called ensembling, in which the model is called up to 100 times, with each call reattempting to solve the problem. Ensembling provides a substantial boost in accuracy but at a significant increase in compute cost increase, due to multiple calls to the model. 

This research aims to explore how far we can push the native ability of smaller language models when they are specialized to solve math problems, without the use of external tools, verifiers or ensembling. More specifically, we focus on two directions:

AgentInstruct

Previous work on synthetic data creation often uses frontier models to generate similar problems based on a seed problem. Providing paraphrases of the seed with different numbers and attributes can be useful for creating training data for the smaller model. We propose employing multi-agent flows, using AutoGen, to create new problems and solutions, which can not only create more demonstrations of the problem but also increase the diversity and range of difficulty of the problems. 

To generate more challenging problems, we create a setup with a team of agents working collaboratively to create a dataset geared toward a predefined objective. For example, we can use two agents, namely Suggester and Editor . The Suggester examines a problem and proposes several methods for increasing its complexity, while the Editor takes the original word problem and the Suggester’s recommendations to generate an updated, more challenging problem. This iterative process can occur over multiple rounds, with each round further increasing the complexity of the previously generated problem. A third agent can then verify that the problem is solvable and create the solution.

Iterative learning

Using high-quality training data that may elicit richer learning signals (e.g. explanations) has been shown to significantly improve SLM’s abilities in acquiring skills that had only emerged before at much larger scale.

This paradigm fits under a teacher-student approach where the large model (the teacher) is  creating demonstrations for the SLM (the student) to learn from. In this work, we extend the teacher-student paradigm to iterative learning settings as follows:

  • Teaching by demonstration : In this stage, we train the SLM by using AgentInstruct to demonstrate problems and their solutions.
  • Practice and feedback: We let the SLM practice solving problems on its own. For every problem, we allow the SLM to create multiple solutions. We then use the teacher model to provide feedback on the SLM solutions. If the SLM is unable to correctly solve the problem, even after multiple attempts, we use a solution provided by the teacher.
  • Iterative improvement: We use the teacher feedback to create preference data showing the SLM both good and bad solutions to the same problem, and then retrain the SLM.

The practice, feedback, and iterative improvement steps can be repeated multiple times.

Our findings show that smaller models are valuable in specialized settings where they can match the performance of much larger models but with a limited scope. By training Orca-Math on a small dataset of 200,000 math problems, we have achieved performance levels that rival or surpass those of much larger models.

The relatively small size of the dataset also shows the potential of using multi-agent flows to simulate the process of data and feedback generation. The small dataset size has implications for the cost of training and highlights that training data with richer learning signals can improve the efficiency of the learning process. Our findings also highlight the potential of continual learning and the improvement of language models, where the model iteratively improves as it receives more feedback from a person or another model.

Related publications

Orca: progressive learning from complex explanation traces of gpt-4, orca-2: teaching small language models how to reason, meet the authors.

Portrait of Arindam Mitra

Arindam Mitra

Senior Researcher

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Hamed Khanpour

Senior Program Manager

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Corby Rosset

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Ahmed Awadallah

Partner Research Manager

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Satya Nadella on stage at Microsoft Ignite 2023 announcing Phi-2.

Phi-2: The surprising power of small language models

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LLMLingua: Innovating LLM efficiency with prompt compression

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Orca 2: Teaching Small Language Models How to Reason

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COMMENTS

  1. PDF Open problems in commutative ring theory

    Open problems in commutative ring theory Paul-Jean Cahen, Marco Fontanay, Sophie Frisch zand Sarah Glaz x December 23, 2013 Abstract This article consists of a collection of open problems in commuta-tive algebra. The collection covers a wide range of topics from both Noetherian and non-Noetherian ring theory and exhibits a variety of re-

  2. reference request

    1 Answer Sorted by: 2 The solutions to Atiyah-MacDonald, Introduction to Commutative Algebra can be found online ( http://dangtuanhiep.files.wordpress.com/2008/09/papaioannoua_solutions_to_atiyah.pdf) and the problems of Matsumura's Commutative Ring Theory come with hints/solutions at the end. Share Cite Follow answered Mar 26, 2014 at 20:10 Manos

  3. Commutative Property

    Examples: If 6 \times 4 = 24 6 × 4 = 24 is known, then 4 \times 6 = 24 4 × 6 = 24 is also known. (Commutative property of multiplication.)

  4. Commutative property of multiplication review

    The commutative property is a math rule that says that the order in which we multiply numbers does not change the product. Example: 8 × 2 = 16 1 3 5 7 9 11 13 15 2 4 6 8 10 12 14 16 2 × 8 = 16 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 So, 8 × 2 = 2 × 8 . Want to learn more about the commutative property? Check out this video.

  5. Steps in Commutative Algebra

    The core of the book discusses the fundamental theory of commutative Noetherian rings. Affine algebras over fields, dimension theory and regular local rings are also treated, and for this second edition two further chapters, on regular sequences and Cohen-Macaulay rings, have been added. This book is ideal as a route into commutative algebra.

  6. Commutative algebra

    Commutative algebra is the main technical tool of algebraic geometry, and many results and concepts of commutative algebra are strongly related with geometrical concepts. The study of rings that are not necessarily commutative is known as noncommutative algebra; it includes ring theory, representation theory, and the theory of Banach algebras .

  7. Commutative Algebra

    Can commutative algebra solve real-world problems? "When we first study advanced math, we learn to solve linear and quadratic equations, generally a single equation and in one variable," said Srikanth Iyengar, Professor of Mathematics at the U. "But most real-world problems aren't quite so easy—they often involve multiple equations in multiple variables."

  8. Commutative Algebra

    Download Course. In this course students will learn about Noetherian rings and modules, Hilbert basis theorem, Cayley-Hamilton theorem, integral dependence, Noether normalization, the Nullstellensatz, localization, primary decomposition, DVRs, filtrations, length, Artin rings, Hilbert polynomials, tensor products, and dimension theory.

  9. Applications of DG Homological Algebra in Commutative Algebra

    Absract: Homological algebra comes from algebraic topology and provides valuable techniques in studying commutative ring theory. Auslander-Buchsbaum and Serre's solution to the localization conjecture of regular local rings firmly established homological algebra as a powerful tool to solve problems in commutative algebra. Solving this conjecture by homological methods was a

  10. Commutative Algebra

    problem and solution problems and solutions in commutative algebra mahir bilen can disclaimer: this file contains some problems and solutions in commutative. Skip to document. University; High School. Books; ... Commutative Algebra. Course: Comutative algebra (comut 2009, blw00908) University: Tulane University. Info More info. Download. 0 0 ...

  11. 9.3.1: Associative, Commutative, and Distributive Properties

    The associative, commutative, and distributive properties of algebra are the properties most often used to simplify algebraic expressions. You will want to have a good understanding of these properties to make the problems in algebra easier to solve.

  12. Applications of commutative algebra

    6 Answers. The book "Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra" by Cox, Little & O'Shea, contains some "real world" applications, specifically chapter 6 (of the 3rd edition) is titled "Robotics and Automatic Geometric Theorem Proving". Google "Gunnar Carlsson" and "Rob Ghrist ...

  13. Commutative property

    Commutative property. An operation (especially a binary operation) is said to have the commutative property or to be commutative if the order of its arguments does not affect the value. For example, the operation addition is commutative on the most commonly used number systems (the complex numbers and its subsets such as the real numbers ...

  14. What Is Commutative Property? Definition, Formula, Examples

    Example 1: Fill in the missing numbers using the commutative property. _________ + 27 = 27 + 11 45 + 89 = 89 + _________ 84 × ______ = 77 × 84 118 × 36 = ________ × 118 Solution: 11; by commutative property of addition 45; by commutative property of addition 77; by commutative property of multiplication

  15. Commutative Property in Algebra

    Why the word "switchable"? The commutative property can be used with addition or multiplication and it says that you can add or multiply numbers in any order and you will still end up with the same answer. Let's first take a closer look at the commutative property for addition. Commutative Property of Addition

  16. Commutative Property in Maths ( Definition and Examples)

    As per the commutative property of multiplication, when we multiply two integers, the answer we get after multiplication will remain the same, even if the position of the integers are interchanged. Let A and B be the two integers, then; A × B = B × A

  17. What Is Commutative Property ⭐ Definition, Formula, Examples

    a + b = b + a For example: 3 + 5 = 5 + 3 8 + 2 = 2 + 8 This property can be extended to more than two numbers as well: 3 + 5 + 7 = 5 + 3 + 7 = 7 + 5 + 3 Commutative Property of Multiplication: The Commutative Property of Multiplication works similarly to the property of addition.

  18. Commutative Algebra: Comprehensive Explanation

    Commutative Algebra Commutative Algebra Explained: In-Depth Examples, Equations, and Expert Insights Commutative Algebra Answered question mugenziwzn6 2023-02-21 The permutations and combinations of a b c d taken 3 at a time are respectively. Commutative Algebra Answered question Ghillardi4Pi 2022-12-04

  19. University Algebra Through 600 Solved Problems

    University Algebra Through 600 Solved Problems. N. S. Gopalkrishnan. New Age International, 1997 - Algebra - 188 pages. Prof. Gopalakrishnan Passed His B.Sc. (Hons) In Mathematics From Vivekananda College Madras In 1955 And His M.A. In The Same Subject From The University Of Madras In 1956.

  20. Algebra Calculator

    To solve an algebraic expression, simplify the expression by combining like terms, isolate the variable on one side of the equation by using inverse operations. Then, solve the equation by finding the value of the variable that makes the equation true. What are the basics of algebra?

  21. Solve math problems with the commutative law of addition

    Solve math problems with the commutative law of addition. From Ramanujan to calculus co-creator Gottfried Leibniz, many of the world's best and brightest mathematical minds have belonged to autodidacts. And, thanks to the Internet, it's easier than ever to follow in their footsteps (or just finish your homework or study for that next big test).

  22. Google's new math app solves nearly any problem with AI: Here's ...

    It works on a wide range of math, from basic elementary school problems like division and multiplication to advanced math like trigonometry and calculus. Once a problem is scanned with the app ...

  23. Orca-Math: Demonstrating the potential of SLMs with model

    Orca-Math is another step in this direction, where we explore the capabilities of small language models (SLMs) when specialized in a certain area, in this case solving grade school math problems, which has long been recognized as a complex task for SLMs. Orca-Math is a 7 billion parameters model created by fine-tuning the Mistral 7B model.