# PROPERTIES OF MULTIPLICATION OF INTEGERS

In Math, the whole numbers and negative numbers together are called integers. The set of all integers is denoted by Z.

Z  =  {... - 2, - 1,0,1,2, ...}, is the set of all integers

Here, we are going to see the following three properties of multiplication of integers.

(i)  Closure property

(ii)  Commutative property

(iii)  Associative property

(iv)  Multiplicative identity

## Closure Property of Multiplication of Integers

Observe the following:

– 10 × (– 5) = 50

40 × (– 15) = – 600

In general, a × b is an integer, for all integers a and b.

Therefore, integers are closed under multiplication.

## Commutative Property of Multiplication of Integers

Observe the following :

5 × (– 6) = – 30 and (– 6) × 5 = – 30

Therefore,

5 × (– 6) = (– 6) × 5

In general, for any two integers a and b,

a × b = b × a

Therefore, multiplication is commutative for integers. ## Associative Property of Multiplication of Integers

Consider the integers 2, – 5, 6.

Look at

[2 x (-5)] x 6  =  -10 x 6  =  -60

2 x [(- 5) x 6]  =  2 x (-30)  =  -60

Thus,

[2 x (-5)] x 6  =  2 x [(- 5) x 6]

In general, for any integers a, b, c,

(a × b) × c  =  a × (b × c)

So, we can say that integers are associative under multiplication.

## Multiplication by Zero

The product of any nonzero integer with zero is zero.

Observe the following:

5 × 0 = 0

– 8 × 0 = 0

In general, for any nonzero integer a

a × 0 = 0 × a = 0 ## Multiplicative Identity

Observe the following:

5 x 1  =  5

1 x (- 7)  =  -7

This shows that ‘1’ is the multiplicative identity for integers.

In general, for any integer a we have

a x 1  =  1 x a  =  a ## Distributive Property of Multiplication of Integers

Multiplication is Distributive Over Addition :

Consider the integers 12, 9, 7.

Look at

12 x (9 + 7)  =  12 x 16  =  192

12 x (9 + 7)  =  12 x 9  +  12 x 7  =  108 + 84  =  192

Thus 12 x (9 + 7)  =  (12 x 9) + (12 x 7)

In general, for any integers a, b, c.

a x (b + c)  =  (a x b) + (a x c)

Therefore, multiplication is distributive over addition of integers.

Multiplication is Distributive Over Subtraction :

Consider the integers 12, 9, 7.

Look at

12 x (9 - 7)  =  12 x 2  =  24

12 x (9 - 7)  =  12 x 9  -  12 x 7  =  108 - 84  =  24

Thus 12 x (9 - 7)  =  (12 x 9) - (12 x 7)

In general, for any integers a, b, c.

a x (b - c)  =  (a x b) - (a x c)

Therefore, multiplication is distributive over subtraction of integers. Apart from the stuff given above, if you need any other stuff in math, please use our google custom search here.

If you have any feedback about our math content, please mail us :

v4formath@gmail.com

You can also visit the following web pages on different stuff in math.

WORD PROBLEMS

Word problems on simple equations

Word problems on linear equations

Algebra word problems

Word problems on trains

Area and perimeter word problems

Word problems on direct variation and inverse variation

Word problems on unit price

Word problems on unit rate

Word problems on comparing rates

Converting customary units word problems

Converting metric units word problems

Word problems on simple interest

Word problems on compound interest

Word problems on types of angles

Complementary and supplementary angles word problems

Double facts word problems

Trigonometry word problems

Percentage word problems

Profit and loss word problems

Markup and markdown word problems

Decimal word problems

Word problems on fractions

Word problems on mixed fractrions

One step equation word problems

Linear inequalities word problems

Ratio and proportion word problems

Time and work word problems

Word problems on sets and venn diagrams

Word problems on ages

Pythagorean theorem word problems

Percent of a number word problems

Word problems on constant speed

Word problems on average speed

Word problems on sum of the angles of a triangle is 180 degree

OTHER TOPICS

Profit and loss shortcuts

Percentage shortcuts

Times table shortcuts

Time, speed and distance shortcuts

Ratio and proportion shortcuts

Domain and range of rational functions

Domain and range of rational functions with holes

Graphing rational functions

Graphing rational functions with holes

Converting repeating decimals in to fractions

Decimal representation of rational numbers

Finding square root using long division

L.C.M method to solve time and work problems

Translating the word problems in to algebraic expressions

Remainder when 2 power 256 is divided by 17

Remainder when 17 power 23 is divided by 16

Sum of all three digit numbers divisible by 6

Sum of all three digit numbers divisible by 7

Sum of all three digit numbers divisible by 8

Sum of all three digit numbers formed using 1, 3, 4

Sum of all three four digit numbers formed with non zero digits

Sum of all three four digit numbers formed using 0, 1, 2, 3

Sum of all three four digit numbers formed using 1, 2, 5, 6

1. Click on the HTML link code below.

Featured Categories

Math Word Problems

SAT Math Worksheet

P-SAT Preparation

Math Calculators

Quantitative Aptitude

Transformations

Algebraic Identities

Trig. Identities

SOHCAHTOA

Multiplication Tricks

PEMDAS Rule

Types of Angles

Aptitude Test 