Properties of Multiplication of Integers

Properties of Multiplication of Integer

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

Multiplication means the addition of an integer to itself a specific number of times. Multiplication of integers is one of the most intriguing operations on integer amongst the other four fundamental operations. Before diving deep into the concept of multiplication operations of integers, let’s take a brief overview of the topic.

What Is The Meaning of An Integer? 

The word integer is derived from a Latin word which specifically means Whole. Integer in Latin means whole. The Latin meaning of the word justifies why an integer is known to be a whole number too.  

For instance, 2,3,4,55,66,87, etc. are all integers whereas ½, ¾ are not as they are in fraction form. Also, whole numbers with a negative sign like -2, -4, -67 are also termed as integers only. 

Apart from this, there are rational and algebraic integers which can be thoroughly understood after learning about the number theory. 

Examples

The number 5 can be termed as an integer as well as a rational number. Since it can be written without putting a decimal component, it qualifies as an integer and it is also a rational number because it can be written as 

2/1

or

4/2

or even

-4/-2

 Why are there infinite numbers between two integers?”

Well, if you draw a number line, and mark the integers on it, it is clear that each point on the line represents a particular number.

Basic Properties of the Integers 

Integers have different properties that need to be studied thoroughly while preparing the lesson. Let us go through some basic properties of the integer numbers. 

Arithmetic Properties

Let us begin with the arithmetic properties which are the properties related to basic mathematical operations like addition, multiplication, and the relation between these two properties. 

For carrying out multiplication you need to always multiply the exact values of integers and there are certain rules to keep in mind to determine the sign of the final answer. 

1. Two positive or two negative integers when multiplied always give a positive integer as the product.

2. The quotient of a negative integer is negative. Similarly, a positive integer has a positive quotient. 

Closure Property

Closure Property Under Multiplication:

Integers are closed under multiplication, meaning that for any two integers x and y, xy is an integer.

Ex: 7 × 4 = 28; (– 4) × (– 5) = 20.

Closure Property Under Addition:

Integers are enclosed under addition (+), meaning that for any two integers x and y, x + y is termed as an integer. 

Take for example: 5 + 9 = 7; (– 8) + 6 = – 2.

Closure Property Under Subtraction:

Integers are enclosed under subtraction (-), meaning that for any two integers x and y, x – y is an integer.

Ex: (– 22) – (– 10) = (– 12); 18 – 3 = 15.

Closure Property Under Division:

Integers are not bound or enclosed under division operation, meaning that for any two integers x and y, xyxy may not be an integer.

Ex:(– 2) ÷ (– 4) = 1212

Commutative Property

Multiplication is commutative for integer numbers. 

 In case of any two integers x and y, xy = yx.

Ex: 10 × (– 7) = – (10 × 7) = –70;

(– 7) × 10 = – (7 × 10) = – 70

∴ 10 × (– 7) = (– 7) × 10.

Commutative Property Under Addition:

The addition is commutative for integers.  In case of two integers x and y, x + y = y + x.

Ex: 4+ (– 5) = 4 – 5 = – 1;

(– 7) + 6 = – 7 + 6 = –1

∴ 6 + (– 7) = (– 7) + 6.

Commutative Property Under Subtraction:

Subtraction is not considered to be commutative for integers.  In the case of two integers x and y, x – y ≠ y – x.

Ex: 9 – (– 7) = 9 + 7 = 16;

(– 7) – 9 = – 7 – 9 = – 16

∴ 9 – (– 7) ≠ – 7 – 9.

Commutative Property Under Division:

The division is not considered to be a commutative for integers just like subtraction.  In case of any two integers x and y, x ÷ y ≠ y ÷ x.

Ex: (– 15) ÷ 3 = – 15

3 ÷ (–15) = – 1/5

Therefore, (– 15) ÷ 3 ≠ 3 ÷ (–15).

Associative Property

Associative property under multiplication:

Multiplication is associative for integers. In case of any three integers x, y and z, (x × y) × z = x × (y × z)

Ex: [(– 3) × (– 2)] × 4 = (6 × 4) = 24

(– 3) × [(– 2) × 4] = (– 3) × (– 8) = 24

∴ [(– 3) × (– 2)] × 4 = [(– 3) × (– 2) × 4].

Associative Property Under Addition:

Addition is an associative property for integers. In case of any three integers x, y and z, x + (y + z) = (x + y) + z

Ex: 5 + (– 6 + 4) = 5 + (– 2) = 3;

(5 – 6) + 4 = (– 1) + 4 = 3

∴ 5 + (– 6 + 4) = (5 – 6) + 4.

Associative Property Under Subtraction:

Subtraction is associative for integers.  For any three integers x, y and z, x – (y – z) ≠ (x – y) – z

Ex: 5 – (6 – 4) = 5 – 2 = 3;

(5 – 6) – 4 = – 1 – 4 = – 5

∴ 5 – (6 – 4) ≠ (5 – 6) – 4.

Associative Property Under Division:

The division is not considered to be an associative property for integers.

Distributive Property

Distributive Property of Multiplication Over Addition:

In case of any three integers x, y and z, x × (y + z) = (x × y) + (x × z).

Ex: – 3 (4 + 5) = –3 (9)  = –27

                      = (– 3 × 4) + (– 3 × 4)

                      = (– 12) + (– 12)

                      = – 24.

Distributive Property of Multiplication Over Subtraction:

In case of any three integers, x, y and z, x × (y - z) = (x × y) – (x × z).

Ex: – 3 (5 – 4) = – 3 (1) = – 3

                     = (–3 × 5) – (– 3 × 4)

                     = (– 15) – (– 12)

                     = – 3.

The distributive property of multiplication over the addition and subtraction holds true in the case of integers. 

Identity Under Multiplication:

The integer 1 depicts the identity under multiplication. Which means that for any integer x, 1 × x = x × 1 = x.

Ex: (– 5) × 1 = 1 × (– 5) = – 5.

When an integer number is multiplied by –1, the number obtained in the result carries the sign of the negative which is the additive identity of the integer.

For any integer x, x × –1 = –1 × x = –x.

Identity Under Addition:

Integer 0 is the identity under addition.  That is, for any integer x, x + 0 = 0 + x = x.

Ex: 5 + 0 = 0 + 5 = 5.