Integers

We all know about natural numbers and whole numbers. The natural numbers denoted by N are the set of all positive numbers starting from one up to infinity. It is written N = { 1, 2, 3, 4, 5, 6………… ∞ }. Whole numbers denoted by W are the set of all the natural numbers with the addition of zero. It is written as W = {0, 1, 2, 3, 4, 5, 6………….. ∞ }. But where do the numbers below zero come under? All numbers below zero are negative numbers. They are written as natural numbers with a negative sign, or -N. The set of all numbers consisting of N, 0, and -N is called integers. Integers are basically any and every number without a fractional component. It is represented by the letter Z. The word integer comes from a Latin word meaning whole. Integers include all rational numbers except fractions, decimals, and percentages. If integers are represented on a number line, the positive numbers occupy the right side while the negative numbers occupy the left side. Integers represented by Z are a subset of rational numbers represented by Q. In turn rational numbers Q is a subset of real numbers R. Hence, integers Z are also a subset of real numbers R.

Symbol Representation 

The symbol Z stands for integers. For different purposes, the symbol Z can be annotated. Z+, Z+, and Z> are the symbols used to denote positive integers. The symbols Z-, Z-, and Z< are the symbols used to denote negative integers. Also, the symbol Z≥ is used for non-negative integers, Z≠ is used for non-zero integers. Z* is the symbol used for non-zero integers.

Operation of Integers 

Addition Rule of Integers :

In case of addition of numbers of the same sign (either positive or negative) simply add the two numbers and put the sign before it.

If the numbers that are supposed to be added have different signs, subtract the smaller number from the larger number ignoring the sign, and then put the sign of the larger number before it.

Example : 2 + 2 = 4, ( -3 ) + (-6) = - 9, ( -8 ) + 4 = - 4

Subtraction Rule of Integers :

If we are supposed to subtract one integer from another, first change the sign of the subtrahend. After this add the two numbers, with the sign of the subtrahend changed and perform according to the addition rule of integers.

Example : 7 - 3 = 4, ( -4 ) - (-5) = ( -4 (+) 5 ) = 1, 8 - (-6) = 8(+) 6 = 14

Multiplication Rule of Integers :

While multiplying any two integers with each other, first find the product of the integers without considering the signs. After you get a product, see the signs of the two numbers you just multiplied. If the sign of both the numbers is the same, the product is positive. On the other hand, if the sign of both the numbers is different, the product is negative.

Example : 9 × 5 = 45, -9 × ( -4 ) = 36, -7 × 5 = ( -35 )

Division Rule of Integers:

Division of integers works the same way as the multiplication of integers. While dividing any integer with another, first find the quotient of the division of integers without considering the signs. After you get a quotient, see the signs of the numbers you just divided. If the sign of both the numbers is the same, the quotient is positive. On the other hand, if the sign of both the numbers is different, the quotient is negative.

Example : 12 ÷ 4 = 3, -16 ÷ 4 = ( -4 ), -36 ÷ ( -12 ) = 3

Algebraic Properties of Integers

The different algebraic properties that apply to numbers apply to integers as well.

Closure Property of Integers :

Integers follow the closure property under the operations of addition, subtraction, and multiplication. This means that for any two integers which are represented by p and q,

p + q is an integer

p – q is an integer

p × q is an integer

Integers are not closed under division, since p/q need not be an integer and can be a fraction. Integers are also not closed under exponentiation as the result can be a fraction if the exponent is negative.

Associative Property of Integers:

Associative property of integers applies to multiplication and addition. This means for any three integers p, q, and r,

p + ( q + r ) = ( p + q ) + r

p × ( q × r ) = ( p × q ) × r

The associative property does not apply to division and subtraction.

The existence of Additive Identity and Multiplicative Identity of integers :

Additive identity is the number which when added to an integer gives the same integer.

The additive identity of integers like the additive identity of any other number is zero.

p + 0 = p

A multiplicative identity is a number which when multiplied to an integer gives the same integer.

The multiplicative identity of integers like the multiplicative identity of any other number is one.

p × 1 = p

The existence of Additive Inverse and Multiplicative Inverse of integers :

Additive inverse is the number which when added to an integer gives zero as the sum.

The additive inverse of a positive number is the negative of the same number, while the additive inverse of a negative number is the positive of the same number.

p + ( -p ) = 0

-p + ( p ) = 0

A multiplicative inverse is a number which when multiplied to an integer gives the answer as one.

The multiplicative inverse of integers like the multiplicative identity of any other number is the reciprocal of the same number.

p × 1/p = 1

-p × ( -1/p ) = 1

Distributive Property of Integers :

Distributive property of integers applies to multiplication over addition or multiplication over subtraction of integers. This means for any three integers p, q, and r,

p × ( q + r ) = ( p × q ) + ( p × r ) or ( p + q ) × r = ( p × r ) + ( q × r )

p × ( q - r ) = ( p × q ) - ( p × r ) or ( p - q ) × r = ( p × r ) - ( q × r )

Integers are the set of all whole numbers and their additive inverses. So, the integer 0 is in the middle of the number line. -1,-2,-3… are the additive inverses of 1,2,3,... respectively. 

Importance of Laws of Integers:

The laws of integers are the rules that help in simplifying a mathematical expression. These rules can be applied to any type of integer and factors. If the law is correctly applied, it helps simplify the resulting terms. In this article you will learn how to apply these laws when solving simple equations or problems involving integers.

Linear Combination: This law states that an integral factor cannot have a variable coefficient unless it is non-zero. It means all coefficients used in linear expressions must either be 1, 0, or -1. It also tells us that we can add/subtract any multiple of one linear expression to another linear expression even if they have different variables. For example:

Factor Law: This law tells us that if two factors divide evenly into each other then the result is either a multiple or factor of each factor used in the problem.

Distributive Law: This law tells us that we can distribute a product over an addition/subtraction problem as long as we distribute each term of the first factor to every term in the second factor. The distributive law also ensures that we can remove parentheses by distributing them to all terms within parenthesis.

Please note that there are two types of Distributive Law which include,

Commutative Laws: These laws tell us how to change the order of factors without affecting the resulting value. There are three basic commutative laws which include,