Integers Formulas

Before learning integers formulas, let us recall what are integers. An integer is a number that has no decimal or fractional part. The term “integer” was derived from the Latin word “integer”, which means "whole". Let us learn the integers formulas in detail in the next section. The set of integers is represented by 'Z' and they include:

• All natural numbers
• The negatives of all natural numbers
• The number zero (0)

What Are Integers Formulas?

The integers formulas are formulas/rules that are used to do the operations of integers. The integers formulas are different for finding the sum/difference and the product/quotient. Here are the formulas.

Addition/Subtraction of Integers Formulas

To add two integers of same signs, we just add their absolute values and use the same sign as the given integers for the result as well. To add two integers of different signs, we subtract their absolute values (in the order bigger number minus smaller number) and use the sign of the bigger number to the result as well. We perform the subtraction also just in the same way as addition except using the rule, a - b = a + (-b). Thus, the formulas of addition/subtraction of integers are:

1. (+) + (+) = +
2. (-) + (-) = -
3. (+) + (-) = + (The absolute value of positive number is bigger)
4. (+) + (-) = - (The absolute value of negative number is bigger)

Examples:

• 2 + 3 = 5
• (-2) + (-3) = -5
• 3 + (-2) = +1 (or) 1
• 2 + (-3) = -1 

Multiplication/Division of Integers Formulas

The product/quotient of two integers of the same signs is always positive and the product or quotient of two integers of different signs is always negative. Thus, the formulas of multiplication/division of integers are:

1. (+) × (+) = +;   (+) ÷ (+) = +
2. (-) × (-) = +;   (-) ÷ (-) = +
3. (+) × (-) = -;   (+) ÷ (-) = -
4. (-) × (+) = -;   (-) ÷ (+) = -

Examples

• 2 × 3 = 6
• -2 × -3 = 6
• 2 × -3 = -6
• -2 × 3 = -6

Note: The set of integers is closed, associative, and commutative under addition and multiplication. The additive identity, 0 and the multiplicative identity, 1 are present in the set of integers. All integers have their additive inverses in the set of integers. None of the integers except 1 and -1 have their multiplicative inverses in the set of integers.

Let us see the applications of integers formulas in the following solved examples.

Examples Using Integers Formulas

Example 1: Evaluate the following: a) 12 + (-3) b) 12 × -3

Solution:

a) In 12 + (-3), 12 is with + sign and 3 is with - sign. So we subtract the absolute values of 12 and -3 (which are 12 and 3), subtract them (12 - 3 = 9) and give the sign of the greatest absolute value number (the sign of 12 which is +) to the result.

Thus, 12 + (-3) = +9 (or) 9

b) 12 × -3

Using one of the integers formulas, (+) × (-) = - and 12 × 3 = 36. 

Thus, 12 × -3 = -36.

Answer: a) 9 b) -36.

Example 2: Write a pair of integers in each of the following cases using the integers formulas: a) whose sum is -6   b) whose product is -8.

Solution:

a) Consider 2 and -8. Their absolute values are 2 and 8.

We know that 8 - 2 =6 and among 2 and -8, 8 has the biggest absolute value. So the result has a minus sign (as -8 has a minus sign).

Thus, 2 + (-8) = -6.

b) Consider 2 and -4. Since they are of opposite signs their product is negative. Thus,

2 × -4 = -8.

Answer: a) 2 and -8;  b) 2 and -4.

Example 3: Evaluate the following: a) (-13) + (-12) + 2   b) -4 + (-2) × (3) + (-4) ÷ 2.

Solution:

We will evaluate the given expressions using the integers formulas.

a) (-13) + (-12) - 2 

= -25 + 2

= -23

b)  -4 + (-2) × (3) + (-4) ÷ 2

= -4 + -6 + (-2)

= -10 + (-2)

= -12

Answer: a) -23   b) -12