Integers Formula: Definition, Properties, Solved Examples

Before we get into the details of integer formulas, let's take a moment to revisit our understanding of integers. Integers are whole numbers without any parts that involve fractions or decimals. The word "integer" comes from Latin, where "integer" means "whole" or "untouched." In math, we often use the symbol 'Z' to represent integers, and this category of numbers includes:

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

Integers Formulas

When performing addition and subtraction with integers, the rules depend on the signs of the numbers. For addition, when both integers have the same sign, you add their absolute values and maintain that sign. If they have different signs, subtract their absolute values and use the sign of the larger number. Subtraction follows the same principles as addition, accounting for signs. In multiplication, two integers with the same sign yield a positive product, while those with different signs result in a negative product

Now, let's explore the formulas that govern operations with integers.

Addition and Subtraction of Integers Formulas

When adding or subtracting integers, the formulas differ based on the signs of the numbers involved.

Addition:

When adding two integers with the same signs, add their absolute values and retain the same sign for the result.

When adding two integers with different signs, subtract their absolute values (larger number minus the smaller number) and use the sign of the larger number for the result. Below are some rules.

a−b=a+(−b).

The formulas for addition are as follows:

(+)+(+) = +

(+)+(+) = +

(−)+(−) = −

(−)+(−) = −

(+)+(−) = +

(+)+(−) = +    (When the absolute value of the positive number is larger)

(+)+(−) = − (When the absolute value of the negative number is larger)

Examples:

2+3 = 5

2+3 = 5

(−2)+(−3) = −5

(−2)+(−3)= −5

3+(−2) = +1

3+(−2)=+1 (or simply 1)

2+(−3) = −1

Subtraction:

Subtraction of integers follows the same rules as addition, taking into account the signs of the numbers involved.

Multiplication and Division of Integers Formulas

Multiplication and division of integers also have distinct rules based on the signs of the integers being multiplied or divided.

Multiplication:

The product of two integers with the same signs is always positive.

The product of two integers with different signs is always negative.

The formulas for multiplication are as follows:

(+)×(+) = +

(−)×(−) = +

(+)×(−) = −

(−)×(+) = −

Examples:

2×3 = 6

2×3 =6

−2×−3 =6

2×−3 = −6

−2×3 = − 6

−2×3 = −6

Division:

Division of integers follows the same rules as multiplication, considering the signs of the integers involved.

Properties of Integers

It's important to note that the set of integers possesses several properties:

• Closure
• Associativity
• Commutativity
• Additive identity (0)
• Multiplicative identity (1)
• Additive inverses (Every integer has an additive inverse)
• Multiplicative inverses (Only 1 and -1 have multiplicative inverses)

Closure: Integers maintain their integer status when added, subtracted, multiplied, or divided.

Associativity: The grouping of integers in operations doesn't change the final result (e.g., (a + b) + c = a + (b + c)).

Commutativity: The order of adding or multiplying integers doesn't affect the outcome (e.g., a + b = b + a).

Additive identity (0): Adding 0 to any integer leaves it unchanged (e.g., a + 0 = a).

Multiplicative identity (1): Multiplying any integer by 1 retains the integer's value (e.g., a * 1 = a).

Additive inverses: Each integer has a counterpart that, when added, equals zero (e.g., a + (-a) = 0).

Multiplicative inverses: Only integers 1 and -1 have counterparts that, when multiplied, yield 1 (e.g., a * 1 = 1, a * (-1) = -1).

Examples Using Integers Formulas

Example 1:

1. a) Evaluate

12+(−3)

Solution:

12+(−3) = 9

1. b) Evaluate

12×−3

Solution:

12×− = −36

Example 2:

a) Find a pair of integers whose sum is -6.

Solution:

2+(−8)= −6

2+(−8)=−6

b) Find a pair of integers whose product is -8.

Solution:

2×−4= −8

2×−4=−8

Example 3:

a) Evaluate

(−13)+(−12)+

Solution:

(−13)+(−12)+2 = −23

(−13)+(−12)+2=−23

b) Evaluate

−4+(−2)×(3)+(−4)÷2

Solution:

−4+(−6)+(−2)

= - 12

−4+(−6)+(−2)=−12

Example 4:

a) Evaluate

5+(−8)

Solution:

5+(−8) = −3

b) Evaluate

(−10)+6

Solution:

(−10)+6 = − 4

Example 5:

a) Find a pair of integers whose sum is 0.

Solution:

3+(−3) =0

3+(−3)=0

b) Find a pair of integers whose product is 12.

Solution:

4×3=12

4×3=12

Example 6:

a) Evaluate

−7×4

−7×4

Solution:

−7×4 = −28

−7×4=−28

b) Evaluate

9÷(−3)

Solution:

9÷(−3) = −3

Example 7:

a) Calculate

(−2)+6+(−5)+3

Solution:

(−2)+6+(−5)+3 =2

b) Calculate

4×(−2)×3

Solution:

4×(−2)×3 =−24

Example 8:

a) Find a pair of integers whose sum is -10.

Solution:

7+(−17) = −10

b) Find a pair of integers whose product is -15.

Solution:

5×(−3) = −15

In this section, we have covered the fundamentals of integers and their associated formulas, which are essential for performing operations with integers.Integers Formula: An integer is a whole number that can be positive, negative, or zero, without any fractions or decimals, often represented by 'Z' in mathematics. Integers are used in various mathematical operations and calculations.