Integers

Introduction – What are integers?

In the field of mathematics, integers are defined as a set of whole numbers and negative numbers with the inclusion of zero. Although fractions and decimal numbers are not included in integers. The various operations that can be performed on integers are addition, division, multiplication, and subtraction. Integers are addressed by the alphabet Z. ‘Integer’ was inspired by a Latin word that translates to the whole. Integers are illustrated as Z = { …..-1,0, 1, 2, 3,…….}. Integers are an infinite set of numbers with no smallest and largest value (countable infinity). Integers and rational numbers are related in such a way that integers are known as a subset of rational numbers. The further sections will elucidate the properties, types, and applications of integers

The types of integers

There are three fundamental types of integers. They are 

• Positive integers – any integer having a value greater than zero is termed a positive integer. Examples – 2,4,6,…..

• Negative integers – any integer having a value lesser than zero is termed a negative integer. Examples: -1, -2, -3,……. 

• Zero – This number is neither considered positive nor negative. However, it is used to distinguish between positive and negative integers. It is considered as a whole number

Operations of integers

Four major operations are performed by integers

• Addition – the final value of the integers depends on the sign of the integers being added. There are two fundamental rules for the addition of two integers. They are 

1. If both the integers have a positive sign, the absolute value is obtained by adding both the integers.

2. If one integer is positive and the other integer is negative, the difference between the two integers is the answer.

Examples: 2 + 3 = 5 

                 -3 + 6 = 3

• Subtraction –  the final value of the integers depends on the sign of the integers being subtracted. The rules are similar to that of the addition operation

1. If both the integers have a negative sign, the absolute value is obtained by adding both the integers and adding a negative sign to the value. 

2. If one integer is positive and the other integer is negative, the difference between the two integers is the answer.

Examples: -3 -5 = -8 

                  4 – 2 = 2 

• Multiplication – The rules are similar to that of adding integers. The following tabular column will help you understand better

• Division – The following table will give us information about the division operation performed by integers

Integer properties

The seven prime properties of integers are elucidated in this segment 

• Closure property 

Any operation performed between two integers will always result in an integer. The operations that are included in this property are addition, subtraction, division, and multiplication. 

c + d = e

Example 3 + 1 = 4

c – d  = e 

Example  4 -1 = 3

c  x d  = e

Example = 2 x 4 = 8

c÷d = e

Example = -12÷4 = -3 

In all the four situations, c, d, and e are all integers

• Commutative property 

Interchanging the positions of the operand will not change the result. Only multiplication and addition operations follow this property.

c + d = d + c – example: 6 + 4 = 4 + 6 = 10

c x d = d x c – example:  5 x 3 = 3 x 5 = 15

• Distributive property 

This property states that any expression of the c x ( d + e) can be distributed over the addition operation to (c x d) + (c x e) 

Example: 3 x ( 4 + 5) = (3 x 4) + (3 x 5) = 27 

• Associative property 

Altering the group sequence of three integers will not result in any change in the result. This property is true for addition and multiplication operations only 

1. Addition – (c + d) + e = c + (d + e) 

2. Multiplication – (c x d) x e = c x (d x e)

• Additive inverse 

This property says that when you add two integers with opposite signs and have the same number, the result will be zero. 

c + (-c) = 0

Example 2 + (-2) = 0

• Multiplicative inverse 

This property says that when you multiply an integer with its reciprocal, the result will be one. 

c x 1/c = 1 

Example 2 x ½ = 1 

• Additive identity 

This property says that when you add an identity element (0) to an integer, the result will be an integer itself. 

c + 0 = c 

Example 3 + 0 = 3

Uses of Integers

Integers are not only used in mathematics, they are used in real-life applications. 

• Integers are used to illustrate contradicting situations

• Positive and negative integers are used to show temperatures. Positive temperatures are used to illustrate hot temperatures. Negative temperatures are used to illustrate freezing temperatures

• Integers are used to rate movies, songs, credit, and debit cards

• Integers are used in bonuses and penalty scores in quizzes and games

• Zero is used as a reference point

Conclusion

Integers are defined as a set of whole numbers and negative numbers with the inclusion of zero. The illustration of integers is Z = { …..-1,0, 1, 2, 3,…….}. There are three types of integers – zero, positive, and negative integers. The four prime operations of integers are – addition, division, multiplication, and subtraction. There are seven properties of integers – closure property, commutative property, associative property, distributive property, additive inverse, multiplicative inverse, and additive identity property. The use of integers is not limited to mathematics; it is used in everyday life as well. Integers are used to illustrate contradicting situations, positive and negative integers are used to show temperatures.