Integers- Definition, Properties And Examples

The word integer originated from the Latin word “Integer” which means whole or intact. Integers are a special set of numbers comprising zero, positive numbers, and negative numbers.
Examples of Integers: – 1, -12, 6, 15.
 

Symbol

The integers are represented by the symbol ‘Z’. 
Z= {……-8,-7,-6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8,……}

Types of Integers
Integers come in three types:

• Zero (0)
• Positive Integers (Natural numbers)
• Negative Integers (Additive inverse of Natural Numbers)

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Zero
Zero is neither a positive nor a negative integer. It is a neutral number i.e. zero has no sign (+ or -).
Positive Integers
The positive integers are natural numbers also called counting numbers. These integers are also sometimes denoted by Z+. The positive integers lie on the right side of 0 on a number line.

Z+ → 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30,….

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Negative Integers
The negative integers are the negative of natural numbers. They are denoted by Z–. The negative integers lie on the left side of 0 on a number line.

Z– → -1, -2, -3, -4, -5, -6, -7, -8, -9, -10, -11, -12, -13, -14, -15, -16, -17, -18, -19, -20, -21, -22, -23, -24, -25, -26, -27, -28, -29, -30,…..

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Rules of Integers

Rules defined for integers are:

• The sum of two positive integers is an integer
• The sum of two negative integers is an integer
• The product of two positive integers is an integer
• The product of two negative integers is an integer
• Sum of an integer and its inverse is equal to zero
• Product of an integer and its reciprocal is equal to 1

Arithmetic Operations on Integers

The basic Maths operations performed on integers are:

• Addition of integers
• Subtraction of integers
• Multiplication of integers
• Division of integers

While adding two integers with different signs, subtract the absolute values, and write down the difference with the sign of the number that has the largest absolute value.
For example, (-4) + (+2) = -2;  (+6) + (-4) = +2.

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Subtraction of Integers
While subtracting two integers, change the sign of the second number that is being subtracted, and follow the rules of addition.
For example, (-7) – (+4) = (-7) + (-4) = -11;   (+8) – (+3) = (+8) + (-3) = +5

 

Multiplication of Integers

While multiplying two integer numbers, the rule is simple.

• If both the integers have the same sign, then the result is positive.
• If the integers have different signs, then the result is negative.

For example,

• (+2) x (+3) = +6
• (+3) x (-4) = – 12

Thus, we can summarise the multiplication of two integers with examples in the below table.

Multiplication of Signs	Resulting Sign	Examples
+ × +	+	3 × 4 = 12
+ × –	–	3 × -4 = -12
– × +	–	-3 × 4 = -12
– × –	+	-3 × -4 = 12

 

Division of Integers

The rule for dividing integers is similar to multiplication.

• If both the integers have the same sign, then the result is positive.
• If the integers have different signs, then the result is negative.

Similarly

• (+6) ÷ (+2) = +3
• (-16) ÷ (+4) = -4

Division of Signs	Resulting sign	Examples
+ ÷ +	+	15 ÷ 3 = 5
+ ÷ –	–	15 ÷ -3 = -5
– ÷ +	–	-15 ÷ 3 = -5
– ÷ –	+	-15 ÷ -3 = 5

Properties of Integers

The major Properties of Integers are:

1. Closure Property
2. Associative Property
3. Commutative Property
4. Distributive Property
5. Additive Inverse Property
6. Multiplicative Inverse Property
7. Identity Property

 

Closure Property

According to the closure property of integers, when two integers are added or multiplied together, it results in an integer only. If a and b are integers, then:

• a + b = integer
• a x b = integer

 Examples: 

• 2 + 5 = 7 (is an integer)
• 2 x 5 = 10 (is an integer)

Applications of Integers

Integers are not just numbers on paper; they have many real-life applications. The effect of positive and negative numbers in the real world is different. They are mainly used to symbolize two contradicting situations.

For example, when the temperature is above zero, positive numbers are used to denote temperature, whereas negative numbers indicate the temperature below zero. They help one to compare and measure two things like how big or small or more or fewer things are and hence can quantify things.

Some real-life situations where integers come into play are players’ scores in golf, football, and hockey tournaments, the rating of movies or songs, and in banks credits and debits are represented as positive and negative amounts respectively.

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Examples on Integers

Example 1:

Solve the following:

• 5 + 3 = ?
• 5 + (-3) = ?
• (-5) + (-3) = ?
• (-5) x (-3) = ?

Solution:

• 5 + 3 = 8
• 5 + (-3) = 5 – 3 = 2
• (-5) + (-3) = -5 – 3 = -8
• (-5) x (-3) = 15

Example 2:

Solve the following product of integers:

• (+5) × (+10)
• (12) × (5)
• (- 5) × (7)
• 5 × (-4)

Solution:

• (+5) × (+10) = +50
• (12) × (5) = 60
• (- 5) × (7) = -35
• 5 × (-4) = -20

Example 3:

Solve the following division of integers:

• (-9) ÷ (-3)
• (-18) ÷ (3)
• (4000) ÷ (- 100)

Solution:

• (-9) ÷ (-3) = 3
• (-18) ÷ (3) = -6
• (4000) ÷ (- 100) = -40