What is an Integer

All Integer Numbers

In Mathematics we deal with different types of numbers. Each number is different than another, but they may share some common characteristics. To understand these numbers they are categorized in the different groups according to their characteristics. Some of the groups in the system are:

  • Natural numbers

  • Whole numbers

  • Integers

  • Rationals

  • Real numbers

In this article let us study what is an integer? And all about integers.

Integers are numbers, which includes positive numbers, negative numbers, and also zero but no fractions are allowed.


So, integers can be positive {1, 2, 3, 4, 5, 6 ... }, negative {−1, −2,−3, −4, -5, -6. .... }, or zero {0}


We can represent this all integer numbers together as:

Integers( Z ) = { ...6, -5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, 6... }.. The dots represent the infinite numbers.


All About Integers

What is an integer?

An integer is any number that can be either 0, or positive number, or negative number. An integer can never be a fraction, a decimal, or a percent. Some examples of integers include 2, 3, -4, 8, 99, -81, -56, etc.

The integers can be represented as:

Z = {……-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5 ……….}

On the number, line integers are represented as follows


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Here 0 is at the center of the number line and is called the origin.

All integers to the left of the origin (0) are negative integers prefixed with a minus(-) sign and all numbers to the right are positive integers prefixed with positive(+) sign, they can also be written without + sign.


Types of Integers

All integer numbers are basically of three types:

  • Zero

  • Positive Numbers and

  • Negative Numbers


Positive Numbers

Positive numbers are those numbers that are prefixed with a plus sign (+). Most of the time positive numbers are represented simply as numbers without the plus sign (+). Every positive number is greater than zero, negative numbers, and also to the number to its left. On a number line, positive numbers are represented to the right of origin( zero).

Example: 1, 5, 500, 66 ,89, etc.


Negative Numbers

Negative numbers are those numbers that are prefixed with a minus sign (-). It is mandatory to mention the sign of negative numbers. Negative numbers are represented to the left of the origin(zero) on a number line.

Example: -8 , -10, -1000, -1919, etc.


Zero

Zero is a neutral integer because it can neither be a positive nor a negative integer, i.e. zero has no +ve sign or -ve sign.


Arithmetic Rules for Integers

  • Addition Rule

To add two integers with the same sign:

If the sign of both the integers is the same, then add the absolute value and the result is assigned the same sign as both the values.


Rules:

  • (+) + (+) = +

  • (-) + (-) = –

For Example:

  • 4 + 9 = 13

  • (-4) + (-9) = -(4+9) = -13


To add two integers with different signs:

If one of the numbers has a different sign, then it will lead to subtraction, and output will have a sign of the larger number.

Find the difference of the absolute values and give the difference the same sign as the number with the largest absolute value.

Let us understand it with following steps

For example: (-7) + (4)

  1. Find the absolute values of -7  and 4.

  2. Find the difference between 7 and 4 i.e is 7 – 4 = 3.

  3. Find the sign of the largest absolute value. Here the largest number is 7 it has a negative sign.

  4. So the result obtained is -7 + 4 = -3

  • Subtraction Rule: 

Subtracting means adding the opposite of a given number or changing the signs of the number. The sign of the first number remains the same, change subtraction to addition and change the sign of the second number. Once you have applied this rule, follow the rules for subtracting integers


Rules:

  • (+) – (+) = (+) + (-)

  • (-) – (-) = (-) + (+)

  • (+) – (‐) = (+) + (+)

  • (‐) – (+) = (‐) + (‐)

Always consider the greater number sign for the result.

For example:

Subtracting -5 from 10, we get:

10 – (– 5) 

Changing the signs or taking the opposite and adding

= 10 + 5 (5 is the opposite of -5) 

= 15.

  • Multiplication Rule

To multiply or divide an integer, use the absolute value. If the signs of the integers are the same, multiply, or divide the answer is always positive. And if the two integers have different signs, the answer is negative.


Rules:

  • (+) x (+) = + and (+) ÷ (+) = +

  • (‐)  x  (‐) = + and (‐)  ÷  (‐) = +

If the signs are different:

  • (+) x (‐) = – and (+) ÷ (‐) = ‐

  • (‐) x (+) = – and (‐) ÷ (+) = ‐

For Examples: 

  •  5 x 2 = 10 and 8 ÷ 2 = 4

  • (-5) x (-2) = 10 and (-8) ÷ (-2) = 4

  • (5) x (-2) = -10 and 8 ÷ (-2) = -4

  • (-5) x (2) = -10 and (-8) ÷ (2) = -4 


Comparing the Integers

We can compare integers as we compare positive whole numbers. By using a number line we can easily compare the integers. The farther the number is to the right, the larger the value it has.


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  • Compare -9 and -2

We will say that -9 is greater than -2, but no smaller the negative numbers greater the value. As you move towards the right the value increases.

Here, -2 is  to the right. Therefore, -2 is greater than -9 i.e -2 > -9.


  • Compare -8 and 3

At first look, you will say -8 is greater than 3, but no negative values are always smaller than the positive ones.

Here we can see that 3 is larger than -8 i.e 3 > -8.

This was all about integers. 


Solved Examples

  1. Compare -1 and 3

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Solution: -1 < 3 as -1 is to the left of the number 3


  1. Compare -35 and 67

Solution: -35 < 67 as negative numbers are always smaller than positive numbers.


Quiz Time

Identify the integers

  1. 67 

  2. 2.98

  3. 589000

  4. 0.00008

Compare the integers

  1. 45 and -89

  2. 67 and -5678

FAQ (Frequently Asked Questions)

1. What are the Opposites of Integers?

Answer:

Integers are the set of positive whole numbers and their opposites, including zero.

The opposite of 4 is -4.

The opposite of 242 is -242.

The opposite of -213 is 213.

The opposite of 0...it is just 0 because 0 is a neutral number it is neither positive nor negative.

We can list the integers on a number line and then add arrows to show that integers continue on and on in both directions.

2. Is 0 an Integer?

Answer: 

Yes, 0 is an integer. By the definition of integers, an integer is any number that can be either 0, or positive number, or negative number. And it does not have a fractional part or component. If 0 is added to an integer we get an integer. Example: 0 + 7 = 7.