Integers on Number Line: Except for fractions, the set of integers consists of zero, positive numbers, and negative numbers. When arithmetic operations are applied to integers, the rules for integer addition and subtraction are not the same.

A number line is a tool for comparing and organising numbers. Each number on a number line is plotted in relation to the origin (zero), and the position of a number on a number line defines its value. The numerals on the number are negative or positive depending on where they are in relation to zero 0.  This indicates that every number from one to zero is a positive number, and any number from one to zero is a negative number.

Integers on Number Line Definition

On a number line, integers are written at equal intervals. The integers are categorized as positive integers and negative integers. If you observe the number line shown below, you can see that the origin is \(0\) (zero), which stands in the middle of the number line. There are positive numbers are on the right side of zero, and on the left side, there are negative numbers.

So, this number line represents the integers which include positive and negative numbers. The integers extend endlessly in both directions, which is why the arrows are marked at both ends to show that the line is infinite.

If you see the above number line carefully, the numbers increase and decrease as the direction changes. This is because any number on the left side is always less than the number to its right. 

For example, \(3\) is less than \(4;\, – 5\) is less than \( – 2.\)
Similarly, any number on the right side is always more than the number to its left.
Example: \(5\) is greater than the number \(3;\, – 1\) is greater than \( – 4.\)

Representing Integers on Number Line

The number line is a straight line with numbers placed at equal intervals along its length. However, a number line can be extended infinitely in any direction and represented horizontally.
The integers are marked on the number line as given below:

The positive integers are marked on the right side of zero on the number line, and negative integers are placed on the left.

The farther numbers move to the right from zero on the number line, the value of the numbers increases, and the more distant the numbers move to the left on the number line from zero, the value decreases.

When we compare any two numbers on a number line, the number on the right side of the number line will be more significant.

Operations on the Number Line

The four fundamentals operations of mathematics are addition, subtraction, multiplication and division. There are different methods to perform these operations.
The arithmetic operations of integers can be explained in a better way on the number line.

Addition of Integers on Number Line

Addition on number line for integers is explained below:

Positive Integers: When you add any two positive numbers, the result will always be positive. Thus, while adding positive numbers, the direction of the movement will always be to the right side.
Example: Addition of \(4\) and \(4\left( {4 + 4 = 8} \right)\)

So the first number is \(4,\) and the second number is also \(4;\) both are positive numbers.
Locate the number \(4\) on a number line, then move \(4\) steps to the right side to reach \(8.\)

Negative Integers: When you add two negative numbers, the result you get is always negative. Thus, when you add negative numbers, the direction of the move will always be on the left side.
Example: The addition of the negative numbers \( – 3\) and \( – 5\)

The first number is \( – 3,\) and the second number is \( – 5;\) both are negative. Now, point out the number \( – 3\) on a number line, then move \(5\) steps to the left side to reach \( – 8.\)

Subtraction of Integers on Number Line

Subtraction on number line for integers is explained below:

Positive Integers: When you want to subtract two positive numbers, move to the left as far as the value of the second number.
Example: Subtract \(5\) from \(4\)

Here the first number is \(4,\) and the second number is \(5;\) both are positive. So, mark the number \(4,\) on a number line. Then, move \(5\) steps to the left will give \( – 1.\)

Negative Integers: When you want to subtract the two negative numbers, move towards the right side as far as the value of the second number.
Example: Subtract \( – 4\) from \( – 2\)

First, locate \( – 2\) on the number line, later move \(4\) steps to the right to reach \(2.\)

Multiplication of Integers of Number Line

You are aware that multiplication is also called repeated addition. To perform multiplication on a number line, you have to move towards the right side of the number line for the given number of times.

Example: Multiply: \(4 \times 3\) on a number line.
Here, you will move towards the right side of the number line.
You will get four groups of \(3\) at equal intervals on a number line. So, starting from \(0,\) you will move \(4\) times, combining \(3\) intervals at a time.
So, you will reach the number \(12\) by forming \(4\) individual groups. You can see the number line below that shows \(4 \times 3 = 12.\)

Division of Integers of Number Line

When you divide the integers on a number line, you move towards the number line’s left side and split a given number into equal groups. You know that the numbers decrease as you move towards the left.

Example: To solve \(8 \div 4,\) you count how many jumps of \(4\) it takes from \(8\) to reach \(0\)—these groups or jumps of \(4\) mean \(8 \div 4 = 2.\) The number line shows that it takes \(2\) groups of \(4\) to reach \(0.\)

Solved Examples – Integers on Number Line

Q.1. Which integers lie between \( – 8\) and \( – 2?\) Write the greatest integer and smallest integer among them?

Ans: Integers between \( – 8\) and \( – 2\) are \( – 7,\, – 6,\, – 5,\, – 4,\, – 3.\)
The integer \( – 3\) is the largest, and \( – 7\) is the smallest integer.

Q.2. One button is kept at \( – 3.\) Write the direction and how many steps we move to reach at \( – 9?\)
Ans: We have to move to six steps to the left side of the number \( – 3.\)

Q.3. Which number will you reach if you move \(4\) steps to the right of \( – 6?\)

Ans: We reach \( – 2\) when we move \(4\) steps to the right side of \(- 6\).

Q.4. Subtract \(\left( { + 3} \right)\) from \(\left( { – 3} \right).\)
Ans: \(\left( { – 3} \right) – \left( { + 3} \right) = \left( { – 3} \right) + \) (additive inverse of \( + 3\))
\( = \left( { – 3} \right) + \left( { – 3} \right) = – 6\)

Q.5. Subtract \(\left( { – 4} \right)\) from \(\left( { – 10} \right)\)
Ans: \(\left( { – 10} \right) – \left( { – 4} \right) = \left( { – 10} \right) + \) (additive inverse of \( – 4\))
\( = – 10 + 4 = – 6\)

Summary

In the given article, we discussed the definition of integers on the number line and how to represent the Integers on the number line. Also, we covered the addition, subtraction, multiplication and division of integers on the number line. Then we covered a few solved examples along with a few FAQs. This knowledge is helpful in understanding the number line.

Frequently Asked Questions (FAQ) – Integers on Number Line

Q.1. How do you add or subtract integers on the number line?
Ans: You have to follow the given steps to add or subtract the integers on the number line:
Add the positive integer by moving on to the right side of the number line.
Add the negative integer by moving on to the left side of the number line.
When you subtract an integer, you subtract by adding its opposites.

Q.2. How do you solve integers on a number line?
Ans: The example is provided below, showing how to subtract the integers on the number line. In the same way, you can solve the integers using different operations on the number line
We will Subtract the numbers \(5\) from \(4.\)
Here the first number is \(4,\) and the second number is \(5;\) both are positive. So, mark the number \(4\) on a number line. Then move \(5\) steps to the left will give \( – 1.\)

Q.3. What are the rules for integers?
Ans: The rules of the Integers are give below:
Positive \( + \) Positive \( = \) Positive
Negative \( + \) Negative \( = \) Negative
Positive \( + \) Negative \( = \) Positive or negative depending on the value of the integer.
Negative \( + \) Positive \( = \) Negative or positive depending on the value of the integer.

Q.4. How do you add integers without a number line?
Ans: While you add the integers, you keep the same symbol if the symbols are the same.
\(\left( + \right) + \left( + \right) = + \)
\(\left( – \right) + \left( – \right) = \)you keep the symbol of the greater number.
In the case of different symbols:
\(\left( + \right) + \left( – \right) = \)you keep the symbol of the greater number.
\(\left( – \right) + \left( + \right) = \)you keep the symbol of the greater number.

Q.5. What is an integer formula?
Ans: We do not have any particular formula for the integer as it is nothing but a set of numbers. However, there are specific rules when you perform any mathematical operations like addition, subtraction, etc. on integers:
(i) The addition of two positive integers will result in a positive integer.
(ii) The addition of the two negative integers results in the negative integer.
(iii) The addition of one positive and one negative integer will result in the following ways:
(a) We get a positive number if the positive integer is greater.
(b) We get a negative number if the negative integer is greater.

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