Class 6 — Mathematics (NCERT)

Integers

Class 6

✓By the end of this lesson students will be able to identify and define integers.
✓By the end of this lesson students will be able to represent integers on a number line.
✓By the end of this lesson students will be able to perform addition of integers using a number line and rules.
✓By the end of this lesson students will be able to perform subtraction of integers using a number line and rules.
✓By the end of this lesson students will be able to solve simple problems involving addition and subtraction of integers.

Key concepts

What are Integers?

Integers are a collection of numbers that include whole numbers (0, 1, 2, 3, ...) and their negative counterparts (-1, -2, -3, ...). They can be positive integers (1, 2, 3, ...), negative integers (-1, -2, -3, ...), or zero (which is neither positive nor negative). The set of integers is denoted by 'Z'.

Number Line for Integers

A number line is a straight line on which integers are marked at equal distances. Zero is marked at the centre. Positive integers are marked to the right of zero, and negative integers are marked to the left of zero. As we move to the right on a number line, the value of the numbers increases. As we move to the left, the value decreases. For example, -3 is smaller than -1, and 5 is greater than 2.

Addition of Integers

To add integers using a number line:\n1. To add a positive integer, move to the right on the number line.\n2. To add a negative integer, move to the left on the number line.\n\nRules for addition without a number line:\n1. When adding two positive integers, add their absolute values and the result is positive. (e.g., 3 + 5 = 8)\n2. When adding two negative integers, add their absolute values and the result is negative. (e.g., -3 + (-5) = -8)\n3. When adding a positive and a negative integer, find the difference between their absolute values and the result takes the sign of the integer with the larger absolute value. (e.g., 5 + (-3) = 2; -5 + 3 = -2)

Subtraction of Integers

Subtracting an integer is the same as adding its additive inverse (or opposite). The additive inverse of a number 'a' is '-a', such that a + (-a) = 0. For example, the additive inverse of 5 is -5, and the additive inverse of -7 is 7.\n\nUsing this concept, a - b = a + (-b).\n\nTo subtract integers using a number line:\n1. To subtract a positive integer, move to the left on the number line.\n2. To subtract a negative integer, move to the right on the number line (because subtracting a negative is equivalent to adding a positive).

a - b = a + (-b)

Key facts to remember

1Integers include positive numbers, negative numbers, and zero.
2Zero is an integer but is neither positive nor negative.
3On a number line, numbers to the right are always greater than numbers to the left.
4Adding a positive integer means moving to the right on the number line.
5Adding a negative integer means moving to the left on the number line.
6Subtracting a positive integer means moving to the left on the number line.
7Subtracting a negative integer means moving to the right on the number line (same as adding a positive integer).
8Every integer has an additive inverse; for example, the additive inverse of 'a' is '-a'.

Worked examples

Example 1

Add 5 and (-8) using a number line.

IStep 1: Start at the first integer, which is 5, on the number line.

IIStep 2: Since we are adding a negative integer (-8), we move 8 steps to the left from 5.

IIIStep 3: Moving 8 steps left from 5, we reach -3.

Answer

5 + (-8) = -3

Adding a negative integer means moving towards the left on the number line.

Example 2

Subtract 6 from 2 using a number line.

IStep 1: The problem is 2 - 6.

IIStep 2: Start at the first integer, which is 2, on the number line.

IIIStep 3: Since we are subtracting a positive integer (6), we move 6 steps to the left from 2.

IVStep 4: Moving 6 steps left from 2, we reach -4.

Answer

2 - 6 = -4

Subtracting a positive integer means moving towards the left on the number line.

Example 3

Evaluate -7 + 10 - (-4).

IStep 1: Simplify the expression by converting subtraction to addition of the additive inverse. -7 + 10 - (-4) becomes -7 + 10 + 4.

IIStep 2: Perform the operations from left to right. First, calculate -7 + 10.

IIIStep 3: To calculate -7 + 10: Start at -7 on the number line. Since 10 is positive, move 10 steps to the right. We reach 3.

IVStep 4: Now, we have 3 + 4.

VStep 5: To calculate 3 + 4: Start at 3 on the number line. Since 4 is positive, move 4 steps to the right. We reach 7.

Answer

-7 + 10 - (-4) = 7

Remember that subtracting a negative integer is equivalent to adding a positive integer.

Common mistakes

✗Confusing the direction of movement on the number line for addition and subtraction (e.g., moving left when adding a positive integer).
✗Incorrectly applying rules for signs when adding or subtracting integers without a number line.
✗Forgetting that 'minus a minus' becomes 'plus' (e.g., -5 - (-3) is often incorrectly calculated as -8 instead of -2).
✗Not understanding that subtracting a number is the same as adding its additive inverse.
✗Misinterpreting the value of negative numbers (e.g., thinking -5 is greater than -2).

Exam tips

★Always draw a simple number line to visualise the movement, especially for initial practice or if you are unsure.
★Remember the rule: 'Subtracting an integer is the same as adding its additive inverse.' This simplifies subtraction problems.
★Pay close attention to the signs of the integers. A small error in sign can lead to a completely wrong answer.
★Practice a variety of problems regularly to master the addition and subtraction of integers.

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