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Integers and Their Arithmetic

Integers are numbers without fractional or decimal parts, including negative numbers, zero, and positive numbers.

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Student-friendly explanation

Integer arithmetic depends strongly on signs. Adding a negative means moving left on the number line, subtracting a negative means adding the opposite, and multiplication or division of two same signs gives positive while different signs give negative.

How to write this in exams

  1. 1

    Start with the exact idea

    Integers are numbers without fractional or decimal parts, including negative numbers, zero, and positive numbers.

  2. 2

    Then show how to use it

    Remove brackets carefully, replace subtraction by addition of the opposite integer, group positives and negatives, then subtract smaller total from larger total and keep the sign of the larger total.

  3. 3

    Add one concrete example

    -7 + 3 = -4, -7 - (-3) = -4, (-6) × (-2) = 12, and (-15) ÷ 3 = -5.

  4. 4

    Avoid this incomplete answer

    For -12 + 7 - (-5), writing -10 is wrong because the second negative has been ignored.

Definition

Integers are numbers without fractional or decimal parts, including negative numbers, zero, and positive numbers.

Example

-7 + 3 = -4, -7 - (-3) = -4, (-6) × (-2) = 12, and (-15) ÷ 3 = -5.

Rule to remember

Same signs in multiplication or division give positive; different signs give negative. Subtracting an integer means adding its additive inverse.

Memory hook

Minus a minus becomes a plus, but only at that operation.

Examples and method

Worked example

Simplify -12 + 7 - (-5). First change -(-5) to +5. Then -12 + 7 + 5 = 0.

Method to apply

Remove brackets carefully, replace subtraction by addition of the opposite integer, group positives and negatives, then subtract smaller total from larger total and keep the sign of the larger total.

Diagram support

A number line can show movement left for negative addition and movement right when subtracting a negative.

How CBSE asks it

Questions usually ask simplification of signed expressions, number-line movement, or identification of wrong sign steps.

Avoid common mistakes

Common confusion

A common mistake is writing -8 - (-5) as -13 instead of changing subtraction of a negative into addition.

Common wrong answer

For -12 + 7 - (-5), writing -10 is wrong because the second negative has been ignored.

Exam tip

In integer simplification, handle brackets and sign changes before doing the final arithmetic.

Quick check

Why is -9 - (-4) equal to -5?

Because subtracting -4 is the same as adding 4, so -9 - (-4) becomes -9 + 4, which equals -5.

Study the integers and their arithmetic diagram carefully

Use the labelled diagram to keep integers and their arithmetic clear in short answers and revision.

What this diagram makes clear

This diagram keeps the labels and direction of integers and their arithmetic in the right order.

Where this helps in exams

Use this for labelled diagram work and short exam answers on integers and their arithmetic.

Revision cue

Revise integers and their arithmetic through the labels before writing the answer.

Answer writing and exam use

1-mark answer

Integers are numbers without fractional or decimal parts, including negative numbers, zero, and positive numbers.

2-mark answer

Integers are numbers without fractional or decimal parts, including negative numbers, zero, and positive numbers. Same signs in multiplication or division give positive; different signs give negative. Subtracting an integer means adding its additive inverse. -7 + 3 = -4, -7 - (-3) = -4, (-6) × (-2) = 12, and (-15) ÷ 3 = -5.

3-mark answer

Integer arithmetic depends strongly on signs. Adding a negative means moving left on the number line, subtracting a negative means adding the opposite, and multiplication or division of two same signs gives positive while different signs give negative. Same signs in multiplication or division give positive; different signs give negative. Subtracting an integer means adding its additive inverse. Simplify -12 + 7 - (-5). First change -(-5) to +5. Then -12 + 7 + 5 = 0. Questions usually ask simplification of signed expressions, number-line movement, or identification of wrong sign steps. For -12 + 7 - (-5), writing -10 is wrong because the second negative has been ignored.
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