Important Formulas: Integers | Mathematics Class 6 ICSE PDF Download

Table of Contents
1. Introduction of INTEGERS
2. Number line and sign
3. Opposite numbers and additive inverse
4. Rules for adding integers
5. Subtraction of integers
6. Algebraic and arithmetic properties
7. Order, successor and predecessor
8. Summary of key formulas and facts
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Introduction of INTEGERS 

Integers are the numbers ... , -4, -3, -2, -1, 0, 1, 2, 3, 4, ... .

• Positive integers are 1, 2, 3, 4, 5, ... .
• Negative integers are -1, -2, -3, -4, ... .
• Zero (0) is an integer that is neither positive nor negative.

Number line and sign

On an integer number line, numbers to the right of 0 are positive and numbers to the left of 0 are negative. As we move towards the right the values increase and as we move towards the left the values decrease. Thus the order of integers is written as ..., -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, ... .

• 0 is less than every positive integer and greater than every negative integer.
• Every positive integer is greater than every negative integer.

Opposite numbers and additive inverse

• Two integers that are at the same distance from 0 but on opposite sides are called opposite numbers (for example, 7 and -7).
• The additive inverse of an integer a is the integer which when added to a gives 0; the additive inverse of a is -a. Thus a and -a are additive inverses of each other.
• The sum of an integer and its opposite is zero: a + (-a) = 0.
• The greater the number, the lesser is its opposite (for example, 9 > 2 but -9 < />

Rules for adding integers

Use the following rules when adding two integers:

• If the integers have the same sign, add their absolute values and give the result the common sign.
  • Example: (-5) + (-3) = -(5 + 3) = -8.
  • Example: 4 + 6 = 10.
• If the integers have different signs, subtract the smaller absolute value from the larger absolute value and give the sign of the integer with the larger absolute value.
  • Example: 7 + (-4) = +(7 - 4) = +3.
  • Example: (-9) + 2 = -(9 - 2) = -7.
• Adding zero: a + 0 = a.

Worked examples - addition

Example 1. Find (-12) + (-8).

Sol. Both integers have the same sign (both negative).
Add their absolute values: 12 + 8 = 20.
Attach the common sign (negative): (-12) + (-8) = -20.
Ans. -20

Example 2. Find 15 + (-9).

Sol. Integers have different signs.
Subtract the smaller absolute value from the larger: 15 - 9 = 6.
Give the result the sign of the number with larger absolute value (15 is
positive): 6.
Ans. 6

Subtraction of integers

To subtract an integer b from another integer a, change the sign of b and add it to a. In symbol form:
a - b = a + (-b)
Use the addition rules above after changing the sign of b.

Worked example - subtraction

Example. Compute 8 - (-5).

Sol. Change the sign of the subtracted integer: -(-5) becomes +5.
Now add: 8 + 5 = 13.
Ans. 13

Algebraic and arithmetic properties

• Closure: The sum or difference of two integers is always an integer. If a and b are integers, then a + b and a - b are integers.
• Commutative property of addition: a + b = b + a.
• Associative property of addition: (a + b) + c = a + (b + c).
• Distributive law links multiplication with addition and subtraction (applicable when multiplication is introduced): a(b + c) = ab + ac.
• All properties that apply to whole-number addition apply to integer addition as well.

Order, successor and predecessor

• On the number line the order of integers increases to the right and decreases to the left.
• Successor of an integer is the integer one more than the given number.
• Predecessor of an integer is the integer one less than the given number.
• For any integer n, the successor is n + 1 and the predecessor is n - 1.

Summary of key formulas and facts

• Set of integers: ..., -3, -2, -1, 0, 1, 2, 3, ...
• a + (-a) = 0
• a - b = a + (-b)
• If signs are the same, add absolute values and keep the sign.
• If signs are different, subtract smaller absolute value from larger and keep the sign of the larger.
• Successor of n = n + 1; predecessor of n = n - 1.
• Integers are closed under addition and subtraction.