Important Formulas Integers - (Maths) Class 7 (Old NCERT)

Integers

The set of integers consists of all whole numbers together with their negative counterparts and zero. Examples are ..., −4, −3, −2, −1, 0, 1, 2, 3, 4, ...

Important formulas and properties

(1) The list ..., −4, −3, −2, −1, 0, 1, 2, 3, 4, ... represents the set of integers.

(2) The integers 1, 2, 3, 4, 5, ... are called positive integers. The integers −1, −2, −3, ... are called negative integers.

(3) 0 is an integer which is neither positive nor negative.

(4) On a number line, all integers to the right of 0 are positive and all integers to the left of 0 are negative.

(5) 0 is less than every positive integer and greater than every negative integer.

(6) Every positive integer is greater than every negative integer.

(7) Two integers that are at the same distance from 0 but on opposite sides of it are called opposite numbers (or additive inverses).

(8) If a is an integer, then its opposite is −a. The greater the number a, the lesser is its opposite −a (for a > 0).

(9) The sum of an integer and its opposite is zero: a + (−a) = 0.

(10) The absolute value of an integer is its distance from 0 on the number line, written |a|. It is defined piecewise as:

(10 - continued) |a| = a if a ≥ 0, and |a| = −a if a < 0.

(11) The sum of two integers with the same sign is an integer with that same sign whose absolute value is the sum of the absolute values of the addends.

(12) The sum of two integers with opposite signs is an integer whose absolute value is the difference of the absolute values; its sign is the sign of the addend with the greater absolute value.

(13) To subtract an integer b from another integer a, change the sign of b and add: a − b = a + (−b).

(14) All algebraic properties of addition and multiplication that hold for whole numbers (commutative, associative, distributive laws, and identity elements) also hold for integers.

(15) If a and b are integers, then (a − b) is also an integer (integers are closed under subtraction).

(16) a and −a are additive inverses of each other.

(17) To find the product of two integers, multiply their absolute values and give the result a positive sign if both integers have the same sign, or a negative sign if their signs are different.

(18) To find the quotient of one integer divided by another non‐zero integer, divide their absolute values and give the result a positive sign if the two integers have the same sign, or a negative sign otherwise.

(19) Integers are closed under addition, subtraction and multiplication. Integers are not closed under division (quotient need not be an integer).

(20) Any integer multiplied or divided by 1 gives itself. Any integer multiplied or divided by −1 gives its opposite (for division, dividing by −1 yields an integer with opposite sign).

(21) When an expression has different types of operations, operations are performed in a standard order. The correct order is given by the BODMAS rule: Brackets → Orders (powers and roots) → Division and Multiplication (from left to right) → Addition and Subtraction (from left to right).

(22) Brackets are used when we want a group of operations to be performed before the others.

(23) While simplifying an expression with brackets, perform the operations in the innermost brackets first, then remove those brackets and continue outward.

(24) While simplifying arithmetic expressions involving various brackets and operations, follow the BODMAS rule carefully. Division and multiplication have equal priority and are carried out left to right; addition and subtraction have equal priority and are carried out left to right.

Number line and opposites

The number line is a straight line on which every point corresponds to an integer. Zero divides the line into positive integers to the right and negative integers to the left. For any integer a, the opposite number −a is located the same distance from 0 on the other side.

Operations on integers - rules and examples

Addition

Rules:

• If both integers have the same sign, add their absolute values and keep the common sign.
• If the integers have opposite signs, subtract the smaller absolute value from the larger absolute value and take the sign of the integer with the larger absolute value.

Example: Add −3 and −5.

Sol.

−3 + (−5) = −(3 + 5)

= −8

Example: Add 7 and −4.

Sol.

7 + (−4) = 7 − 4

= 3

Subtraction

Rule: To subtract b from a, add the opposite of b: a − b = a + (−b).

Example: Compute 5 − (−2).

Sol.

5 − (−2) = 5 + 2

= 7

Multiplication

Rule: Multiply absolute values; the sign of the product is positive when the factors have the same sign, negative otherwise.

Example: Compute (−3) × 4.

Sol.

|−3| × |4| = 3 × 4

Since signs are different, product = −12

Example: Compute (−3) × (−4).

Sol.

|−3| × |−4| = 3 × 4

Since signs are same, product = 12

Division

Rule: Divide absolute values; the sign of the quotient is positive when dividend and divisor have the same sign, negative otherwise. Division of integers may produce a non‐integer quotient.

Example: Compute (−12) ÷ 3.

Sol.

|−12| ÷ |3| = 12 ÷ 3

Since signs differ, quotient = −4

Example: Compute 12 ÷ (−5).

Sol.

|12| ÷ |−5| = 12 ÷ 5

Since signs differ, quotient = −12/5 (not an integer)

Properties of integer operations (summary)

• Closure: Integers are closed under addition, subtraction and multiplication; not closed under division.
• Commutative: a + b = b + a and a × b = b × a.
• Associative: (a + b) + c = a + (b + c) and (a × b) × c = a × (b × c).
• Distributive: a × (b + c) = a × b + a × c.
• Identities: Additive identity is 0 (a + 0 = a). Multiplicative identity is 1 (a × 1 = a).
• Additive inverse: For every a there exists −a such that a + (−a) = 0.

Order of operations (BODMAS) - example

Apply the BODMAS rule: Brackets → Orders (powers/roots) → Division and Multiplication (left to right) → Addition and Subtraction (left to right).

Example: Simplify 3 + 6 × (5 + 4) ÷ 3 − 7.

Sol.

Evaluate the bracket first: (5 + 4) = 9

Replace into expression: 3 + 6 × 9 ÷ 3 − 7

Perform multiplication and division left to right: 6 × 9 = 54

Now: 3 + 54 ÷ 3 − 7

Now: 54 ÷ 3 = 18

Now: 3 + 18 − 7

Perform addition and subtraction left to right: 3 + 18 = 21

21 − 7 = 14

More worked examples for practice

Example: Find |−15| + (−6) + 8.

Sol.

|−15| = 15

15 + (−6) + 8

15 − 6 + 8

9 + 8 = 17

Example: Simplify (−4) × [2 + (−3)] − (−6).

Sol.

Evaluate inner bracket: 2 + (−3) = −1

(−4) × (−1) − (−6)

(−4) × (−1) = 4

4 − (−6) = 4 + 6

= 10

Concluding notes

Understanding integers, their properties and the correct order of operations is essential. Use the number line to visualise size and opposites. Remember the sign rules for addition, subtraction, multiplication and division, and always apply BODMAS when simplifying mixed expressions.