Basic Operations: Addition, Subtraction, Multiplication and Division

Basic Operations include addition, subtraction, multiplication and division. This topic explains the meaning, signs, operation rules and calculation methods for working with whole numbers, integers, fractions and decimals. It also covers the correct order of operations and useful arithmetic shortcuts for simplifying numerical expressions accurately.

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What Are Basic Operations?

Basic Operations are the four fundamental arithmetic processes: addition, subtraction, multiplication and division. They are used to combine, compare, scale or share quantities.

In addition, quantities are combined; in subtraction, one quantity is taken away from another; in multiplication, equal groups are combined; and in division, a quantity is split into equal parts. For an expression containing several operations, use brackets first, then multiplication or division from left to right, and finally addition or subtraction from left to right. For example, 8 + 3 × 2 = 8 + 6 = 14.

Basic Operations Formula & Tricks

Important Formulas

Addition
a + b = sum

The numbers a and b are called addends, and their result is the sum.

Subtraction
a − b = difference

Here, a is the minuend, b is the subtrahend and the result is the difference.

Multiplication
a × b = product

The numbers a and b are factors, and their result is the product.

Division
a ÷ b = quotient, where b ≠ 0

The number being divided is the dividend, b is the divisor and the result is the quotient. Division by zero is undefined.

Division with remainder
Dividend = Divisor × Quotient + Remainder

For whole-number division, the remainder is non-negative and less than the divisor.

Quick Tricks

Use left-to-right order for equal precedence

Multiplication and division have equal precedence, so calculate them from left to right. Addition and subtraction also have equal precedence and are calculated from left to right.

Example: 24 ÷ 6 × 3 = 4 × 3 = 12, not 24 ÷ 18.
Apply the distributive property

Multiplication can be distributed over addition or subtraction: a(b + c) = ab + ac and a(b − c) = ab − ac.

Example: 48 × 19 = 48 × (20 − 1) = 960 − 48 = 912.
Use compensation for nearby numbers

Round one number to a convenient value and adjust the result by the same amount.

Example: 399 + 248 = 400 + 248 − 1 = 647.
Check division by multiplication

Multiply the quotient by the divisor and add the remainder. The result must equal the dividend.

Example: 157 ÷ 12 gives quotient 13 and remainder 1 because 12 × 13 + 1 = 157.

Basic Operations Concepts

Addition and Subtraction of Integers

For integers, add numbers with the same sign by adding their absolute values and retaining the sign; for different signs, subtract the smaller absolute value from the larger and retain the sign of the number with the larger absolute value.

Subtraction can be changed into addition of the additive inverse: a − b = a + (−b). Therefore, subtracting a negative number becomes addition. For example, (−9) + 14 = 5 and 7 − (−3) = 7 + 3 = 10.

Example: (−18) − 7 = (−18) + (−7) = −25.

Multiplication and Division Sign Rules

The product or quotient is positive when both numbers have the same sign and negative when their signs are different.

The sign rules are (+) × (+) = +, (−) × (−) = +, (+) × (−) = − and (−) × (+) = −. The same rules apply to division. Any number multiplied by zero is zero, but division by zero is undefined.

Example: (−36) ÷ (−6) = 6, while 36 ÷ (−6) = −6.

Order of Operations

Evaluate brackets first, followed by exponents if present, multiplication or division from left to right, and addition or subtraction from left to right.

The sequence is often written as BODMAS or PEMDAS. Multiplication does not always come before division; both have the same precedence. Similarly, addition and subtraction have equal precedence. Brackets may be nested, so solve the innermost bracket first.

Example: 18 − 4 × (6 + 2) ÷ 2 = 18 − 4 × 8 ÷ 2 = 18 − 16 = 2.

Fractions in Basic Operations

To add or subtract fractions, convert them to a common denominator; to multiply fractions, multiply numerators and denominators; to divide by a fraction, multiply by its reciprocal.

For non-zero denominators, a/b + c/d = (ad + bc)/bd and a/b − c/d = (ad − bc)/bd. Also, (a/b) × (c/d) = ac/bd and (a/b) ÷ (c/d) = ad/bc, where c is not zero. Reduce the final fraction to its lowest terms.

Example: 2/3 + 1/4 = 8/12 + 3/12 = 11/12, and 3/5 ÷ 2/7 = 3/5 × 7/2 = 21/10.

Properties of Arithmetic Operations

Addition and multiplication are commutative and associative, but subtraction and division are generally neither commutative nor associative.

Commutative property: a + b = b + a and ab = ba. Associative property: (a + b) + c = a + (b + c) and (ab)c = a(bc). Distributive property: a(b + c) = ab + ac. The additive identity is 0, and the multiplicative identity is 1.

Example: 3 + 8 = 8 + 3 and 2 × (5 + 4) = 2 × 5 + 2 × 4 = 18. However, 10 − 4 ≠ 4 − 10.

Basic Operations Video Lessons

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13 Lessons
Lesson 1 of 13 Quick Revision

BODMAS with Overline Expressions

Learn how to apply BODMAS rules to simplify expressions containing overlines, following the correct order of operations step by step.

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Practice Basic Operations Questions

Practise published questions related to this topic.

11/(6 + √3) + 1/(6 - √3) = ?→ 2Simplify: ((3⁵ × 6⁴ × 4³))/((3² × 6² × 4¹))→ 3Simplify √(√(√256 + √144 + (2√2)²) + √841 × (√2)²)→ 4Simplify: (√98 + √200 - 2√8)²→ 5Simplify: (∛4096/⁴√10000) × 5/48 × 186→ 6(√6+√2)+1/((√6-√2)) = ?→ 7Simplify: √2 + 1/√2 + 1/(2+√2) - 1/(2-√2)→ 8(√32 + √162)/√18 = ?→ 9If x2 × x3 = xa and x²/(x^b) = x1, what is the value of a + b?→ 10Choose the incorrect relation from the following: I. √6 + √2 = √5 + √3 II. √6 + √2 √5 + √3→

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Quick Revision Notes

Basic Operations Quick Revision

Use these rules to simplify arithmetic expressions correctly.

  • Addition combines quantities, subtraction finds the difference, multiplication forms equal groups and division splits a quantity into equal parts.
  • For integers with the same sign, add absolute values and retain the sign; for different signs, subtract absolute values and retain the sign of the larger absolute value.
  • A product or quotient is positive for equal signs and negative for different signs.
  • Use brackets first, then multiplication or division from left to right, followed by addition or subtraction from left to right.
  • For fractions, use a common denominator for addition or subtraction, multiply corresponding terms for multiplication and use the reciprocal for division.
  • Division algorithm: Dividend = Divisor × Quotient + Remainder, with remainder less than the divisor.
  • Division by zero is undefined; multiplication by zero always gives zero.
  • Addition and multiplication are commutative and associative. Subtraction and division generally are not.

Basic Operations FAQs

What is the correct order of operations?

Solve brackets first, then exponents if present, multiplication and division from left to right, and addition and subtraction from left to right.

Is multiplication always performed before division?

No. Multiplication and division have equal precedence and are evaluated from left to right. For example, 36 ÷ 6 × 2 = 6 × 2 = 12.

How do you subtract a negative number?

Change subtraction into addition of the opposite number: a − (−b) = a + b. Thus, 9 − (−4) = 13.

What is the division algorithm?

Dividend = Divisor × Quotient + Remainder. For example, 29 ÷ 5 has quotient 5 and remainder 4 because 5 × 5 + 4 = 29.

How do you divide one fraction by another?

Multiply the first fraction by the reciprocal of the second: (a/b) ÷ (c/d) = (a/b) × (d/c) = ad/bc, provided b, c and d are non-zero.

Which arithmetic operations are commutative?

Addition and multiplication are commutative: a + b = b + a and ab = ba. Subtraction and division are generally not commutative.

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