🔢 Math Formulas
39 formulas in 18 sections. For each one you get the formula, what every symbol means, where it is used, an explanation in plain words and a worked example.
Sections: HCF & LCM · Exponents · Standard Form · Squares · Cubes · Fractions · Divisibility · Surds · Order of Operations · Rational Numbers · Divisibility Rules · Factors · Number Types · Decimals · Unit Digit · Place Value · Consecutive Numbers · Comparing
HCF × LCM = a × b
| HCF | highest common factor |
|---|---|
| LCM | lowest common multiple |
| a, b | the two numbers |
Where it is used: To find the LCM quickly when the HCF is known (or vice versa) for two numbers.
In simple words: For any two numbers, the biggest number that divides both, times the smallest number both divide into, equals the two numbers multiplied together.
Example: For 12 and 18: HCF=6, LCM=36 → 6×36 = 12×18 = 216
LCM = (a × b) ÷ HCF
| a, b | the two numbers |
|---|---|
| HCF | highest common factor |
Where it is used: To calculate the LCM of two numbers using their product and their HCF — used in adding fractions and solving timing problems.
In simple words: Multiply the two numbers and divide by the biggest number that divides both.
Example: a=12, b=18, HCF=6 → LCM = 216 ÷ 6 = 36
HCF = HCF(numerators)/LCM(denominators); LCM = LCM(num)/HCF(den)
| num | numerators |
|---|---|
| den | denominators |
Where it is used: To find the HCF or LCM of fractions.
In simple words: Special rule: swap where HCF and LCM go for the tops and bottoms.
Example: HCF of 2/3, 4/9 = HCF(2,4)/LCM(3,9) = 2/9
aᵐ × aⁿ = aᵐ⁺ⁿ
| a | the base |
|---|---|
| m, n | the powers |
Where it is used: To multiply powers of the same base by adding the exponents — used all over algebra and science.
In simple words: When you multiply the same number raised to powers, just add the little numbers on top.
Example: 2³ × 2² = 2⁵ = 32
aᵐ ÷ aⁿ = aᵐ⁻ⁿ
| a | the base (a ≠ 0) |
|---|---|
| m, n | the powers |
Where it is used: To divide powers of the same base by subtracting the exponents.
In simple words: When you divide the same number raised to powers, subtract the little numbers on top.
Example: 2⁵ ÷ 2² = 2³ = 8
(aᵐ)ⁿ = aᵐⁿ
| a | the base |
|---|---|
| m, n | the powers |
Where it is used: To simplify a power raised to another power by multiplying the exponents.
In simple words: A power of a power: multiply the two little top numbers together.
Example: (2³)² = 2⁶ = 64
a⁰ = 1 (a ≠ 0)
| a | any non-zero number |
|---|
Where it is used: A rule used constantly when simplifying algebraic and scientific expressions.
In simple words: Any number (except 0) raised to the power 0 equals 1.
Example: 7⁰ = 1, 100⁰ = 1
a⁻ⁿ = 1 ÷ aⁿ
| a | the base (a ≠ 0) |
|---|---|
| n | the power |
Where it is used: To rewrite negative powers as fractions — used in standard form and algebra.
In simple words: A negative power means 'one divided by' that power — it flips it into a fraction.
Example: 2⁻³ = 1 ÷ 2³ = 1/8
a^(1/n) = ⁿ√a
| a | the base |
|---|---|
| n | the root |
Where it is used: To rewrite roots as powers and vice versa — used in surds, indices and calculus.
In simple words: A power of one-over-n means the nth root: power 1/2 is a square root, 1/3 is a cube root.
Example: 27^(1/3) = ³√27 = 3
(aᵐ)ⁿ = a^(mn), aᵐ × aⁿ = a^(m+n)
| a | the base |
|---|---|
| m, n | the powers |
Where it is used: The core index laws used to simplify powers.
In simple words: Multiply powers of a power; add powers when multiplying the same base.
Example: (2²)³ = 2⁶ = 64
N = k × 10ⁿ, 1 ≤ k < 10
| k | a number between 1 and 10 |
|---|---|
| n | an integer power of 10 |
Where it is used: To write very big or very small numbers compactly — used in science for distances, sizes and masses.
In simple words: Write the number as a value between 1 and 10 times a power of ten, so huge or tiny numbers are easy to read.
Example: 300000 = 3 × 10⁵
Square = n × n = n²
| n | the number |
|---|
Where it is used: To find a number multiplied by itself — used in area, Pythagoras and algebra.
In simple words: Squaring means multiplying a number by itself.
Example: 6² = 6 × 6 = 36
√n = m, where m × m = n
| n | the number |
|---|---|
| m | its square root |
Where it is used: To find the number which, multiplied by itself, gives n — used in Pythagoras, areas, statistics.
In simple words: The square root asks: what number times itself makes this number?
Example: √64 = 8 because 8 × 8 = 64
(n5)² = n(n+1) followed by 25
| n | the digits before the 5 |
|---|
Where it is used: A quick trick to square any number ending in 5.
In simple words: Multiply the front part by the next number, then stick 25 on the end.
Example: 35² → 3×4=12, so 1225
Cube = n × n × n = n³
| n | the number |
|---|
Where it is used: To find a number multiplied by itself three times — used in volume and algebra.
In simple words: Cubing means multiplying a number by itself three times.
Example: 4³ = 4 × 4 × 4 = 64
a/c + b/c = (a + b)/c
| a, b | the numerators |
|---|---|
| c | the common denominator |
Where it is used: To add fractions that share the same denominator — sharing, measuring, part-work problems.
In simple words: When the bottoms are the same, just add the tops and keep the bottom.
Example: 2/7 + 3/7 = 5/7
a/b × c/d = (a × c)/(b × d)
| a, c | numerators |
|---|---|
| b, d | denominators |
Where it is used: To multiply two fractions — scaling recipes, finding a fraction of a fraction.
In simple words: Multiply the top numbers together and the bottom numbers together.
Example: 2/3 × 3/4 = 6/12 = 1/2
a/b ÷ c/d = a/b × d/c
| a/b | first fraction |
|---|---|
| c/d | second fraction (flipped) |
Where it is used: To divide by a fraction by multiplying with its reciprocal (flip) — a key algebra and arithmetic skill.
In simple words: To divide by a fraction, flip the second one upside down and multiply instead.
Example: 1/2 ÷ 1/4 = 1/2 × 4/1 = 2
a/c − b/c = (a − b)/c
| a, b | numerators |
|---|---|
| c | common denominator |
Where it is used: To subtract fractions with the same denominator.
In simple words: Same bottoms: subtract the tops and keep the bottom.
Example: 5/8 − 3/8 = 2/8 = 1/4
a/b + c/d = (ad + bc) ÷ bd
| a/b, c/d | the two fractions |
|---|
Where it is used: To add fractions with different denominators.
In simple words: Cross-multiply and add the tops over the product of the bottoms.
Example: 1/2 + 1/3 = (3+2)/6 = 5/6
a b/c = (a × c + b) ÷ c
| a | whole part |
|---|---|
| b/c | fraction part |
Where it is used: To convert a mixed number into an improper fraction before calculating.
In simple words: Multiply the whole number by the bottom, add the top, and keep the bottom.
Example: 2¾ = (2×4+3)/4 = 11/4
Dividend = Divisor × Quotient + Remainder
| Dividend | number being divided |
|---|---|
| Divisor | number you divide by |
| Quotient | how many times it fits |
| Remainder | what is left over |
Where it is used: The basic relationship behind all division — used to check answers and in number theory.
In simple words: The number you start with equals how many times the divisor fits, times the divisor, plus whatever is left over.
Example: 17 = 5 × 3 + 2
√a × √b = √(ab)
| a, b | non-negative numbers |
|---|
Where it is used: To multiply or simplify square roots (surds) in algebra.
In simple words: Multiply what's inside the roots and keep it under one root.
Example: √2 × √8 = √16 = 4
√a ÷ √b = √(a/b)
| a, b | non-negative, b ≠ 0 |
|---|
Where it is used: To simplify a ratio of square roots.
In simple words: Divide what's inside the roots and keep it under one root.
Example: √18 ÷ √2 = √9 = 3
Brackets → Orders(powers) → Division/Multiplication → Addition/Subtraction
| BODMAS | the order to calculate in |
|---|
Where it is used: To evaluate an expression in the correct order so everyone gets the same answer.
In simple words: Do brackets first, then powers, then × and ÷, and finally + and −.
Example: 2 + 3 × 4 = 2 + 12 = 14 (not 20)
Between a and b: (a + b) ÷ 2
| a, b | the two numbers |
|---|
Where it is used: To find a number lying between any two given numbers — shows there are infinitely many.
In simple words: The average of two numbers always sits exactly between them.
Example: Between 1/2 and 1: (1/2 + 1)/2 = 3/4
by 2: last digit even · by 5: ends in 0 or 5 · by 10: ends in 0
| last digit | the units digit |
|---|
Where it is used: To check quickly whether a number divides by 2, 5 or 10 without dividing.
In simple words: Just look at the last digit to test these three.
Example: 340 is divisible by 2, 5 and 10
by 3: digit sum ÷ 3 · by 9: digit sum ÷ 9
| digit sum | add all the digits |
|---|
Where it is used: To test divisibility by 3 or 9 using the sum of the digits.
In simple words: Add all the digits — if that total divides by 3 (or 9), so does the number.
Example: 927 → 9+2+7=18, divisible by both 3 and 9
by 4: last 2 digits ÷ 4 · by 8: last 3 digits ÷ 8
| last digits | the ending digits |
|---|
Where it is used: To test divisibility by 4 or 8 using only the ending digits.
In simple words: Check the last two digits for 4, the last three for 8.
Example: 1236 → 36 ÷ 4, so divisible by 4
Difference of alternate digit sums is 0 or a multiple of 11
| — | odd-place sum − even-place sum |
|---|
Where it is used: To test divisibility by 11.
In simple words: Add the digits in odd spots and in even spots; if the difference is 0 or 11, it divides by 11.
Example: 918082 → (9+8+8)−(1+0+2)=22, divisible by 11
If N = aᵖ × bᑫ × …, factors = (p+1)(q+1)…
| a, b | prime factors |
|---|---|
| p, q | their powers |
Where it is used: To count how many divisors a number has from its prime factorisation.
In simple words: Add one to each power in the prime factorisation and multiply them.
Example: 12 = 2²×3¹ → (2+1)(1+1) = 6 factors
Sum = [(aᵖ⁺¹−1)/(a−1)] × [(bᑫ⁺¹−1)/(b−1)] …
| a, b | prime factors |
|---|---|
| p, q | their powers |
Where it is used: To find the total of all divisors of a number from its prime factorisation.
In simple words: A formula that adds up every divisor of a number without listing them.
Example: 12 = 2²×3 → (7)(4) = 28 (1+2+3+4+6+12)
Natural ⊂ Whole ⊂ Integers ⊂ Rational ⊂ Real
| ⊂ | is contained within |
|---|
Where it is used: To classify numbers — the foundation of the number system.
In simple words: Counting numbers sit inside whole numbers, inside integers, inside fractions, inside all real numbers.
Example: −3 is an integer and a rational and a real number
Prime: exactly 2 factors · Composite: more than 2 · Co-prime: HCF = 1
| factors | numbers that divide it exactly |
|---|
Where it is used: To identify prime, composite and co-prime numbers in number theory.
In simple words: A prime has only 1 and itself as factors; two co-primes share no common factor but 1.
Example: 7 is prime; 8 is composite; 8 and 9 are co-prime
0.ā = a/9, 0.āb̄ = ab/99
| a, ab | the repeating digits |
|---|
Where it is used: To turn a repeating decimal into an exact fraction.
In simple words: Put the repeating digits over as many 9s as there are repeating digits.
Example: 0.333… = 3/9 = 1/3
Unit digits repeat in cycles (2,4,8,6…), period 4
| — | the last digit of the base |
|---|
Where it is used: To find the last digit of a large power without full calculation.
In simple words: The last digit of powers repeats in a short cycle, so you only need the remainder of the power ÷ 4.
Example: 2¹⁰ ends in 4 (cycle 2,4,8,6)
Place value = digit × its place; Face value = the digit itself
| digit | a single figure in the number |
|---|
Where it is used: To distinguish a digit's position-based value from the digit itself.
In simple words: Face value is the digit as-is; place value depends on where it sits.
Example: In 57, place value of 5 is 50, face value is 5
Sum = (number of terms) × (average of first and last)
| average | (first + last) ÷ 2 |
|---|
Where it is used: To add a run of consecutive numbers quickly.
In simple words: Multiply how many numbers there are by the average of the ends.
Example: 5+6+7+8 = 4 × 6.5 = 26
a/b vs c/d: compare a×d with b×c
| a/b, c/d | the two fractions |
|---|
Where it is used: To decide which of two fractions is larger without a common denominator.
In simple words: Cross-multiply; the bigger product sits above the bigger fraction.
Example: 3/4 vs 5/7 → 21 vs 20, so 3/4 is bigger