A fraction
$\frac{a}{b}$
numerator over denominator, b ≠ 0
Equivalent fractions
$\frac{a}{b} = \frac{a \cdot k}{b \cdot k}$
expanding: multiply both numerator and denominator by the same k
Adding fractions
$\frac{a}{b} + \frac{c}{d} = \frac{a \cdot d + c \cdot b}{b \cdot d}$
bring both to a common denominator first
Multiplying fractions
$\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}$
multiply the numerators, multiply the denominators
Reducing a fraction
$\frac{a}{b} = \frac{a \div \mathrm{gcd}(a,b)}{b \div \mathrm{gcd}(a,b)}$
gcd = greatest common divisor — divide numerator and denominator by it

Decimals

Place value
$3.45 = 3 + \frac{4}{10} + \frac{5}{100}$
the digits after the point are tenths, hundredths, thousandths
A fraction as a decimal
$\frac{3}{4} = \frac{75}{100} = 0.75$
when the denominator expands to a power of 10
Adding decimals
$3.25 + 1.40 = 4.65$
line up the decimal points, then add like whole numbers
Multiplying decimals
$2.5 \cdot 1.2 = 3.00$
multiply like whole numbers, place the point at the end
Rounding to hundredths
$3.465 \approx 3.47$
look at the first digit you drop — here 5, so it rounds up

Percentages

Percentage notation
$p\% = \frac{p}{100}$
a percentage is a hundredth of a whole
Percentage of a number
$p\% \text{ of } c = \frac{p}{100} \cdot c$
how much p percent of the whole c is
Whole from a known part
$c = \text{part} \div \frac{p}{100}$
given the part and its percentage, recover the whole
Percentage change
$\Delta\% = \frac{\text{new} - \text{old}}{\text{old}} \cdot 100\%$
positive is an increase, negative is a decrease

Powers

Definition of a power
$a^n = \underbrace{a \cdot a \cdot \ldots \cdot a}_{n \text{ factors}}$
a is the base, n is the exponent
Multiplying powers with the same base
$a^m \cdot a^n = a^{m+n}$
add the exponents
Dividing powers with the same base
$\frac{a^m}{a^n} = a^{m-n}$
subtract the exponents
A power of a power
$(a^m)^n = a^{m \cdot n}$
multiply the exponents
A power of a product
$(a \cdot b)^n = a^n \cdot b^n$
raise each factor separately
Zero and negative exponents
$a^0 = 1, \quad a^{-n} = \frac{1}{a^n} \quad (a \neq 0)$
a negative exponent means a reciprocal

Roots

Square root
$\sqrt{a} = b \iff b^2 = a \quad (a, b \ge 0)$
the inverse of squaring
Cube root
$\sqrt[3]{a} = b \iff b^3 = a$
defined for negative numbers too
Product of roots
$\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$
multiply under a single radical
Quotient of roots
$\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} \quad (b > 0)$
the same law for division
Taking a factor out of a radical
$\sqrt{a^2 \cdot b} = a\sqrt{b} \quad (a \ge 0)$
a square leaves the radical as its base
Rationalising a denominator
$\frac{1}{\sqrt{a}} = \frac{\sqrt{a}}{a} \quad (a > 0)$
multiply top and bottom by the same root

GCD and LCM

Divisibility
$a \mid b \iff b = a \cdot k, \; k \in \mathbb{Z}$
a divides b when b is a multiple of a
Greatest common divisor
$\gcd(a, b) = \max\{d : d \mid a \text{ and } d \mid b\}$
the largest number dividing both
Lowest common multiple
$\operatorname{lcm}(a, b) = \min\{w > 0 : a \mid w \text{ and } b \mid w\}$
the smallest positive number both divide
The identity tying GCD to LCM
$\gcd(a, b) \cdot \operatorname{lcm}(a, b) = a \cdot b$
know one and you can compute the other
Euclidean algorithm
$\gcd(a, b) = \gcd(b, a \bmod b)$
repeat until the remainder is zero

Prime numbers

Definition of a prime
$p \text{ is prime} \iff p > 1 \text{ and } p \text{ has exactly 2 divisors}$
its only divisors are 1 and itself
Fundamental theorem of arithmetic
$n = p_1^{a_1} \cdot p_2^{a_2} \cdot \ldots \cdot p_k^{a_k}$
the prime factorisation is unique
Primality test
$n \text{ composite} \implies \exists\, p \le \sqrt{n} : p \mid n$
divisors up to the square root are enough
GCD and LCM from factorisations
$\gcd = \prod p_i^{\min(a_i, b_i)}, \quad \operatorname{lcm} = \prod p_i^{\max(a_i, b_i)}$
lowest exponents for the GCD, highest for the LCM

$A fraction bar and a percent sign rendered as geometric forms, with a circle split into equal orbital slices, against a cosmic nebula in violet and magenta$

Numbers

A percentage is a fraction in disguise — three ways to write one number

1/2, 0.5 and 50% are not three different things but three ways of writing the same number. Here is where the fraction bar, the decimal point and the % sign came from, why 1/3 never ends, why +20% and −20% do not return you to the start, and why fractions are the first serious wall in school mathematics.

11 min

Numbers

Topics in this branch

Branch formulas

Fractions