Cooking
The recipe asks for 3/4 cup of milk and you are making half a batch: 3/4 · 1/2 = 3/8 of a cup — an actual mark on the jug, halfway between 1/4 and 1/2, instead of a guess.
Fractions
Grocery shopping
The deli scale reads 0.384 kg of cheese at 41.90 a kilo: 0.384 · 41.90 = 16.0896, and the receipt prints 16.09 — you round the answer, never the numbers going into it.
Decimals
The sales rack
A 349 jacket at 30% off costs 349 · 0.7 = 244.30, and the promised "another 20% at the till" is not a 50% discount but 244.30 · 0.8 = 195.44 — 44% off altogether.
Percentages See the lesson: Decimals
Drives and memory sticks
A "64 GB" stick shows up in the file explorer as 59.6 GB: the maker counts 64 · 10⁹ bytes, the system divides by 2³⁰ = 1,073,741,824. Nobody stole the missing 4.4 GB — two different powers did.
Powers Convert it here: Data (IEC)
The garden
You bought 20 m² of paving for a square patio, so the side comes out at √20 = 2√5 ≈ 4.47 m. Rounding up to a "convenient" 4.5 m needs 20.25 m² — a quarter of a square metre you do not have.
Roots
Party planning
Sausages come ten to a pack, buns eight. To leave nothing in the fridge you make lcm(10, 8) = 40 hot dogs: four packs of sausages and five of buns.
GCD and LCM
Online banking
The padlock beside your bank's address rests on multiplying being easy and factoring not: you read 91 = 7 · 13 at a glance, but no machine today can split a 617-digit RSA key back into the two 300-plus-digit primes it was built from.
Prime numbers
Shopping and planning
A trip takes 47 people and one car seats 6 — the division gives 7.83, but the answer has to be a natural number, so 8 cars are needed. That is not rounding: the result must live in ℕ, because the eighth car either goes or it does not.
Number sets
Shopping and complaints
A bag of flour should hold 1000 g; the shop scale reads 977 g. The deviation is |977 − 1000| = 23 g — and that is what decides the complaint, because against an allowance of 15 g it makes no difference whether the bag is lighter or heavier.
Absolute value
Clothing sizes
A size chart assigns size M to a chest measurement of [96, 104) cm. A chest of 104 cm is already an L — and that is what the round bracket on the right end says, not the small print under the table.
Intervals
Cooking
A cake recipe for 4 people calls for 300 g of flour and 6 guests turn up. The proportion 300 : 4 = x : 6 gives x = 450 g. Every other ingredient scales the same way — one multiplier of 1.5 for the whole recipe.
Proportions and scale
Photocopiers and paper sizes
A-series sheets have a side ratio of √2 = 2^(1/2), which is why enlarging A4 to A3 is set at 141% (since 2^(1/2) ≈ 1.414) and reducing A3 to A4 at 71%. Two enlargements of 141% give exactly 200%, because 2^(1/2) · 2^(1/2) = 2.
Rational exponents
Noise and hearing
A conversation runs at about 60 dB and a busy street at 90 dB. The 30 dB gap does not mean half again as loud — it means 10³ = 1000 times the sound power, because the decibel is a logarithmic unit rather than an ordinary scale.
Logarithms Convert it here: Decibels
Savings account
A deposit of 1000 at 5% a year compounded monthly comes to 1000 · (1 + 0.05/12)²⁴ ≈ 1104.94 after two years, against 1102.50 with annual compounding. Those 2.44 of difference are the entire meaning of the word compounding in the terms and conditions.
Compound interest

Branch formulas

Branch: Numbers

Fractions

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
Subtracting fractions
$\frac{a}{b} - \frac{c}{d} = \frac{a \cdot d - c \cdot b}{b \cdot d}$
the same common denominator as for adding
Multiplying fractions
$\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}$
multiply the numerators, multiply the denominators
Dividing fractions
$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c} \quad (c \neq 0)$
multiply by the reciprocal of the divisor
A mixed number as an improper fraction
$c\frac{a}{b} = \frac{c \cdot b + a}{b}$
multiply the whole part by the denominator and add the numerator
Comparing fractions
$\frac{a}{b} > \frac{c}{d} \iff a \cdot d > c \cdot b \quad (b, d > 0)$
cross-multiplication — compare two products instead of two fractions
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
A decimal as a fraction
$0.35 = \frac{35}{100} = \frac{7}{20}$
the digits after the point over a power of ten, then reduce
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
Dividing decimals
$14.4 \div 1.2 = 144 \div 12 = 12$
shift the point in both numbers by the same amount until the divisor is whole
Rounding to hundredths
$3.465 \approx 3.47$
look at the first digit you drop — here 5, so it rounds up
A purely repeating expansion
$0.(3) = \frac{3}{9} = \frac{1}{3}$
the period over as many nines as it has digits
A mixed repeating expansion
$0.1(6) = \frac{16 - 1}{90} = \frac{1}{6}$
nines for the repeating digits, zeros for the ones before them

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
Change multiplier
$\text{new} = \text{old} \cdot \left(1 \pm \frac{p}{100}\right)$
a rise of p% multiplies by 1 + p/100, a discount by 1 − p/100
Two changes in a row
$\text{new} = \text{old} \cdot \left(1 + \frac{p_1}{100}\right)\left(1 + \frac{p_2}{100}\right)$
multiply the multipliers; never add the percentages

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
A power of a quotient
$\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n} \quad (b \neq 0)$
raise the numerator and the denominator 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
Monotonicity in the base
$0 \le a < b \implies a^n < b^n \quad (n \ge 1)$
with the same exponent, the larger base gives the larger power
Monotonicity in the exponent
$m < n \implies \begin{cases} a^m < a^n & \text{for } a > 1 \\ a^m > a^n & \text{for } 0 < a < 1 \end{cases}$
a base above 1 makes the power grow, a base between 0 and 1 makes it shrink

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
Absorbing a factor under a radical
$a\sqrt{b} = \sqrt{a^2 \cdot b} \quad (a \ge 0)$
a factor in front goes under the sign squared
Estimating a root
$n^2 \le x < (n+1)^2 \implies n \le \sqrt{x} < n + 1$
find the two consecutive squares the number lies between
Comparing roots
$0 \le a < b \iff \sqrt{a} < \sqrt{b}$
the larger number under the radical gives the larger root
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

Number sets

Natural numbers
$\mathbb{N} = \{0, 1, 2, 3, \ldots\}$
the numbers you count items with — no sign, no fractional part
Integers
$\mathbb{Z} = \{\ldots, -2, -1, 0, 1, 2, \ldots\}$
the natural numbers together with their opposites
Rational numbers
$\mathbb{Q} = \left\{ \frac{p}{q} : p \in \mathbb{Z},\ q \in \mathbb{Z},\ q \neq 0 \right\}$
anything writable as a fraction of two integers
Chain of inclusions
$\mathbb{N} \subset \mathbb{Z} \subset \mathbb{Q} \subset \mathbb{R}$
each set contains the previous one whole
Irrationality
$\sqrt{2} \notin \mathbb{Q}$
no fraction whatsoever equals the square root of 2
Even and odd
$a = 2k \quad\text{or}\quad a = 2k + 1, \quad k \in \mathbb{Z}$
the notation every divisibility proof rests on

Absolute value

Definition
$|a| = \begin{cases} a & \text{for } a \ge 0 \\ -a & \text{for } a < 0 \end{cases}$
a non-negative number is left alone, a negative one has its sign flipped
Non-negativity
$|a| \ge 0$
a distance is never negative
Symmetry
$|-a| = |a|$
opposite numbers sit equally far from zero
Distance between two numbers
$|a - b| = |b - a|$
the order of subtraction does not change the distance
Products and quotients
$|a \cdot b| = |a| \cdot |b| \qquad \left|\frac{a}{b}\right| = \frac{|a|}{|b|}$
the modulus distributes over multiplication and division
Root of a square
$\sqrt{a^2} = |a|$
not a, because a square root is always non-negative
Triangle inequality
$|a + b| \le |a| + |b|$
the modulus of a sum never exceeds the sum of the moduli

Intervals

Closed interval
$[a, b] = \{ x \in \mathbb{R} : a \le x \le b \}$
both endpoints belong to the set
Open interval
$(a, b) = \{ x \in \mathbb{R} : a < x < b \}$
neither endpoint belongs to the set
Half-closed interval
$[a, b) = \{ x \in \mathbb{R} : a \le x < b \}$
the left end belongs, the right one does not
Unbounded interval
$(-\infty, b) = \{ x \in \mathbb{R} : x < b \}$
infinity always takes a round bracket
Intersection
$A \cap B = \{ x : x \in A \ \text{and} \ x \in B \}$
the numbers belonging to both intervals at once
Union
$A \cup B = \{ x : x \in A \ \text{or} \ x \in B \}$
the numbers belonging to at least one of the intervals
Modulus as an interval
$|x - a| < r \iff x \in (a - r, a + r)$
the numbers less than r away from a

Proportions and scale

Property of a proportion
$\frac{a}{b} = \frac{c}{d} \iff a \cdot d = b \cdot c$
the diagonal products are equal — hence cross-multiplication
The unknown in a proportion
$x = \frac{b \cdot c}{d}$
from x : b = c : d, after cross-multiplying
Direct proportionality
$y = k \cdot x, \quad k = \frac{y}{x}$
the ratio of the two quantities stays constant
Dividing in a ratio
$a : b \ \Rightarrow \ \text{one part} = \frac{S}{a + b}$
the whole splits into a + b equal parts
Reducing scale
$\text{scale } 1 : n \ \Rightarrow \ \text{real} = n \cdot \text{on the drawing}$
on a 1 : 50 000 map one centimetre is 500 metres

Rational exponents

Unit fraction exponent
$a^{\frac{1}{n}} = \sqrt[n]{a} \quad (a > 0)$
the n-th root written as a power
Rational exponent
$a^{\frac{m}{n}} = \sqrt[n]{a^m} = \left(\sqrt[n]{a}\right)^m \quad (a > 0)$
the denominator is the degree of the root, the numerator is the power
Negative rational exponent
$a^{-\frac{m}{n}} = \frac{1}{a^{\frac{m}{n}}} = \frac{1}{\sqrt[n]{a^m}}$
the minus inverts, the denominator takes the root
Laws of exponents
$a^{r} \cdot a^{s} = a^{r+s} \qquad \left(a^{r}\right)^{s} = a^{r \cdot s}$
unchanged for rational exponents
Monotonicity for a base above 1
$a > 1 \ \text{and} \ r < s \ \Rightarrow \ a^{r} < a^{s}$
a larger exponent gives a larger power
Monotonicity for a base between 0 and 1
$0 < a < 1 \ \text{and} \ r < s \ \Rightarrow \ a^{r} > a^{s}$
the direction reverses

Logarithms

Definition of a logarithm
$\log_a b = c \iff a^c = b \quad (a > 0,\ a \neq 1,\ b > 0)$
a logarithm is the exponent the base is raised to
Two logarithms straight from the definition
$\log_a 1 = 0 \qquad \log_a a = 1$
because a⁰ = 1 and a¹ = a
Logarithm of a product
$\log_a (x \cdot y) = \log_a x + \log_a y$
multiplication turns into addition
Logarithm of a quotient
$\log_a \frac{x}{y} = \log_a x - \log_a y$
division turns into subtraction
Logarithm of a power
$\log_a x^k = k \cdot \log_a x$
the exponent comes out in front
Inverse operations
$a^{\log_a x} = x \qquad \log_a a^k = k$
raising to a power and taking a logarithm cancel each other
Common and natural logarithm
$\log x = \log_{10} x \qquad \ln x = \log_{e} x$
two bases used so often that they have their own notation

Compound interest

Balance after n periods
$K_n = K_0 \left(1 + \frac{p}{100}\right)^{n}$
one period multiplier raised to the power n
Compounding m times a year
$K_n = K_0 \left(1 + \frac{p}{100 \cdot m}\right)^{m \cdot n}$
the rate is divided by m and the number of periods multiplied by m
Compound decline
$K_n = K_0 \left(1 - \frac{p}{100}\right)^{n}$
the same formula with a minus — depreciation, loss of value
Interest earned
$O = K_n - K_0$
what the bank added, that is the difference of the balances
Percentage points
$\text{p.p.} = p_2 - p_1$
the plain difference between two figures given as percentages

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

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