📚 Mathematics

Line types, geometric shapes, formulas, symbols, and the history behind the world of numbers.

Line types in mathematics

A line is rarely just a line. Its shape, thickness and pattern signal something specific in mathematical notation.

Horizontal

Used for fraction bars, the two strokes of the equals sign, and as the x-axis in a coordinate system.

Vertical

Used for absolute value bars |x|, as the y-axis, and in vertically stacked division.

Diagonal

Used as a slanted fraction bar (a/b), a division slash, and as the stroke through "=" to mean not equal (≠).

Thick (bold)

A bold stroke highlights vectors, matrices, and selected curves or edges in a graph.

Thin

Thin lines are used for construction lines and gridlines that aren't part of the figure itself.

Dotted

Used for projection lines down to an axis, and to signal that a pattern continues infinitely.

Dashed

Marks construction lines, hidden edges in 3D figures, and asymptotes a graph approaches but never touches.

Wavy

The tilde (~) means "approximately equal to" or "proportional to", and a wavy line over an axis marks a break in scale.

Zigzag

Marks that a section has been skipped on a number line or axis, e.g. when the gap to the next value is very large.

Spiral

Seen in the logarithmic spiral and the golden spiral, closely tied to Fibonacci numbers and the golden ratio φ.

Geometric shapes

From flat figures to solid shapes – with their key properties and formulas.

Triangle

3 sides and an angle sum of 180°. Area = ½ · base · height.

Square

4 equal sides and four right angles. Area = side².

Rectangle

4 sides with equal opposite pairs and four right angles. Area = length · width.

Circle

Every point on the edge is the same distance (radius) from the centre. Area = πr², circumference = 2πr.

Ellipse

An "elongated circle" with two foci and a major and minor semi-axis.

Parallelogram

4 sides where opposite sides are parallel and equal in length. Area = base · height.

Trapezoid

4 sides with only one pair of parallel sides. Area = ((a + b)/2) · height.

Rhombus

4 equal sides whose diagonals cross at a right angle.

Pentagon

5 sides and an angle sum of 540°.

Hexagon

6 sides and an angle sum of 720° – the shape bees use for honeycombs.

Sphere

Every point on the surface is the same distance from the centre. Volume = 4/3 · πr³.

Cube

6 square faces, 12 edges and 8 corners. Volume = side³.

Cylinder

Two parallel circular ends connected by a curved surface. Volume = πr² · height.

Cone

A circular base that tapers to a single point. Volume = ⅓ · πr² · height.

Pyramid

A polygonal base that tapers to a single apex, like the Egyptian pyramids. Dashed edges are the ones hidden from view.

Basic formulas

Area, volume, surface area and trigonometric ratios – gathered in three reference tables.

2D shapes and areas

Name	Shape	Area
Square		A = s²
Rectangle		A = b · h
Parallelogram		A = b · h
Trapezoid		A = ((a + b) · h) / 2
Circle		A = π · r²
Ellipse		A = π · a · b

3D shapes, volume & surface area

Name	Volume	Lateral surface area	Total surface area
Cube	V = s³	4s²	6s²
Cuboid	V = l · b · h	2h(l + b)	2(lb + bh + hl)
Cylinder	V = πr²h	2πrh	2πr(r + h)
Cone	V = ⅓ πr²h	πrl	πr(r + l)
Sphere	V = ⁴⁄₃ πr³	4πr²	4πr²
Hemisphere	V = ⅔ πr³	2πr²	3πr²

Basic trigonometry formulas

The six basic ratios in a right-angled triangle.

Ratio	Definition
sin θ	Opposite / Hypotenuse
cos θ	Adjacent / Hypotenuse
tan θ	Opposite / Adjacent
sec θ	Hypotenuse / Adjacent
cosec θ	Hypotenuse / Opposite
cot θ	Adjacent / Opposite

Fraction rules

The basic arithmetic rules for fractions.

Rule	Formula
Whole number + fraction	a + b/c = (a · c + b) / c
Addition, same denominator	a/b + d/b = (a + d) / b
Addition, different denominators	a/b + c/d = (a · d + b · c) / (b · d)
Multiplication	a/b · c/d = (a · c) / (b · d)
Division	a/b ÷ c/d = a/b · d/c

Algebra formulas

The distance formula and the most commonly used algebraic identities.

Name	Formula
Distance formula	d = √((x₂ − x₁)² + (y₂ − y₁)²)
Quadratic formula	x = (−b ± √(b² − 4ac)) / 2a
Square of a sum	(a + b)² = a² + 2ab + b²
Square of a difference	(a − b)² = a² − 2ab + b²
Difference of squares	a² − b² = (a + b)(a − b)
Cube of a sum	(a + b)³ = a³ + 3a²b + 3ab² + b³

Mathematical symbols

The most common symbols, grouped by topic.

Arithmetic

Plus

Addition – adds two numbers

−

Minus

Subtraction

Times

Multiplication

Division

Equals

Two expressions have the same value

≠

Not equal to

Two expressions have different values

Plus-minus

Two possible values, one positive and one negative

Percent

Per hundred – 1% = 1/100

Comparison

Less than

Greater than

≤

Less than or equal to

≥

Greater than or equal to

≈

Approximately equal to

∝

Proportional to

Set theory

∈

Element of

x ∈ A: x is an element of set A

∉

Not an element of

⊂

Proper subset of

⊆

Subset of

∪

Union

∩

Intersection

∅

Empty set

Algebra & calculus

√

Square root

Ratio of a circle's circumference to its diameter, ≈ 3.14159

∞

Infinity

∑

Summation

Sum of a series of terms

∫

Integral

Area under a curve, the "reverse" of differentiation

∂

Partial derivative

∇

Nabla

Gradient of a function

Delta

Change in a quantity, e.g. Δx

Geometry

Degree

∠

Angle

⊥

Perpendicular to

∥

Parallel to

≅

Congruent to

Same shape and size

∼

Similar to

Same shape, different size

Logic

∀

For all

∃

There exists

Not

∧

And

∨

⇒

Implies

⇔

If and only if

Statistics

x̄

Sample mean

The average of the values in a sample

Mu (population mean)

The average of the entire population

Sigma (standard deviation)

A measure of how spread out values are around the population mean

σ²

Variance

The standard deviation squared

Sample standard deviation

Spread measure calculated from a sample

Sample size

Number of observations in a sample

Population size

Number of individuals in the entire population

P(A)

Probability

The probability of event A

Correlation coefficient

A measure of the linear relationship between two variables, between -1 and 1

Median

The middle value in a sorted dataset

The history of mathematics

A crash course in how mathematics has evolved over roughly 5000 years.

c. 3000–500 BCE

Egypt and Babylon

The first number systems emerge for trade, taxation and land surveying. The Babylonians use a base-60 system still visible today in 60 minutes per hour and 360 degrees in a circle.

c. 600–300 BCE

Ancient Greece

Pythagoras proves his theorem about right triangles, and Euclid compiles the era's geometry in "Elements" – one of the most widely used textbooks ever. Archimedes calculates π and areas/volumes with great precision.

c. 500–1200

India and the Arab world

Indian mathematicians invent the number zero and the decimal place-value system we still use. Al-Khwarizmi in Baghdad writes the book that gives "algebra" its name (al-jabr), and "Arabic numerals" spread to Europe.

1200–1500

Middle Ages and Renaissance

Fibonacci introduces the Hindu-Arabic numeral system to Europe and describes the sequence that bears his name. Algebraic notation starts to resemble what we use today.

1600s

The Scientific Revolution

Newton and Leibniz independently develop calculus, describing change and motion – the foundation of modern physics and engineering.

1700s–1800s

Euler and Gauss

Leonhard Euler is history's most prolific mathematician, introducing the number e and much of modern notation. Carl Friedrich Gauss, the "Prince of Mathematics", contributes to nearly every branch of the field.

1800s

New foundations

Non-Euclidean geometry shows that Euclid's parallel postulate isn't the only possibility. Georg Cantor founds set theory and describes different sizes of infinity.

1900–present

The modern era

Kurt Gödel's incompleteness theorems show that every consistent mathematical system has limits. Alan Turing's concept of "computability" lays the groundwork for the computer, and mathematics becomes a tool in everything from cryptography to artificial intelligence.

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