This is the second half of a course in school mathematics you can teach yourself. The first half, Numbers and algebra, built the number line, fractions, powers, and the whole of algebra up to the quadratic formula. This guide spends all of it freely — ratios, square roots, negative numbers, the line y = mx + b, completing the square — so wherever a step here leans on something you would rather see built from scratch, that guide is where it is built. What follows is the part of mathematics that has a shape. We begin with the shapes themselves: angles, triangles, circles. We pin them to a grid, where algebra and geometry turn out to be one subject seen twice. On that grid a function draws its own behaviour as a curve, and three kinds of change each get their own chapters — growth that multiplies, waves that repeat, and the spread and chance of real data. The last chapter reaches the edge of calculus, the mathematics of change itself, and finds you already speaking its language. Read it in sittings, with paper and a pencil.
An angle is an amount of turn
The first half of this course was about number. This half is about shape, and something new arrives with it.
Until now, every rule could be checked by calculating. Here, mathematics starts to prove: to show that a fact must hold, everywhere and forever, by reasoning from simpler facts. Your first proof is one chapter away, and it is three lines long. This chapter lays the one paving stone it stands on.
Start with the thing being measured. An angle measures how far one direction has been turned from another. It has nothing to do with how long the arms of the corner are — a wide-open door and a wide-open gate stand at the same angle, though one is bigger than the other.
The unit is the degree, written with a small circle: a full turn is 360°. The number is a convention, but a well-chosen one. 360 divides evenly by 2, 3, 4, 5, 6, 8, 9, 10 and 12, so the common fractions of a turn all come out whole.
Half a turn is 180°, the two arms folding flat into a single straight line. A quarter turn is 90°, a right angle — the corner of this page.
From the one fact that a straight line is 180°, the everyday angle facts come free. Angles that sit side by side along a straight line must add to 180°: if one is 116°, its neighbour is 64°, because together they fill the straight line.
When two straight lines cross, the angles opposite each other are equal. Each of them teams up with the same neighbour to make 180°, so neither has any choice but to match the other.
One last ingredient, and it is the one the proof needs. Two lines are parallel when they point in exactly the same direction and never meet — the two rails of a train track. When a third line cuts across a pair of parallel lines, it meets both at the same tilt.
That equal tilt shows up as equal angles. The two angles tucked between the parallel lines on opposite sides of the crossing line — the two crooks of a Z shape — are equal. They are called alternate angles, and they are the whole of what the next chapter needs.
That is the entire toolkit: a straight line is 180°, and alternate angles are equal. Watch how far it reaches.
Why a triangle's three angles make a straight line
Here is a fact you may have been told to memorise: the three angles inside any triangle add up to 180°. Not most triangles — every triangle that could ever be drawn. Being told it is one thing. Seeing why it cannot be otherwise is the moment geometry becomes a different subject, so let us earn it.
Take any triangle at all. Label its corners A at the top, B and C along the bottom. Now add a single new line: through the top corner A, draw the line parallel to the base BC.
Two of the angle facts from the last chapter now do all the work.
The base BC and the new line are parallel, cut by the left side of the triangle. So angle 1, tucked at corner A, and the angle at B are alternate angles — the crooks of a Z — and therefore equal.
The same on the right: the base and the new line are cut by the right side of the triangle, so angle 2 and the angle at C are alternate angles, and equal.
Now look at corner A. Angles 1, A and 2 sit side by side along the new straight line, so they add to 180°. Replace angle 1 with the angle at B, and angle 2 with the angle at C — each equals its partner — and you have B + A + C = 180°.
Nothing in that argument ever asked which triangle you started with. It holds for all of them at once. This is Euclid's proof, more than two thousand years old, and notice what it quietly did: a fact about triangles turned out to be a fact about parallel lines.
Put to work, it hands you missing angles for free. A triangle with angles of 65° and 40° must have a third of 180 − 65 − 40 = 75°.
Theorems have fine print
The proof leaned entirely on parallel lines, and parallel lines behave that way only on a flat surface. On a curved one the theorem simply fails.
Draw a triangle on a globe: start at the North Pole, go straight down to the equator, turn and run a quarter of the way along it, then turn and go straight back up to the pole. Every one of those three turns is a right angle. The angles sum to 270°, not 180°.
So the rule is really "the angles of a triangle drawn on a flat surface sum to 180°." Every theorem carries conditions like this, and part of learning mathematics is learning to read the fine print.
When two triangles are the same
One more idea belongs in this chapter, because the circle chapter will need it. Two triangles are congruent if one could be picked up, turned or flipped, and set down exactly on the other — same shape, same size.
You never have to check all six measurements (three sides, three angles) to be sure. Any one of these three shorter checks settles it:
- three pairs of sides equal (called SSS, side-side-side);
- two sides and the angle between them equal (SAS);
- two angles and the side between them equal (ASA).
SAS shows the idea most plainly: hold two sticks joined at a fixed angle, and there is exactly one way to close off the triangle by joining their far ends. The shape is forced. The other two checks pin it down just as tightly.
What is deliberately not on the list: two sides and an angle that is not between them. That combination can leave two different triangles standing, and the gap it opens will come back to bite in the trigonometry chapters. Hold the thought.
One consequence is needed almost at once. A triangle with two equal sides is called isosceles, and its two base angles — the ones facing the equal sides — are equal. Flip the triangle across its line of symmetry and it lands on itself, so those two angles have to match.
Cut it apart, and the area survives
Before the most famous theorem in mathematics, one humble idea, because the proof runs on it: what area is, and the one move that computes almost all of it.
Area answers a single question: how many unit squares fit inside a shape? For a rectangle the answer is there to be counted. A rectangle 3 units by 5 holds three rows of five squares, 15 in all, so its area is length × width. Every other area formula in this chapter is that one fact, in a cleverer disguise.
The disguise is always the same trick: move a piece without changing the area. Cutting a shape apart and sliding the pieces around rearranges the area but never adds or removes any. Watch it turn a rectangle into three other shapes.
A parallelogram is a rectangle pushed over. Slice the overhanging triangle off one end and slide it across to the other, and it straightens back into a rectangle — same area, rearranged. So its area is base × height, where the height is the straight-across distance between the two parallel sides, not the length of the slanted edge.
A triangle is half a parallelogram. Take two copies of any triangle, turn one upside down, and the pair snaps together into a parallelogram. So the triangle is exactly half of it: area = ½ × base × height.
A trapezoid (called a trapezium in Britain) is a four-sided shape with one pair of parallel sides, of lengths a and b. Two copies, one rotated, again make a parallelogram — this time with base a + b — so the trapezoid is half of that: area = ½(a + b) × height. Parallel sides 6 and 10, height 4, gives ½ × 16 × 4 = 32.
Every one of these was the rectangle again, either moved into shape or doubled and halved. Moving pieces never changes the total, and that single sentence is about to prove something far less obvious.
Pythagoras, seen and not just stated
A right triangle contains a right angle. The side facing that angle — always the longest — is the hypotenuse; the other two are the legs.
The most famous theorem in mathematics is a fact about the areas of squares built on those three sides. Square each leg, and the two squares together have exactly the area of the square on the hypotenuse. Writing the legs as a and b and the hypotenuse as c: a² + b² = c².
Test it on the triangle with legs 3 and 4: the leg-squares are 9 and 16, adding to 25, and 25 is 5². So the hypotenuse is exactly 5 — no measuring, no approximation.
Why must this be true? Here is a proof you can see, and it needs nothing but the last chapter's one idea: moving pieces does not change area.
Take four identical copies of a right triangle with legs a, b and hypotenuse c. Take a large square whose side is a + b. There are two different ways to pack the four triangles into the corners of that large square, and comparing the leftover space is the whole proof.
On the left, the four triangles are turned so their hypotenuses face inward. They fence off a tilted square in the middle, and its side is the hypotenuse c, so the leftover area is c².
On the right, the same four triangles pair up into two rectangles pushed into opposite corners. Now the leftover space is two upright squares, one a by a and one b by b — areas a² and b².
Both large squares are the same size, and both contain the same four triangles. Subtract the four triangles from each, and the leftover areas must be equal: c² = a² + b². That is the entire proof, and it never once gave a and b particular values, so it holds for every right triangle at once.
In practice the theorem runs both ways across its equals sign. Given the two legs, it produces the hypotenuse. Given the hypotenuse and one leg, it produces the other.
legs 5 and 12: c² = 25 + 144 = 169 c = √169 = 13
hyp 10, leg 6: b² = 100 − 36 = 64 b = √64 = 8Same shape, only bigger
Congruent triangles were identical twins. This chapter is about the family resemblance between a shape and its photograph at a different size — same proportions, different scale. Two figures like that are called similar.
In similar figures, every length in one is the same fixed multiple of the matching length in the other. That multiple is the scale factor, written k, and matching angles stay exactly equal — enlarging a photo does not bend its corners.
For triangles there is a wonderfully cheap test. If two angles of one triangle equal two angles of another, the triangles are similar. This is called AA, for angle-angle. The third pair of angles costs nothing to check, because the 180° theorem forces it to agree once the first two do.
Similarity is how you measure things you cannot climb. A 1.2-metre stick stands in the sun and casts a shadow 1.8 metres long. Nearby, a tree casts a shadow 21 metres long. How tall is the tree?
The sun's rays arrive as good as parallel, so the stick with its shadow and the tree with its shadow form two right triangles with equal angles — similar, by AA. Matching sides therefore sit in the same ratio, and the tree's height falls out:
height ÷ shadow is the same for both: 1.2 ÷ 1.8 = 2/3
tree height = 21 × 2/3 = 14 mWhat scaling does to area and volume
Here is where scaling holds a genuine surprise, and it is worth slowing down for, because almost everyone gets it wrong the first time.
Scaling by k multiplies every length by k. But area is a length times a length, so it picks the factor up twice: areas are multiplied by k². And volume is length times length times length, so it picks it up three times: volumes are multiplied by k³.
Double a shape's sides, and its area becomes four times as big, its volume eight times. Triple the sides, and area is nine times, volume twenty-seven times.
Double the width, double the pizza?
No — and a menu is where it costs you. A 40-centimetre pizza against a 20-centimetre one: the width only doubles, but the area is what you actually eat, and area scales by k², not by k.
Twice the width is four times the pizza. So if the large costs less than four small ones, it is the better value — and it usually is.
The seed of trigonometry
One last consequence of AA, planted here on purpose, to be harvested four chapters on.
Pick any acute angle — any angle under 90°, say 35°. Now draw any right triangle that contains it. Every such triangle is similar to every other one: they all share the right angle and the 35°, and AA finishes the job.
Because they are all similar, the ratio of any two chosen sides — say, the side opposite the 35° divided by the hypotenuse — comes out as the very same number in every one of them, whatever the triangle's size.
That number belongs to the angle itself, not to any particular triangle — a property of "35°" alone. Later, it will get a name.
Round things, flat and solid
Straight-edged shapes gave up their areas to cut-and-slide. A circle has no straight edges to slide, so it needs one new number and one new idea.
Every circle is the same shape as every other — scale one up and you get another — so, exactly as with similar triangles, any matching ratio is shared across all of them.
The ratio that matters is the distance around a circle (its circumference) divided by the distance straight across (its diameter). That ratio is a single fixed number, the same for a coin and for a planet's orbit. It is called π (the Greek letter pi), and π = 3.14159…, an endless non-repeating decimal — one of the irrational numbers from the first guide. Writing C for circumference, d for diameter and r for radius (half the diameter), this says C = πd = 2πr.
The circle's area takes the new idea. Cut the disc into many thin wedges, like a sliced pie, and lay them in a row with their points alternating up and down. They interlock into a shape close to a rectangle. Its height is the radius r; its length is half the circumference (the other half of the pointed edges faces the other way).
So the area is close to half the circumference times the radius, ½ × C × r, and the thinner you cut the wedges, the closer it gets. Hold on to that phrase — the thinner, the closer — because it is the idea the whole last third of this guide runs on, and the final chapter comes back to collect it.
Turn that into a formula and it is the one you know:
A = ½ · C · r = ½ · (2πr) · r = πr²
r = 3: C = 2π·3 ≈ 18.85 A = π·3² ≈ 28.27 = ½ × 18.85 × 3 ✓A slice of a circle is priced by its share of the full turn. A 60° slice is 60/360 = 1/6 of the whole, so its curved edge (an arc) is 1/6 of the circumference, and its area (a sector) is 1/6 of πr². For a circle of radius 6: the arc is 2π × 6 ÷ 6 ≈ 6.28, the sector is π × 36 ÷ 6 ≈ 18.85.
A right angle, free of charge
Draw any diameter across a circle. Pick any point on the circle and join it to the two ends of that diameter. The angle at your chosen point is always exactly 90°.
Here is why. Join your point to the centre as well. Every line from the centre to the circle is a radius, so the two triangles you have just made are both isosceles. Their base angles come in equal pairs — call them a, a and b, b. The big triangle's three angles are then a, b, and a + b at your point. Setting the sum to 180° gives a + b + (a + b) = 180°, so a + b = 90°.
This is Thales' theorem, and the isosceles fact from two chapters ago has just earned its keep. Keep it — the unit circle in the trigonometry chapters relies on it.
Into the third dimension
Volume counts unit cubes instead of unit squares. A prism is any flat shape pushed straight upward into a solid — a cylinder is the circular case. It is a stack of identical layers, so its volume is the base area times the height. A cylinder of radius 3 and height 5 holds π × 9 × 5 ≈ 141.4.
Two more solids come with formulas that this guide will state but cannot yet prove. A cone standing over the same circle at the same height holds exactly one third of the cylinder: πr²h ÷ 3 ≈ 47.1 for that base. A sphere of radius r holds 4/3 × πr³, about 33.5 when r = 2.
Where do that ⅓ and that 4/3 come from? Slice the solid into many thin disks and add up their volumes. Do it for the cone with two million slices and the total comes to 47.1238898…, matching πr²h ÷ 3 to eight decimal places.
But proving that the slices land on exactly ⅓, and not merely close to it, needs "the thinner, the closer" taken with full seriousness. That is calculus, and the last chapter of this guide pays the debt in full.
Every point has an address
So far, shape and number have been two subjects. This chapter fuses them into one, with an idea so simple it is easy to miss how large it is.
Take two number lines. Lay one flat, running left to right, and stand the other upright, running down to up, and cross them at their two zeros. Now every point on the page has an address: a pair of numbers (x, y), where x is how many steps across and y how many steps up. Negative numbers, from the first guide, take care of left and down. The crossing point, where both are zero, is the origin, (0, 0).
This grid is one of the great inventions in mathematics, because of what it does to equations.
An equation linking x and y sorts every point on the page into two camps: the pairs that make it true, and the pairs that do not. Collect all the pairs that make it true, plot them, and they form a shape. Algebra has become geometry — and, read the other way, every geometric question has become a calculation.
Lines, and slope
Try it on the equation y = ½x + 1. Choose any x, work out y, and plot the pair: (0, 1), then (2, 2), (4, 3), (6, 4). The points fall on a perfectly straight line.
Every solution of the equation lies on that line, and every point of the line is a solution. The line is the equation's set of solutions, drawn.
The number that controls a line is its slope: the rise over the run, how much y climbs for each step across in x.
From (2, 2) to (6, 4) the line rises 2 and runs 4, a slope of 2/4 = ½. Take any other two points — (2, 2) to (4, 3), a rise of 1 over a run of 2 — and the slope is ½ again. A straight line has one slope everywhere; that is what makes it straight.
In the form y = mx + b, the number m is the slope, and b is the height at which the line crosses the upright axis. A negative slope falls instead of climbing, and a slope of 0 is dead level.
Distance and midpoint
How far apart are two points? Take (1, 2) and (4, 6). Walk across, then up: 3 across and 4 up. Those two legs frame a right triangle, and the straight-line distance you want is its hypotenuse. Pythagoras, carried straight into the plane.
d = √( (x₂ − x₁)² + (y₂ − y₁)² )
(1, 2) to (4, 6): d = √(3² + 4²) = √25 = 5The halfway point takes even less work: average the two x-values, and average the two y-values. The midpoint of (2, 3) and (8, 7) is ((2 + 8)/2, (3 + 7)/2) = (5, 5).
The circle, by its address
A circle is defined as every point at a fixed distance r from a centre. Put the centre at the origin, and the distance formula turns that sentence straight into algebra: a point (x, y) is on the circle exactly when √(x² + y²) = r, that is, x² + y² = r².
A circle is therefore not the graph of a function-style rule; it is the set of points that pass a distance test. Move the centre to (a, b) and the test becomes (x − a)² + (y − b)² = r².
In the wild, a circle rarely arrives already tidy. What is x² + y² − 4x + 6y − 12 = 0? Completing the square — the technique from the quadratics chapter — reassembles it, once for the x terms and once for the y:
x² − 4x + y² + 6y = 12
(x − 2)² − 4 + (y + 3)² − 9 = 12
(x − 2)² + (y + 3)² = 25
a circle: centre (2, −3), radius √25 = 5From here on, "draw the graph" is a sentence that makes sense. Everything left in this guide happens on this grid.
A function is a machine with one output
Algebra, so far, has asked for particular numbers — the value, or the pair of values, that makes a statement true. The next idea is larger than any single answer.
Look around and one quantity constantly depends on another. The postage depends on the weight. The stopping distance depends on the speed. The temperature depends on the time of day. Mathematics wants to study the dependence itself, as a thing you can hold and reason about.
A function is a rule that assigns to each input exactly one output. That is the whole definition, and both halves matter.
Each input: the rule has to give an answer for everything it is allowed to be fed. Exactly one: it may never reply "well, either 4 or 7." Two different inputs are perfectly welcome to share an output — but one input may never have two.
Think of it as a machine. A number goes in; the machine does its fixed thing; one number comes out. Feed the same input again and the same output must appear, every time.
Notice what the definition never mentions: formulas. A table is a function — a delivery company's price list, weight in and price out, is a perfectly good rule even though no formula generates it. A description is a function — "round the parking time up to the next hour" assigns each duration exactly one fee.
So if you meet a rule given by a table or by cases and catch yourself thinking but that's not a real function, where's the formula?, the definition is on the rule's side. Formulas are simply the best-behaved examples, not the whole species.
Now the notation, which trips people up once and then never again. A function gets a name — f — and f(x) means "the output of f when the input is x." It is one object, the answer; it is not f times x. For the rule f(x) = x² − 4, we get f(0) = −4, f(3) = 5, and f(−3) = 5 as well.
Two different inputs, 3 and −3, sharing the output 5 — exactly what the definition allows. The set of inputs the rule accepts is its domain; the set of outputs it actually produces is its range.
The graph
Plot every pair (input, output) as a point on the coordinate plane, and the whole function appears at once as a picture: its graph. For f(x) = x², a short table of inputs and outputs
x: −3 −2 −1 0 1 2 3
f(x): 9 4 1 0 1 4 9sweeps out a symmetric curve, the parabola: falling, turning at (0, 0), then climbing — the shape every quadratic makes.
The rule "each input exactly one output" leaves a visible fingerprint on a graph: no vertical line may cross it more than once. A vertical line asks "at this one input, what outputs appear?" — and the answer must never be "two." The circle from the last chapter fails this test; it is a fine shape, but not the graph of a function.
Reading a graph is a skill of its own. Where the curve climbs, the function is increasing; where it falls, decreasing; a peak is a high point of the quantity, a trough a low point. The picture tells the quantity's whole story at a glance.
The graph is not a picture of the scene
Suppose a graph shows your distance from home against time, and the curve climbs, runs flat, then climbs again. The flat stretch does not mean you walked along level ground. It means your distance stopped changing — you stood still. A steeper stretch means you moved faster, not uphill.
Most people read a distance–time graph as a picture of the route. It isn't. A graph is a picture of a relationship between two quantities, not a picture of the road.
To hammer home that a function need not be a formula, here is one that cannot be: postage that costs £1 up to 1 kg, £2 up to 2 kg, £3 up to 3 kg. Its graph is a flight of steps. Each step is closed on the left (a 1 kg parcel still costs £1) and open on the right, and the whole thing is one perfectly legal function — each weight has exactly one price.
Sliding, stretching, and running a graph backwards
You already know two families of functions. Linear functions f(x) = mx + b graph as straight lines — equal steps in give equal steps out. Quadratics graph as parabolas, and completing the square (from the algebra guide) locates the turning point. The next two chapters will add two genuinely new families; this short chapter is about what you can do to any family once you have it.
Whole families move together, and the moves are worth knowing because they save you re-plotting from scratch. Compared with the graph of f(x):
- f(x) + k is the same graph shifted up by k (every output is k larger);
- f(x + k) is the same graph shifted left by k — left, not right, because the +k gives x a head start, so the curve reaches each height sooner;
- k · f(x) is the same graph stretched to k times the height.
Check the leftward shift on the parabola, since it is the one that feels backwards: y = (x + 3)² reaches its lowest point where x + 3 = 0, that is at x = −3, three steps to the left of the original bottom at 0. The +3 moved it left.
Running the machine backwards
If the machine f turns 5 into 13, what turns 13 back into 5? That reverse machine is the inverse function, written f⁻¹.
Build it by undoing the steps in reverse order. Take f(x) = 2x + 3, which doubles and then adds 3. To undo it, first subtract 3, then halve: f⁻¹(13) = (13 − 3)/2 = 5. Doing and undoing land you back where you started.
On a graph, an inverse has a beautiful signature: it is the original reflected across the diagonal line y = x. That line is the mirror that swaps the two axes, and swapping input with output is exactly what an inverse does.
There is one catch, and it will matter enormously in the very next chapter. An inverse exists only if no two inputs ever shared an output — otherwise the return journey faces two destinations and cannot choose. The parabola fails this: since f(3) = f(−3) = 5, the question "which input gave 5?" has two answers, so x² has no single inverse.
Keep that condition warm. The next chapter builds a family of functions that always climbs and so never reuses an output — which means it does have an inverse, and that inverse turns out to be one of the most useful tools in all of mathematics.
Growth by multiplying, not adding
Two kinds of growth run the world, and telling them apart is the whole of this chapter. One adds a fixed amount each step. The other multiplies by a fixed amount each step. They start out looking similar and end up in different universes.
Start small, with a list of numbers that follows a rule — a sequence. Two rules matter more than all the rest.
An arithmetic sequence adds the same amount every step: 3, 7, 11, 15, 19, 23, … adds 4 each time. Plot the terms against their positions and they land on a straight line. Adding equal amounts is exactly the "equal steps in, equal steps out" of a linear function — arithmetic growth is linear growth, watched one step at a time.
A geometric sequence multiplies by the same factor every step: 5, 10, 20, 40, 80, 160, … doubles each time. This is a different animal entirely, and its smooth version is this chapter's subject.
That smooth version is the exponential function, written bˣ — a fixed base b raised to a varying power x. Where a linear function turns equal steps of x into equal differences of output, the exponential turns equal steps of x into equal factors of output. Step x up by one, and the output does not gain a fixed amount; it multiplies by b.
The gap this opens is violent. Set the exponential 2ⁿ beside the quadratic n², one of the fastest growers from the algebra guide. At n = 10 they stand at 1024 against 100. At n = 20, at 1,048,576 against 400. Multiplying does to numbers what adding never can.
Compound growth
The multiplier idea from the percentages chapter is what makes exponential growth calculable. Money earning 3% a year multiplies by 1.03 each year, so after ten years it has been multiplied by 1.03 ten times over:
1000 × 1.03¹⁰ = 1343.92… (not 1300 — each year's 3% is charged
on the previous year's larger total)
falling 15% four times = × 0.85⁴ = × 0.522 (a 48% drop, not 60%)A handy summary of any growth rate is its doubling time. At 5% a year, 1.05 multiplied 14 times gives about 1.98, and 15 times about 2.08 — so money doubles in a little over 14 years.
The banker's shortcut is the rule of 72: divide 72 by the percentage rate. Here 72 ÷ 5 = 14.4 years. It is an approximation, not a law — the true figure is 14.2 — but a serviceable one at everyday rates, and worth keeping the two numbers side by side so you remember which is which.
Decay is the same machinery with a multiplier below 1. A quantity falling 15% each step multiplies by 0.85 each step, and it never quite reaches zero — it just keeps taking 15% of an ever-smaller amount.
One curiosity to file away, because it will reappear everywhere. Compound 100% growth all at once and you get ×2. Compound it in two half-year instalments of 50%: ×1.5² = 2.25. Monthly: ×2.613. Daily: ×2.7146. Compounding ever more often does not run away to infinity — it closes in on a ceiling, 2.71828…, a constant important enough to have its own letter, e. Yet another process cornering a value it never quite reaches; the guide keeps meeting this idea, and the last chapter names it for what it is.
The logarithm is the exponent
Now run the exponential machine backwards, and meet the inverse promised two chapters ago.
The equation 2ˣ = 10 asks a plain question: how many doublings does it take to reach 10? An answer must exist. Since 2³ = 8 is a touch too small and 2⁴ = 16 too big, and 2ˣ climbs smoothly through everything between, some x between 3 and 4 lands exactly on 10.
No ordinary algebra pries that x out of the exponent. It takes a new function — the inverse of 2ˣ. And that inverse is guaranteed to exist, because 2ˣ always climbs and so never sends two inputs to one output: exactly the condition the functions chapter set for an inverse to be possible.
That inverse is the logarithm. The expression log₂ c means: the power you must raise 2 to in order to get c. One sentence is worth engraving above the door — the log is the exponent.
So a logarithm simply counts multiplications. Start at 1 and count the doublings:
1 → 2 → 4 → 8 three doublings, so log₂ 8 = 3
1 → 10 → 100 → 1000 three tenfoldings, so log₁₀ 1000 = 3
and the question we started with: x = log₂ 10 = 3.3219…
check: 2 ^ 3.3219 ≈ 10.00Because logs are exponents, every law of exponents from the powers chapter comes back wearing logarithm clothing. Multiplying two powers adds their exponents — so the log of a product is the sum of the logs:
log(a · b) = log a + log b log₁₀ 8 + log₁₀ 125
log(a / b) = log a − log b = 0.903… + 2.097… = 3
log(aⁿ) = n · log a = log₁₀ 1000 ✓That first rule is why logarithms were invented, back in 1614: they turn multiplication into addition. Before calculators, that turned days of multiplying huge numbers into hours of adding them — a gift that, in one astronomer's phrase, "doubled the life of the astronomer."
Reading across scales
Logarithms also cure a picture problem. Try to plot 200, 2,000 and 2,000,000 on one ordinary axis: the first two are crushed into the same speck against the third.
A log scale plots each number's exponent instead of the number itself. Now every step along the axis means "times ten," not "plus ten," and numbers spanning a millionfold range sit comfortably on one line.
Earthquake magnitude is such a scale: one step up means shaking ten times as large, which is how a single axis holds both a tremor and a catastrophe. Loudness in decibels and acidity in pH are the same trick.
Both new functions now earn their keep across the rest of the guide — the exponential as the model of anything that grows or decays by a factor, the logarithm as the tool that answers "how long until…?" Ask when money at 3% doubles, and you are solving 1.03ⁿ = 2, an exponent to extract: take the log of both sides, let the power rule pull the n down in front, and n = log 2 ÷ log 1.03 ≈ 23.4 years. The rule of 72 guessed 24; now you can be exact.
Sine is a ratio the angle owns
Time to harvest the seed planted back in the similarity chapter. It grows into trigonometry — the mathematics of triangles, and then of anything that turns or repeats.
Fix an acute angle, say 35°, and look at every right triangle that contains it. They are all similar to one another: each shares the right angle and the 35°, and AA does the rest. Because they are all similar, the ratio between any chosen pair of sides is the same number in every one of them, whatever the triangle's size.
So the ratio belongs to the angle, not to any particular triangle — a fact about "35°" the way π is a fact about "circle." Trigonometry begins by giving three of these ratios names.
Stand at the acute angle θ (the Greek letter theta, the usual name for an unknown angle) inside a right triangle. The side facing you across the triangle is the opposite; the side along your arm, the one that is not the hypotenuse, is the adjacent; and the hypotenuse is the hypotenuse. The three named ratios are:
sin θ = opposite ÷ hypotenuse (soh)
cos θ = adjacent ÷ hypotenuse (cah)
tan θ = opposite ÷ adjacent (toa)Try them on the 3-4-5 triangle, standing at its smallest angle (the one facing the side of length 3): sin θ = 3/5 = 0.6, cos θ = 4/5 = 0.8, tan θ = 3/4 = 0.75. And because these ratios belong to the angle, you have just learned something about an angle you never measured — the angle whose sine is 0.6 is about 36.9°, in every right triangle that will ever contain it.
Sine is not a button
It is easy to come away thinking sin is a magic key on a calculator that spits out unrelated decimals. It is nothing of the sort. It is a ratio: sin 30° = ½ means the side opposite a 30° angle is always half the hypotenuse — a fact about every right triangle with a 30° corner, exactly as π is a fact about every circle. The calculator only spares you drawing the triangle and measuring.
The angles you can do by hand
Most angles give ugly, endless decimals for their ratios, and a calculator is the honest tool. But a few special angles give exact values, drawn straight from two triangles you can build by hand — and knowing them cold is what lets you check a calculator's work and see the structure underneath.
Cut a 1-by-1 square along its diagonal. You get a right triangle with two legs of length 1 and, by Pythagoras, a hypotenuse of √(1 + 1) = √2. Its two acute angles are both 45°. So sin 45° = 1/√2 = √2/2, exactly.
Now cut an equilateral triangle of side 2 straight down the middle. You get a right triangle with hypotenuse 2, a short leg of 1, and — by Pythagoras — a long leg of √(4 − 1) = √3. Its angles are 30° and 60°. So sin 30° = 1/2 exactly, and cos 30° = √3/2.
Add the easy ends, 0° and 90°, and you have the table worth carrying in your head:
θ: 0° 30° 45° 60° 90°
sin θ: 0 1/2 √2/2 √3/2 1
cos θ: 1 √3/2 √2/2 1/2 0
tan θ: 0 1/√3 1 √3 — (90°: no adjacent to divide by)Read the symmetry down the columns: sin 30° = cos 60°, and sin 60° = cos 30°. The two acute angles of a right triangle add to 90°, and one angle's opposite side is the other's adjacent — so the sine of an angle always equals the cosine of its complement. That is all "cosine" ever meant: complementi sinus, the sine of the complement.
Solving triangles
Here is the practical power: measure what you can reach, and compute what you cannot.
Stand 30 metres from a tree and sight its top at 35° above the horizontal — the angle of elevation. The height of the tree and your distance from it are the opposite and adjacent sides of a right triangle, so the ratio that links them is the tangent:
tan 35° = height ÷ 30
height = 30 × tan 35° = 30 × 0.7002… ≈ 21.0 mLooking downward instead — from a clifftop to a boat — the angle below the horizontal is the angle of depression, and the working is identical.
The reverse question comes up too: the sides are known and you want the angle. A ramp that climbs 3 m over a run of 4 m has tan θ = 3/4, and the angle with that tangent is written tan⁻¹(3/4) — the inverse-function notation from the functions chapter — which is 36.87°. The 3-4-5 triangle again, naturally.
One wall still stands. Everything here lives inside a right triangle, so θ has to stay between 0° and 90°. The next chapter knocks that wall down.
Off the triangle, onto the circle
The triangle definitions stop dead at 90°. A right triangle cannot hold an obtuse angle — one past 90° — so "sin 150°" is, so far, meaningless. Fixing that takes one clean idea, and it turns trigonometry from a fact about triangles into a fact about turning.
Draw a circle of radius 1, centred at the origin of the coordinate plane — the unit circle. Let an angle θ be a rotation: start pointing along the positive x-axis, and turn counterclockwise by θ.
For an acute θ, drop straight down from the point where your turned radius meets the circle. You have made a right triangle with hypotenuse 1 — and its adjacent leg is the point's x-coordinate, its opposite leg the point's y-coordinate. So that point on the circle is exactly (cos θ, sin θ).
Now keep turning past 90°. The little triangle falls apart — but the point is still there, still carrying two coordinates. So let the coordinates take over as the definition:
For any angle whatever, cos θ and sin θ are the x- and y-coordinates of the point at angle θ on the unit circle.
For acute angles this agrees perfectly with the triangle version; everywhere else, it extends it. And the signs now come from geometry, not decree. At 150° the point is up and to the left, so sin is positive and cos is negative. Straight left at 180° gives cos 180° = −1; straight down at 270° gives sin 270° = −1. Values no triangle could ever produce, read off the circle at a glance.
One identity falls straight out. The point (cos θ, sin θ) sits on the unit circle, whose equation from the coordinates chapter is x² + y² = 1. Substitute the coordinates and you have sin²θ + cos²θ = 1, true for every angle (sin²θ is the usual shorthand for (sin θ)²). It is not a new fact to memorise — it is Pythagoras, wearing circle clothing.
The radian
With angles now living on a circle, a more natural unit than the degree suggests itself. Measure an angle by the length of arc it cuts on the unit circle.
A full turn cuts the entire circumference, 2π, so 360° = 2π radians. Half a turn, 180°, is π radians; a right angle, 90°, is π/2. One radian — the angle whose arc is exactly one radius long — is about 57.3°.
Why bother with a second unit? Because it makes the formulas shed their clutter. On a circle of radius r, the arc cut by an angle θ is simply s = rθ — no 360, no π correction to remember. And more deeply: for small angles, sin x and x are very nearly the same number when x is measured in radians (sin 0.001 ÷ 0.001 = 0.9999998…). In degrees that ratio is a lopsided 0.01745, a factor that would rust onto every formula in the next subject. Do not worry if that last step went quickly — the one thing to carry away is that radians are the unit in which sine comes out clean, and calculus will insist on them.
Beyond the right angle
The unit circle freed the angle from the right triangle. That freedom pays off in two directions at once — first as a wave, then as a way to solve any triangle.
Sine becomes a wave
Let θ run freely: around the circle, around again, backwards for negative angles. Track the height of the moving point — that is, sin θ — as it goes. Since the point keeps circling, the height keeps repeating: it climbs to 1, sinks to −1, and returns, over and over. Sine is periodic, with period 360° = 2π.
Cosine, the x-coordinate, traces the identical wave a quarter-turn ahead. These curves are the mathematics of anything that cycles — tides, seasons, sound, alternating current — and the family is genuinely new: no line, parabola, exponential or logarithm ever repeats itself, and this one grew straight out of a circle.
The graph transformations from the functions chapter now have a job. Multiplying sine by a number stretches its height — its amplitude; the number in front of θ squeezes or stretches how fast it repeats — its period. The single machine builds every wave.
Triangles without a right angle
Trigonometry now repays its debt to geometry by cracking triangles that have no right angle at all. In a triangle with sides a, b, c and the angle C sitting between a and b, the law of cosines holds:
c² = a² + b² − 2ab·cos C
a = 5, b = 7, C = 60°: c² = 25 + 49 − 2·5·7·(½) = 39 c = √39 ≈ 6.24Set C = 90° and something lovely happens: cos 90° = 0, the whole last term vanishes, and the law collapses into c² = a² + b² — Pythagoras. So the extra term is exactly the correction for how far the angle bends away from square, and the unit circle is what lets C run past 90° (where cos C goes negative and the far side grows).
Its partner, the law of sines, says a ÷ sin A = b ÷ sin B = c ÷ sin C — the three side-to-sine ratios are all equal. One inherited caution: the gap the congruence tests left open (two sides and a non-included angle) resurfaces here, where the same data can fit two different triangles, so the law of sines will sometimes hand you two valid answers and you must check both. Between them, the two laws let a few known parts of any triangle surrender all the rest.
Describing a pile of numbers
The register changes here. Every question so far had one exact answer. Data — many measurements at once — asks a softer kind of question: what is typical, how much do the values vary, and what goes with what?
The answers are summaries, and every summary earns its keep by throwing information away. The whole skill is knowing what each one discards.
Take the small dataset 2, 4, 4, 4, 5, 5, 7, 9. There are three honest answers to "what is typical?", and they are different numbers.
- The mean shares the total out equally: 40 ÷ 8 = 5.
- The median is the middle of the sorted list; here the middle pair is 4 and 5, so it is 4.5.
- The mode is the most frequent value: 4.
Three legitimate answers, because "typical" is not one idea. When the data are roughly symmetric, the three huddle together and it hardly matters which you quote. When they are not, the choice matters enormously — and this is where averages get used to mislead.
Eight people sit in a room, earning, in thousands: 31, 34, 36, 39, 42, 45, 48, and 250. The mean is 65.6k — a figure larger than what seven of the eight actually earn. The median is 40.5k, and it would not budge if the top earner made 2,500k instead.
The mean is a fair-share calculation, and a single outlier can drag it anywhere. The median only asks who stands in the middle, and shrugs the outlier off. Neither is wrong — but they are very different sentences about the same room, so whenever you hear an "average," ask which one, and what shape the data had.
How spread out?
Two datasets can share a mean and be nothing alike: a room of near-identical salaries, and a room of unpaid interns plus one founder. The average hides the difference; a measure of spread reveals it.
The standard one is the standard deviation (sd): roughly, the typical distance of a value from the mean. The recipe is four steps: find each value's distance from the mean, square them all (which makes them positive and punishes big misses hardest), average the squares, then take the square root to get back to the original units.
distances from mean 5: −3 −1 −1 −1 0 0 2 4
squared: 9 1 1 1 0 0 4 16 sum = 32
average of squares: 32 ÷ 8 = 4
sd: √4 = 2So values here sit typically about 2 away from the mean of 5. A shape is often clearest as a histogram — bars counting how many values fall in each band — with a boxplot beneath it marking the median, the middle-half box, and the range:
(Dividing by n is the convention when you are describing data you actually hold. When you are using a sample to estimate a larger population's spread, statisticians divide by n − 1 instead, for reasons that belong to a statistics course; do not let the fork eat the idea.)
What data doesn't tell you
Two of the most confident errors in all of quantitative reasoning live in this chapter. Both come from reading more into data than the data contains.
What goes with what
Plot two measurements per person — height and shoe size, say — as points on the coordinate plane, and you get a scatterplot. If the cloud of points leans upward, the two variables are positively correlated; if it leans downward, negatively; if it is a shapeless blob, they are uncorrelated. The straight line of best fit through the cloud — the line sitting as close as it can to every point — makes the lean precise.
Correlation measures leaning together. It does not measure causing, and the difference is where fortunes and reputations are lost. Ice-cream sales and drownings rise and fall together across the year — not because ice cream drowns anyone, but because summer drives both. A third thing lurking behind two others is the usual reason correlation appears with no causation, and "correlation is not causation" survives as a slogan precisely because the error it names refuses to die.
The bell-shaped model
Many datasets pile up symmetrically — crowded near the mean, thinning fast in both directions: repeated measurements of one thing, or adult heights within one sex. Mathematics keeps an idealised curve for that shape, the normal curve, pinned down entirely by two numbers: its mean, where the peak sits, and its standard deviation, how far it spreads. For the ideal curve the accounting is exact, and worth memorising in rounded form as the 68–95–99.7 rule:
But the rule is a property of that one ideal curve, not a law that data must obey. A real dataset either resembles the curve or it does not, and checking is part of the method — a caution the school standards themselves spell out.
Not everything is a bell
Incomes are not normal. The salary room from the last chapter has mean 65.6k and sd about 70k — so "mean minus one sd" is negative, and the 68% rule would cheerfully cover salaries below zero.
Skewed data, waiting times, city sizes, wealth: no bell. Even the classic classroom claim that human heights are bell-shaped needs care — the mix of two sexes is not a clean single bell. Reaching for mean ± sd without first looking at the shape is how confident nonsense gets computed.
Chance: counting the ways
Probability puts a number on uncertainty. The scale runs from 0, meaning cannot happen, to 1, meaning certain, with ½ meaning as likely as not. A probability can be written as a fraction, a decimal or a percentage, and the fraction arithmetic from the first guide is about to earn its keep.
The habit that makes the whole subject work is simple: list what can happen before asking how likely anything is. The full list of possible outcomes is the sample space, and when its outcomes are equally likely, probability is just counting — favourable outcomes over total outcomes.
Roll two dice. The sample space is not "the totals 2 through 12"; those are not equally likely. It is the 36 equally likely pairs:
A sum of 7 is six times as likely as a sum of 12, because six pairs make a 7 and only one pair makes a 12. The tempting error is to call outcomes equally likely because they are random — "7 and 12 are both just totals, so surely fifty-fifty." Randomness does not equalise anything; the grid settles it, so always build the grid first.
Two rules then come almost free (writing P(E) for the probability of an event E). If two events cannot both happen at once, their probabilities add: P(sum 7 or sum 11) = 6/36 + 2/36 = 8/36 = 2/9. And an event and its opposite split 1 between them, so P(not 7) = 1 − 1/6 = 5/6 — the complement rule, which shines on "at least one" questions: P(at least one six in four rolls) = 1 − (5/6)⁴ ≈ 0.518, one subtraction in place of a mountain of cases.
When two events are independent — neither one's outcome tells you anything about the other's — their probabilities multiply: P(a six, then another six) = 1/6 × 1/6 = 1/36.
The coin has no memory
Five heads in a row, so tails is surely "due"? No. The sixth toss is a fresh coin, and P(heads after five heads) is exactly ½.
Chance does not correct itself; it swamps. Those five surplus heads are never cancelled out — they are simply drowned by the thousands of tosses that follow (5 extra heads in 1,005 tosses is a proportion of 0.502, near enough a half). Expecting short runs to balance out is the gambler's fallacy.
While the multiplication rule is fresh, one more warning. Adding a condition can only shrink a probability, because "A and B" is a subset of "A alone" — so P(A and B) ≤ P(A), always. Yet describe a woman as thoughtful, outspoken, and active in social causes, and ask which is more likely: that she is a bank teller, or a bank teller and a feminist. Most people pick the second — even though it is a subset of the first, and so cannot be more likely. Detail makes a story feel more plausible while making the event less probable; when a fuller claim feels likelier than one of its own parts, the feeling is simply wrong.
When one fact changes another
New information reshapes the sample space, and the arithmetic of that reshaping is both the most useful and the most misused idea in the subject.
The probability of "the two dice sum to at least 10" is 6/36 = 1/6. But suppose you are told the first die shows a 6. You now live in a smaller world — the six pairs (6,1) through (6,6) — of which three reach 10. The conditional probability is 3/6 = ½. In symbols — with the bar read as "given" — P(A | B) = P(A and B) ÷ P(B), which is just the shrunken sample space written as a formula.
The costliest mistake in the whole subject is to swap a conditional for its reverse. P(positive test, given sick) is not the same as P(sick, given positive test) — and confusing the two has sent people to surgery. Counting whole people, rather than juggling percentages, is what keeps the two apart.
A disease affects 1% of a population. A test catches 99% of the people who have it, but also falsely flags 5% of the people who do not. You test positive. What is the chance you are actually sick? Follow 10,000 people:
A positive result is wrong five times out of six. The sick are rare and the well are many, so 5% of a large number outweighs 99% of a small one. When a version of this was put to Harvard physicians in 1978, most answered "95%"; the true figure was about 2%. The instinct that failed them is the one just fixed by counting — always ask out of how many.
How steep, and how much
This guide has been quietly running up debts. The circle's area leaned on "the thinner the wedges, the closer." The cone's ⅓ and the sphere's 4/3 were stated and never proved. Back in the first guide, 0.999… was shown to equal 1 by an algebraic trick, with a promise of a better reason. And compounding kept closing in on e.
Every one of those is the same debt, and a single idea settles them all: the limit — the value a process closes in on, even if no individual step ever lands there exactly.
Look at 0.9, 0.99, 0.999, 0.9999, … The list never ends, but only one number is being cornered, and the endless decimal is that number: 1, exactly. Calculus is what mathematics becomes when it takes that cornering seriously, and it opens with two questions you already almost have the tools for.
How steep is a curve?
Slope was born on straight lines: rise over run, the same everywhere on the line. But on the parabola y = x², the steepness is different at every point. So what could "the slope at x = 1" even mean?
Slope needs two points, and a single point on the curve gives you only one. The move is to refuse to give up: take a second point a little further along, compute an honest two-point slope between them, and then slide the second point in toward the first.
Watch the two-point slopes as the gap h shrinks:
h (the gap): 1 0.5 0.1 0.01 0.001
slope: 3 2.5 2.1 2.01 2.001At h = 0 the recipe itself collapses — a rise of 0 over a run of 0, and 0 ÷ 0 has no answer. But the limit steps around that. The slopes are cornering one value, 2, and that is the slope of the curve at x = 1 — the slope of the straight line that just grazes it there. That cornered value is the derivative: an instantaneous rate of change, a number that exists only as a limit.
The same move reads a speedometer. A dropped object has fallen 4.9t² metres after t seconds, and its average speed over ever-shorter windows just after t = 2 runs 24.5, 20.09, 19.649, 19.6049… metres per second. The value being cornered, 19.6, is its speed at that instant — again, a number that only a limit can define.
How much lies under a curve?
Second question, same trick turned around. The region under y = x² from x = 0 to x = 1 has a curved roof, and none of the area formulas from Part I fit it. So approximate: slice the base into n thin strips, stand a rectangle up to the roof on each, and add their areas.
The more slices, the closer the total, and the totals corner a value:
n (slices): 10 100 1000 10000
total area: 0.385 0.33835 0.333833 0.333383They corner 1/3, the exact area — delivered by the very same "thinner is closer" that Archimedes used on the circle's wedges. A limit of slice-sums like this is an integral: an accumulated total of something that varies as you sweep across it.
The two are one
Now the astonishment that makes calculus a subject and not two clever tricks. Steepness-at-a-point and area-so-far turn out to be inverses of each other — the way subtraction undoes addition.
You have already seen both halves in miniature. The area under x² out to x = 1 came to 1/3; run the same slicing out to any point x and the total comes to x³/3. And the two-point slopes of x³/3 corner x² (at x = 0.7 they close in on 0.49 = 0.7²). Area-taking and slope-taking undo each other. That pairing is the fundamental theorem of calculus — stated here, proved a course later.
take the slope →
x³ / 3 x²
← take the areaWith the limit in hand, read back through the ledger and every debt clears. The circle's wedge argument was an integral in disguise — slice-sums cornering ½·C·r, so A = πr² exactly, not approximately. The cone's two million slices cornering πr²h/3, the sphere's cornering 4/3·πr³: those thirds and that 4/3 are limits. And 0.999… = 1 is the smallest limit of all, partial sums cornering their target. Every trick this course asked you to take on trust now has a foundation.
Where this goes
Everything before this chapter is the common floor of school mathematics. Calculus sits just past it in both school systems — beyond the shared core in the United States, and beyond GCSE in Britain, where it begins at A-level. This chapter has honestly been a look over the fence: on the other side, derivatives and integrals come with a whole machinery of rules this guide has not touched.
But the country over that fence is one whose language you now speak. Functions and their graphs, slope, area, the exponential whose steepness equals its own height at every point (the property e exists to make exact), the sine wave that radians keep clean — these are the objects calculus works with, and they are already yours. If you carry on, you cross the border speaking the language.
Further reading
Everything below is free, and each was used in building this guide. The one thing none of them can do is the thing this guide keeps insisting on: mathematics is learned by working problems with paper, so whatever you open next, open it with a pencil.
- OpenStax Precalculus 2e — a full, free textbook covering the functions, exponentials, logarithms and trigonometry of Parts II–IV, with worked examples and exercise sets for every section. The natural place to go deeper and to practise.
- OpenStax Introductory Statistics 2e — the data-and-chance material of Part V as a proper course: descriptive statistics, probability, distributions, and the normal curve, with the cautions this guide only had room to gesture at.
- Khan Academy — free video lessons and endless graded practice across geometry, trigonometry, statistics and precalculus. Best when a single idea here needs a slower second pass and a hundred exercises.
- 3Blue1Brown — Essence of Calculus — a short, visual video series that rebuilds the last chapter properly, making derivatives and integrals feel like something you could have discovered yourself. Its opening example is the circle's area, exactly where this guide's debts began.