Proofs

Some facts in these books are proved where you meet them. Others are stated, because stopping to prove everything would leave no room to practise anything. This is where the second kind is written out.

Every page here proves one thing, in the shortest way a reader can follow rather than the shortest way there is. Most of them have something to move, so the reason is visible rather than only readable.

The proofs

• The angles in a triangle add to 180° — Draw the line through one corner parallel to the opposite side, and the three angles fall onto a straight line.
• The sine rule — Drop a perpendicular and the same height comes out as b sin A and as a sin B. Equate them and the rule appears.
• Pythagoras' theorem — Four copies of the triangle make a square with a hole in it, and the hole is the square on the hypotenuse.
• The cosine rule — Pythagoras twice, with a dropped perpendicular. The correction term is what an angle that is not a right angle costs.
• The area of a triangle is half a b sin C — Drop a perpendicular and the triangle's area is half its base times its height. The height is what the sine ratio turns into b sin C.
• The angle at the centre is twice the angle at the circumference — Join the two points to the centre. Two isosceles triangles appear whose base angles must add up to half the straight angle at the centre.
• Completing the square — Why half the coefficient of x, squared, is the number to add: it is the corner of the same square drawn as a picture.
• The discriminant decides how many roots there are — Read it off the completed square. The square root is the whole story, and the graph shows the same thing.
• Differentiating x to the n from first principles — Expand (x + h) to the n, subtract, divide by h, and let h go to zero. The terms that survive are the ones the rule names.
• The product rule — The area of a rectangle whose sides both grow: the change is the two strips plus a corner that vanishes in the limit.
• The chain rule — Rates multiply. If y changes twice as fast as u and u three times as fast as x, y changes six times as fast as x.
• The binomial theorem for a positive whole number n — Count the ways each term can be made. The coefficients are the number of choices, which is why they are the numbers in Pascal's triangle.
• The factor theorem — Dividing a polynomial by (x - a) leaves a remainder you can find by substituting. If that remainder is zero, the division came out exactly.
• The laws of logarithms — A logarithm is an index, so every log law is an index law read backwards.
• The area of a trapezium — Two copies, one turned upside down, make a parallelogram. One trapezium is half of it.
• The sum of an arithmetic series — Write the sum down twice, once backwards, and add. Every pair makes the same total.
• The sum of a geometric series — Multiply the sum by the common ratio and subtract. Everything cancels except the two ends.
• The quadratic formula — Complete the square on ax² + bx + c and solve for x. The formula is what is left when the algebra is done.
• The angle in a semicircle is 90° — Join the corner to the centre. Two isosceles triangles appear, and their angles must fill a straight line.
• The angles in a polygon add to (n − 2) × 180° — Cut the polygon into triangles from one corner. Count how many, and the triangle proof does the rest.
• The sum of the first n squares — Why 1² + 2² + … + n² is n(n + 1)(2n + 1)/6, proved by induction so the formula arrives rather than appears.

About the label

A proof that is not in your course carries a Beyond the course note. That is not a warning. It means you can read it, enjoy it and use it, and that no exam paper will ask you to reproduce it. A proof without the note is one your course does expect — often it is a fact you are asked to use without being shown why.

Where these come from

The proofs are written for this collection, following the standard argument, because a proof copied from a board document is written for examiners rather than for readers. The claims about which course a proof belongs to are not ours to invent, and each one rests on the documents recorded in sources.yaml:

• the DfE subject content for GCSE and for A level, which is where the curriculum statements come from;
• the awarding bodies' specifications, which is where the tier and the assessment of proof come from.