Mathematics from Scratch: Basic to Advanced, Explained Simply

Mathematics is a small set of ideas that build on each other. This post goes in order. Part 1 covers the basics that everything else rests on. Part 2 covers the topics usually called advanced: functions, logarithms, trigonometry, calculus, vectors and matrices, probability and counting. The last sections collect a formula sheet and reference tables: formulas by topic with an example for each, a times table, squares and cubes, primes, divisibility rules, trigonometry values, degrees and radians, and unit conversions.

How to use this post

Every topic has the same parts, in this order:

1. The idea in plain words.
2. How to do it, as numbered steps you can follow every time.
3. Worked examples from easy to harder. Each step is written out, so cover the answer, try it yourself, then compare.
4. A picture or something to try. These work with the keyboard as well as the mouse.
5. The usual mistake, so you can spot it.
6. Practice questions. Open "Show the steps and the answer" only after you have had a go.

Work with a pen and paper. Reading a solution feels like understanding, but only doing it shows whether you have it.

To change the text size or read this post in a handwritten style, use the round Aa button at the bottom right of the page. Code, formulas in the boxes and the interactive parts keep their normal font.

Part 1: The basics

1. Numbers, including negatives

Numbers come in layers. Each layer contains the one before it and adds something.

Negative numbers sit on the other side of zero on the number line. Moving right adds, moving left subtracts.

How to do it

1. Subtracting a negative is the same as adding: a − (−b) = a + b. Think of removing a debt.
2. Adding a negative is the same as subtracting: a + (−b) = a − b.
3. Multiplying or dividing: same signs give a positive, different signs give a negative.
4. Absolute value is the distance from zero, so |−4| = 4 and |4| = 4.

Example 1: 5 − (−3)

• The two minus signs next to each other turn into a plus: 5 + 3.
• Answer: 8.

Example 2: −4 + 7

• Start at −4 on the number line and move 7 steps right.
• You pass zero after 4 steps and move 3 more. Answer: 3.

Example 3: −6 − 2

• Start at −6 and move 2 steps left. Answer: −8.

Example 4: (−3) × (−5)

• Both signs are the same, so the result is positive: 3 × 5. Answer: 15.

Example 5: 4 × (−2) and −12 ÷ 3

• Different signs, so both results are negative.
• 4 × 2 = 8, so the first is −8. 12 ÷ 3 = 4, so the second is −4.

The usual mistake: reading 5 − (−3) as 5 − 3. Two minus signs in a row cancel.

2. Factors, primes, HCF and LCM

A factor of a number divides it exactly: the factors of 12 are 1, 2, 3, 4, 6 and 12. A prime has exactly two factors, 1 and itself: 2, 3, 5, 7, 11. The number 1 is not prime. Every whole number above 1 is a product of primes in only one way, so 12 = 2 × 2 × 3. Splitting a number like this is called prime factorisation.

Two questions come up all the time:

• HCF (highest common factor, also called GCD): the biggest number that divides both. Used to simplify fractions and to cut things into equal pieces.
• LCM (lowest common multiple): the smallest number that both divide into. Used to find a common bottom for fractions and to find when repeating events line up.

How to do it

Method A, with prime factors:

1. Write each number as a product of primes. Divide by 2 as long as you can, then by 3, then 5, and so on.
2. HCF: multiply the primes that appear in both, each at its lowest power.
3. LCM: multiply every prime that appears in either, each at its highest power.

Method B, Euclid's method for the HCF. It works for big numbers where factoring is slow.

1. Divide the larger number by the smaller and note the remainder.
2. Replace the pair by (smaller number, remainder) and repeat.
3. When the remainder is 0, the last divisor is the HCF.

A shortcut for two numbers: HCF × LCM = the two numbers multiplied together. So LCM = (a × b) ÷ HCF.

Divisibility rules help with step 1: a number divides by 2 if it ends in an even digit, by 3 if its digits add up to a multiple of 3, and by 5 if it ends in 0 or 5. The full list is in the reference tables at the end.

Example 1: factorise 60

1. 60 ÷ 2 = 30, and 30 ÷ 2 = 15. Two 2s so far.
2. 15 is odd. 15 ÷ 3 = 5, and 5 is prime.
3. So 60 = 2 × 2 × 3 × 5 = 2² × 3 × 5.

Example 2: HCF and LCM of 12 and 18

1. 12 = 2² × 3 and 18 = 2 × 3².
2. HCF: shared primes at the lowest power. 2 appears as 2² and 2¹, so take 2. 3 appears as 3¹ and 3², so take 3. HCF = 2 × 3 = 6.
3. LCM: every prime at the highest power: 2² × 3² = 4 × 9 = 36.
4. Check with the shortcut: 6 × 36 = 216 and 12 × 18 = 216.

Example 3: HCF of 84 and 126 by Euclid's method

1. 126 = 1 × 84 + 42. The remainder is 42.
2. 84 = 2 × 42 + 0. The remainder is 0.
3. The last divisor is 42, so the HCF is 42.
4. LCM = (84 × 126) ÷ 42 = 10,584 ÷ 42 = 252.

Example 4: simplify 84/126

• Divide the top and bottom by their HCF, 42: 84 ÷ 42 = 2 and 126 ÷ 42 = 3. Answer: 2/3. This is the fastest way to reduce a fraction fully in one step.

Example 5: add 1/6 + 1/8

1. The best common bottom is the LCM of 6 and 8. 6 = 2 × 3 and 8 = 2³, so the LCM is 2³ × 3 = 24.
2. 1/6 = 4/24 and 1/8 = 3/24.
3. 4/24 + 3/24 = 7/24.

Example 6: two buses. One bus leaves the station every 12 minutes and another every 18 minutes. They leave together at 9:00. When do they next leave together?

• The answer is the LCM of 12 and 18, which is 36. So at 9:36.

Example 7: tiles. A floor is 240 cm by 180 cm. What is the biggest square tile that fits exactly, with no cutting?

1. The tile side must divide both 240 and 180, so you want the HCF.
2. 240 = 1 × 180 + 60, then 180 = 3 × 60 + 0. The HCF is 60.
3. Tiles of 60 cm: 240 ÷ 60 = 4 along one side and 180 ÷ 60 = 3 along the other. That is 12 tiles.

Example 8: numbers with nothing in common. 8 = 2³ and 15 = 3 × 5 share no primes, so the HCF is 1. Such numbers are called coprime. Their LCM is just the product: 8 × 15 = 120.

The usual mistakes

• Taking the lowest power for the LCM. The LCM needs the highest power, or it will not divide into the number with the bigger power.
• Mixing the two up. A quick test: the HCF is never bigger than the smaller number, and the LCM is never smaller than the bigger number.
• Listing only some factors. Writing out the prime factorisation first avoids this.

3. Fractions

A fraction is parts of a whole. The bottom number says how many equal parts there are, and the top number says how many you have.

How to do it

• Simplify: divide the top and bottom by the same number until you cannot any more.
• Add or subtract: (1) find a common bottom (the LCM of the two bottoms is the smallest), (2) rewrite each fraction with it, (3) add or subtract the tops, (4) simplify (divide by the HCF).
• Multiply: top times top, bottom times bottom, then simplify.
• Divide: flip the second fraction and multiply.

Example 1: simplify 12/18

• The biggest number that divides both 12 and 18 is 6.
• 12 ÷ 6 = 2 and 18 ÷ 6 = 3. Answer: 2/3.

Example 2: write 7/4 as a mixed number

• 4 goes into 7 once, with 3 left over.
• Answer: 1 and 3/4.

Example 3: 2/3 + 3/4

1. The common bottom is 12, because both 3 and 4 divide 12.
2. 2/3 = 8/12 (multiply top and bottom by 4). 3/4 = 9/12 (multiply by 3).
3. Add the tops: 8/12 + 9/12 = 17/12.
4. 17/12 is 1 and 5/12.

Example 4: 3/4 − 1/6

1. Common bottom: 12.
2. 3/4 = 9/12 and 1/6 = 2/12.
3. 9/12 − 2/12 = 7/12.

Example 5: 2/3 × 3/5

1. Tops: 2 × 3 = 6. Bottoms: 3 × 5 = 15. That gives 6/15.
2. Divide both by 3: 2/5.

Example 6: 3/4 ÷ 2/5

1. Flip the second fraction: 2/5 becomes 5/2.
2. Multiply: 3/4 × 5/2 = 15/8, which is 1 and 7/8.

The usual mistake: adding tops and bottoms. 1/2 + 1/3 is 5/6, not 2/5.

4. Percentages

Percent means per hundred, so 25% is 25/100, or 0.25. You use this for discounts, tax, tips and interest.

How to do it

1. Turn the percentage into a decimal by dividing by 100: 20% becomes 0.20.
2. Percentage of a number: multiply by the decimal.
3. Increase by p%: multiply by 1 + p/100. A 13% increase multiplies by 1.13.
4. Decrease by p%: multiply by 1 − p/100. A 25% decrease multiplies by 0.75.
5. Percentage change: (new − old) ÷ old.
6. Finding the original: divide the final value by the multiplier.

Example 1: 20% of 150

• 0.20 × 150 = 30.

Example 2: a 25% discount on a 40 item

• Discount: 0.25 × 40 = 10. Price: 40 − 10 = 30.
• Same thing in one step: 40 × 0.75 = 30.

Example 3: 80 plus 13% tax

• 80 × 1.13 = 90.40.

Example 4: percentage change from 50 to 65

• Change: 65 − 50 = 15. Divide by the old value: 15 ÷ 50 = 0.30.
• Answer: up 30%.

Example 5: percentage change from 80 to 60

• Change: 60 − 80 = −20. Divide by 80: −0.25.
• Answer: down 25%.

Example 6: after a 20% discount a jacket costs 48. What was the original price?

1. A 20% discount means you pay 80%, a multiplier of 0.8.
2. original × 0.8 = 48.
3. original = 48 ÷ 0.8 = 60.

Example 7: 1000 earning 5% a year for 2 years

1. Year 1: 1000 × 1.05 = 1050.
2. Year 2: 1050 × 1.05 = 1102.50.
3. The second year's interest is taken on 1050, not 1000, which is why this beats 2 × 5% = 10%.

A quick trick: 10% is the number with the decimal point moved one place, and 5% is half of that. So a 15% tip on 40 is 4 + 2 = 6.

The usual mistake: treating an increase and a decrease of the same percentage as cancelling out. Take 80, raise it by 25% to get 100, then cut 20% to get 80. The numbers differ because each percentage is taken of a different base. Up 20% then down 20% ends at 96% of the start, not 100%.

5. Ratios and proportion

A ratio compares amounts. A ratio of 3:2 means for every 3 of one thing there are 2 of another. A proportion says two ratios are equal, which lets you scale.

How to do it

• Share a total in a ratio: (1) add the parts, (2) divide the total by that sum to get the size of one part, (3) multiply by each ratio number.
• Unit method: find the value of one unit, then multiply.
• Equal ratios: if a/b = c/d, then a × d = b × c (cross-multiply).

Example 1: share 30 in the ratio 3:2

1. Parts: 3 + 2 = 5.
2. One part: 30 ÷ 5 = 6.
3. Shares: 3 × 6 = 18 and 2 × 6 = 12. Check: 18 + 12 = 30.

Example 2: 3 pens cost 12. How much do 5 pens cost?

1. One pen: 12 ÷ 3 = 4.
2. Five pens: 5 × 4 = 20.

Example 3: a car travels 150 km in 2 hours. How far in 5 hours at the same speed?

1. Speed: 150 ÷ 2 = 75 km/h.
2. Distance: 75 × 5 = 375 km.

Example 4: a map has the scale 1:50,000. Two towns are 3 cm apart on the map.

1. 1 cm on the map is 50,000 cm on the ground.
2. 3 cm is 150,000 cm.
3. 100 cm is 1 m and 1000 m is 1 km, so divide by 100,000: 1.5 km.

Example 5: solve 3/x = 6/10

1. Cross-multiply: 3 × 10 = 6 × x, so 30 = 6x.
2. x = 30 ÷ 6 = 5.

The usual mistake: dividing the total by one of the ratio numbers instead of by their sum. In 3:2 the total is split into 5 parts, not 3.

6. Order of operations

When an expression has several operations, the order decides the answer. You may see the rule called BODMAS or PEMDAS. It is the same rule.

How to do it

1. Brackets first, innermost first.
2. Then powers and roots.
3. Then multiplication and division, left to right. They have equal rank.
4. Then addition and subtraction, left to right. Also equal rank.

Rewrite the expression after each step so you never skip one in your head.

Example 1: 7 + 3 × 2² − 4

1. Power: 2² = 4, giving 7 + 3 × 4 − 4.
2. Multiply: 3 × 4 = 12, giving 7 + 12 − 4.
3. Left to right: 7 + 12 = 19, then 19 − 4 = 15.

Example 2: (8 − 2) × 3 + 12 ÷ 4

1. Bracket: 8 − 2 = 6, giving 6 × 3 + 12 ÷ 4.
2. Multiply and divide: 18 + 3.
3. Add: 21.

Example 3: 20 − 6 ÷ 2 × 3

1. Divide and multiply have equal rank, so go left to right. 6 ÷ 2 = 3, giving 20 − 3 × 3.
2. 3 × 3 = 9, giving 20 − 9.
3. Subtract: 11.

Example 4: 2 × (3 + (4 − 1)²)

1. Innermost bracket: 4 − 1 = 3, giving 2 × (3 + 3²).
2. Power inside the bracket: 3² = 9, giving 2 × (3 + 9).
3. Bracket: 12, giving 2 × 12.
4. Multiply: 24.

The usual mistake: thinking multiplication always beats division. They have equal rank, so you go left to right. In Example 3, doing 2 × 3 first would give 20 − 1 = 19, which is wrong.

7. Exponents and roots

An exponent says how many times to multiply a number by itself: 2³ = 2 × 2 × 2 = 8. A root undoes it: √9 = 3 because 3² = 9.

How to do it

Example 1: 3⁴

• 3 × 3 = 9, × 3 = 27, × 3 = 81.

Example 2: 2³ × 2⁴

• Same base, add the exponents: 2⁷ = 128.

Example 3: 10⁻²

• A negative exponent flips it: 1 ÷ 10² = 1 ÷ 100 = 0.01.

Example 4: simplify √12

1. Find the biggest square that divides 12. That is 4.
2. √12 = √4 × √3 = 2√3, which is about 3.46.

Example 5: scientific notation

• 4,500,000 is 4.5 × 10⁶: the decimal point moves 6 places.
• 0.00032 is 3.2 × 10⁻⁴: the decimal point moves 4 places the other way.

Example 6: 8^(1/3)

• The 1/3 power is the cube root. 2 × 2 × 2 = 8, so the answer is 2.

Powers of 2 show up everywhere in computing: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024 are 2¹ to 2¹⁰.

The usual mistake: −3² is −9, because the power is done first and the minus is applied after. (−3)² is 9.

8. Algebra

Algebra uses letters for unknown numbers. An equation is a balance: whatever you do to one side, you must do to the other.

How to do it

1. Expand brackets and collect like terms on each side.
2. Get the terms with x on one side and plain numbers on the other, by adding or subtracting the same thing on both sides.
3. Divide both sides by the number in front of x.
4. Put your answer back in the original equation to check.

Example 1: x + 7 = 15

• Subtract 7 from both sides: x = 8.

Example 2: 3x + 5 = 20

1. Subtract 5 from both sides: 3x = 15.
2. Divide both sides by 3: x = 5.
3. Check: 3 × 5 + 5 = 20.

Example 3: x on both sides, 4x − 9 = 2x + 7

1. Subtract 2x from both sides: 2x − 9 = 7.
2. Add 9 to both sides: 2x = 16.
3. Divide by 2: x = 8.
4. Check: 4 × 8 − 9 = 23 and 2 × 8 + 7 = 23.

Example 4: brackets, 2(x + 3) = 18

1. Multiply out: 2x + 6 = 18.
2. Subtract 6: 2x = 12.
3. Divide by 2: x = 6.

Example 5: turn a story into an equation

A phone plan costs 20 a month plus 5 per gigabyte. This month's bill is 65. How many gigabytes?

1. Let g be the gigabytes. The bill is 20 + 5g = 65.
2. Subtract 20: 5g = 45.
3. Divide by 5: g = 9.

Multiplying out two brackets. Multiply every term in one by every term in the other: (x + 2)(x + 3) = x² + 3x + 2x + 6 = x² + 5x + 6.

Quadratic equations have an x² term. There are two ways to solve them.

Example 6: factor, x² − 5x + 6 = 0

1. Find two numbers that multiply to +6 and add to −5. They are −2 and −3.
2. So (x − 2)(x − 3) = 0.
3. A product is zero only if one factor is zero, so x = 2 or x = 3.

Example 7: factor, x² + 2x − 8 = 0

1. Two numbers that multiply to −8 and add to +2: they are 4 and −2.
2. (x + 4)(x − 2) = 0, so x = −4 or x = 2.

Example 8: the formula, x² − 4x − 5 = 0

For ax² + bx + c = 0, the formula is x = (−b ± √(b² − 4ac)) ÷ 2a. Here a = 1, b = −4, c = −5.

1. Inside the root: (−4)² − 4 × 1 × (−5) = 16 + 20 = 36. The root is 6.
2. x = (4 ± 6) ÷ 2.
3. x = 10 ÷ 2 = 5 or x = −2 ÷ 2 = −1.

The formula always works. Factoring is faster when the numbers are small.

The usual mistake: changing only one side of the equation.

9. Geometry

π is about 3.14159. For solids, the volume of a box is length × width × height, and a cylinder is π × r² × height.

How to do it

1. Draw the shape and label what you know.
2. Pick the formula that uses those numbers. Check the units: area is in square units, volume in cubic units.
3. Substitute, calculate, and write the unit.
4. For a shape made of parts, find each part and add or subtract.

Example 1: a rectangle 8 by 5

• Area: 8 × 5 = 40.
• Perimeter: 2 × (8 + 5) = 26.

Example 2: a triangle with base 6 and height 4

• ½ × 6 × 4 = 12.

Example 3: a circle with radius 5

• Area: π × 5² = 25π ≈ 78.54.
• Circumference: 2 × π × 5 = 10π ≈ 31.42.

Example 4: a 6 by 4 room with a 2 by 2 cupboard in one corner. How much floor is free?

1. Whole room: 6 × 4 = 24.
2. Cupboard: 2 × 2 = 4.
3. Free floor: 24 − 4 = 20.

Example 5: a box 3 by 4 by 5

• Volume: 3 × 4 × 5 = 60.

Example 6: a cylinder with radius 2 and height 10

• π × 2² × 10 = 40π ≈ 125.66.

Example 7: angles in a triangle always add to 180°. Two angles are 50° and 60°.

• Third angle: 180 − 50 − 60 = 70°.

Pythagoras. In a right-angled triangle, the square on the longest side (the hypotenuse) equals the sum of the squares on the other two.

Example 8: legs 6 and 8, find the hypotenuse

1. 6² + 8² = 36 + 64 = 100.
2. Square root: 10.

Example 9: hypotenuse 13 and one leg 5, find the other leg

1. The hypotenuse is on its own side: 13² = 5² + b², so 169 = 25 + b².
2. b² = 144, so b = 12.

Example 10: distance between the points (1, 2) and (4, 6)

1. Across: 4 − 1 = 3. Up: 6 − 2 = 4.
2. These are the two legs of a right triangle: 3² + 4² = 25.
3. Distance: 5.

A 5 m ladder with its foot 3 m from a wall reaches 4 m up, because 3² + 4² = 5².

The usual mistake: using the diameter where a formula wants the radius. The radius is half the diameter. A second one: when finding a leg, subtracting the squares (169 − 25), not adding.

10. Averages

• Mean: add everything and divide by how many there are.
• Median: the middle value when the numbers are in order.
• Mode: the value that appears most often.

How to do it

1. Write the numbers in order, smallest to largest.
2. Mean: add them all, divide by the count.
3. Median: pick the middle one. With an even count, take the mean of the two middle ones.
4. Mode: count repeats.

Example 1: the data 4, 8, 6, 5, 3, 8, 9

1. In order: 3, 4, 5, 6, 8, 8, 9.
2. Mean: 3 + 4 + 5 + 6 + 8 + 8 + 9 = 43. Divide by 7: about 6.14.
3. Median: 7 values, so the 4th is the middle. It is 6.
4. Mode: 8 appears twice, the rest once. The mode is 8.

Example 2: an even count, 2, 4, 6, 10

• The middle two are 4 and 6. Their mean is 5, so the median is 5.

Example 3: a weighted average. An exam counts 40% and coursework counts 60%. You scored 80 and 90.

• 0.4 × 80 + 0.6 × 90 = 32 + 54 = 86. A plain mean of 85 would be wrong, because the two parts do not count equally.

Example 4: why the choice matters. Take the salaries 20, 22, 23, 25 and 200 (in thousands). The mean is 290 ÷ 5 = 58. The median is 23. One very large value drags the mean up but barely moves the median.

The usual mistake: forgetting to sort before picking the median.

Part 2: Advanced topics

11. Functions and graphs

A function takes an input x and gives one output y. Writing f(x) = 2x + 1 means: take x, double it, add 1. A graph plots every pair. Two shapes cover most of school and everyday work:

• A line, y = mx + b. The slope m is how much y changes for each step in x. The number b is where the line crosses the y axis.
• A parabola, y = ax² + bx + c. It has a turning point and opens upward when a is positive.

How to do it

• Evaluate: replace x with the number, then follow the order of operations.
• Slope between two points: (y₂ − y₁) ÷ (x₂ − x₁).
• Where a line crosses an axis: set y = 0 for the x intercept, set x = 0 for the y intercept.
• Vertex of a parabola: x = −b ÷ 2a, then put that x back into the function.

Example 1: f(x) = 2x + 1. Find f(3) and f(−2).

• f(3) = 2 × 3 + 1 = 7.
• f(−2) = 2 × (−2) + 1 = −3.

Example 2: slope between (1, 3) and (4, 9)

1. Rise: 9 − 3 = 6. Run: 4 − 1 = 3.
2. Slope: 6 ÷ 3 = 2.

Example 3: where does y = 2x − 6 cross the axes?

1. y intercept: put x = 0, giving y = −6.
2. x intercept: put y = 0, giving 2x = 6, so x = 3.

Example 4: the parabola y = x² − 4x + 3

1. Vertex x: −(−4) ÷ (2 × 1) = 2.
2. Vertex y: 2² − 4 × 2 + 3 = −1. The vertex is (2, −1).
3. Roots (where y = 0): x² − 4x + 3 = 0 factors as (x − 1)(x − 3), so x = 1 and x = 3.

Example 5: a real function. A taxi charges 3 to start plus 2 per km. The fare is f(d) = 3 + 2d. For 7 km: f(7) = 3 + 14 = 17. The slope 2 is the price per km, and the 3 is the fixed start charge.

12. Logarithms

A logarithm answers the question "what power gives this number?". Since 2³ = 8, log₂(8) = 3. It is the undo button for exponents.

How to do it

1. Read log_b(N) as: "b to what power gives N?"
2. The rules turn multiplication into addition:
   • log(a × b) = log a + log b
   • log(a ÷ b) = log a − log b
   • log(aⁿ) = n × log a
3. To solve b^x = N for an unknown exponent, take logs: x = log(N) ÷ log(b). A calculator has log (base 10) and ln (base e).

Example 1: log₂(8)

• 2 × 2 × 2 = 8, so the answer is 3.

Example 2: log₁₀(1000) and log₃(81)

• 10³ = 1000, so the first is 3.
• 3⁴ = 81, so the second is 4.

Example 3: log₅(1/25)

• 5⁻² = 1/25, so the answer is −2.

Example 4: the rules in use

• log₂(8 × 4) = log₂(32) = 5, which is 3 + 2.
• log₂(32 ÷ 4) = log₂(8) = 3, which is 5 − 2.
• log₂(8²) = log₂(64) = 6, which is 2 × 3.

Example 5: solve 2^x = 50

1. Take logs: x = log(50) ÷ log(2).
2. 1.699 ÷ 0.301 is about 5.64.
3. Check: 2⁵ = 32 and 2⁶ = 64, and 50 sits between them.

Example 6: how long to double money at 7% a year?

1. Solve 1.07ⁿ = 2, so n = log(2) ÷ log(1.07).
2. 0.30103 ÷ 0.02938 is about 10.24 years.
3. The rule of 72 gives 72 ÷ 7 ≈ 10.3, which agrees.

Example 7: logarithms count doublings. Binary search on 1,000,000 sorted items needs about 20 steps, because log₂(1,000,000) is about 19.93. Doubling the number of items adds just one more step.

The usual mistake: log(a + b) is not log(a) + log(b). Only multiplication turns into addition.

13. Trigonometry

For a right-angled triangle, with an angle marked, the three ratios are:

• sine = opposite ÷ hypotenuse
• cosine = adjacent ÷ hypotenuse
• tangent = opposite ÷ adjacent

The unit circle shows them as positions. Walk around a circle of radius 1 and the point's horizontal position is cos, and its vertical position is sin.

Where the exact values come from. Two triangles give every value you need. Draw them once and read the ratios off the sides.

The values to remember

Two ways to rebuild this table from memory:

• The counting trick for sine: the sines of 0°, 30°, 45°, 60°, 90° are √0/2, √1/2, √2/2, √3/2, √4/2, which are 0, 0.5, 0.707, 0.866, 1. Cosine is the same list backwards.
• Tangent is sine divided by cosine. At 90° the cosine is 0, and dividing by zero is not allowed, so tan 90° is undefined.

Beyond 90°. Angles go all the way round. The size of sin, cos and tan repeats from the first quarter, and only the sign changes, depending on which quarter the angle is in.

A memory aid for the quarters in order is "All Students Take Calculus": All, Sine, Tangent, Cosine. To find the value, find the matching angle in the first quarter, then apply the sign.

How to do it

1. Mark the angle you are using. Label the three sides from that angle: opposite, adjacent, hypotenuse.
2. Pick the ratio that uses the two sides you care about. A memory aid is SOH CAH TOA: Sine is Opposite over Hypotenuse, Cosine is Adjacent over Hypotenuse, Tangent is Opposite over Adjacent.
3. To find a side, rearrange. To find an angle, use the inverse button (sin⁻¹, cos⁻¹, tan⁻¹).
4. Check the calculator is in degrees mode if the angle is in degrees.

Example 1: a 3-4-5 triangle. The angle at the bottom left has opposite side 3, adjacent side 4, hypotenuse 5.

• sin = 3/5 = 0.6, cos = 4/5 = 0.8, tan = 3/4 = 0.75.
• The angle itself: sin⁻¹(0.6) ≈ 36.87°.

Example 2: how high does a ladder reach? A 5 m ladder leans at 70° to the ground.

1. The height is opposite the angle and the ladder is the hypotenuse, so use sine.
2. height = 5 × sin 70° = 5 × 0.9397 ≈ 4.70 m.
3. The foot is adjacent: 5 × cos 70° = 5 × 0.3420 ≈ 1.71 m from the wall.

Example 3: find an angle. A ramp rises 30 over a run of 40.

1. Opposite 30, adjacent 40, so use tangent: tan θ = 30 ÷ 40 = 0.75.
2. θ = tan⁻¹(0.75) ≈ 36.87°.

Example 4: degrees and radians. A half turn is 180° or π radians. To convert, multiply degrees by π ÷ 180.

• 90° = π/2 radians, about 1.571.
• 60° = π/3 radians, about 1.047.
• Going back: 2 radians × 180 ÷ π ≈ 114.59°.

Programming languages, including JavaScript and Java, expect radians in Math.sin and Math.cos.

Example 5: a value beyond 90°, sin 150°

1. 150° is in the second quarter, where sine is positive.
2. The matching first-quarter angle is 180° − 150° = 30°.
3. sin 150° = sin 30° = 0.5.

Example 6: cos 120°

1. 120° is in the second quarter, where cosine is negative.
2. The matching angle is 180° − 120° = 60°.
3. cos 120° = −cos 60° = −0.5.

Example 7: tan 225°

1. 225° is in the third quarter, where tangent is positive.
2. The matching angle is 225° − 180° = 45°.
3. tan 225° = tan 45° = 1.

Example 8: the diagonal of a square. A square has side 4. The diagonal cuts it into two 45, 45, 90 triangles, so the diagonal is the side times √2.

• 4 × √2 ≈ 5.66.

The usual mistake: feeding degrees to a function that expects radians, or leaving the calculator in the wrong mode.

14. Calculus

Calculus has two halves.

The derivative is the rate of change: the slope of a curve at one point. If position is a function of time, its derivative is speed.

How to find a derivative

A few rules cover most cases:

1. Power rule: bring the exponent down in front and subtract 1 from the exponent. The derivative of xⁿ is n × xⁿ⁻¹.
2. A number in front stays put: the derivative of 5x³ is 5 × 3x² = 15x².
3. Take each term of a sum separately.

Example 1: x⁴

• Bring down 4, reduce the exponent to 3: 4x³.

Example 2: 3x² + 5x − 7

1. 3x² gives 3 × 2x = 6x.
2. 5x gives 5.
3. The constant −7 gives 0.
4. Together: 6x + 5.

Example 3: slope of y = x² at x = 3

1. The derivative is 2x.
2. At x = 3 it is 6. The curve rises 6 for each step right at that point.

Example 4: speed from position. A car's position is s = t² + 2t (metres, with t in seconds).

1. Speed is the derivative: v = 2t + 2.
2. At t = 4: v = 10 m/s.

Example 5: finding the lowest point. Where is x² − 6x + 5 smallest?

1. At the bottom of the curve the slope is zero. The derivative is 2x − 6.
2. Set it to zero: 2x − 6 = 0, so x = 3.
3. Value there: 9 − 18 + 5 = −4. The minimum is −4, at x = 3.

This is how optimisation works: set the derivative to zero to find the best value.

The integral is the area under a curve. If speed is a function of time, its integral is distance. Driving at a steady 60 km/h for 2 hours gives a rectangle of area 120, which is 120 km.

How to find an integral

1. Reverse the power rule: add 1 to the exponent and divide by the new exponent. The integral of xⁿ is xⁿ⁺¹ ÷ (n + 1), plus a constant C.
2. For the area between a and b, put b into the result, then a, and subtract: F(b) − F(a). The constant cancels.

Example 6: the integral of x²

• Add 1 to the exponent: x³. Divide by 3: x³/3 + C.

Example 7: the area under x² from 0 to 3

1. F(x) = x³/3.
2. F(3) = 27/3 = 9. F(0) = 0.
3. Area: 9 − 0 = 9.

Example 8: the area under 2x from 1 to 4

1. F(x) = x² (since the derivative of x² is 2x).
2. F(4) − F(1) = 16 − 1 = 15.

Example 9: distance from speed. A runner's speed is v = 2t m/s for the first 5 seconds. Distance is the area under that line.

1. F(t) = t².
2. F(5) − F(0) = 25.
3. Check with a triangle: ½ × 5 × 10 = 25.

The two halves undo each other: the derivative of the area function gives back the original curve.

The usual mistake: forgetting to subtract 1 from the exponent when differentiating, or forgetting to divide by the new exponent when integrating.

15. Vectors and matrices

A vector is a list of numbers that you can picture as an arrow, with a length and a direction.

How to do it with vectors

1. Add: add the matching parts.
2. Scale: multiply every part by the number.
3. Length: square the parts, add, take the root.
4. Dot product: multiply matching parts and add. If the result is 0, the vectors are perpendicular.

Example 1: (2, 3) + (4, −1)

• (2 + 4, 3 + (−1)) = (6, 2).

Example 2: 3 × (2, −1)

• (6, −3).

Example 3: the length of (5, 12)

• 25 + 144 = 169. The root is 13.

Example 4: the dot product (1, 2) · (3, 4)

• 1 × 3 + 2 × 4 = 3 + 8 = 11.

Example 5: are (2, 3) and (3, −2) perpendicular?

• 2 × 3 + 3 × (−2) = 6 − 6 = 0. Yes.

Example 6: the angle between (1, 0) and (1, 1)

1. Dot product: 1.
2. Lengths: 1 and √2.
3. cos θ = 1 ÷ √2 ≈ 0.707, so θ = 45°.

Matrices. A matrix is a grid of numbers. A 2 × 2 matrix transforms every point on a plane: it can rotate, stretch, shear or mirror a shape. Try it.

How to multiply a matrix by a vector

1. Each row of the matrix is dotted with the vector.
2. The results become the new vector.

Example 7: stretch. The matrix [[2, 0], [0, 3]] times (4, 5).

• Row 1: 2 × 4 + 0 × 5 = 8. Row 2: 0 × 4 + 3 × 5 = 15. Result: (8, 15). The x part doubled and the y part tripled.

Example 8: rotate. The matrix [[0, −1], [1, 0]] times (1, 0).

• Row 1: 0 × 1 + (−1) × 0 = 0. Row 2: 1 × 1 + 0 × 0 = 1. Result: (0, 1).
• The point (1, 0) moved to (0, 1): a 90 degree turn counter-clockwise.

Matrix times matrix. Each entry is a row dotted with a column.

Example 9: [[1, 2], [3, 4]] times [[0, 1], [1, 0]]

1. Top left: row (1, 2) with column (0, 1): 0 + 2 = 2.
2. Top right: row (1, 2) with column (1, 0): 1 + 0 = 1.
3. Bottom left: row (3, 4) with column (0, 1): 0 + 4 = 4.
4. Bottom right: row (3, 4) with column (1, 0): 3 + 0 = 3.
5. Result: [[2, 1], [4, 3]]. The columns of the first matrix were swapped.

The determinant says how much areas are scaled. For [[a, b], [c, d]] it is a × d − b × c. A determinant of 0 means the shape is flattened and cannot be recovered.

Example 10: [[3, 1], [2, 4]]

• 3 × 4 − 1 × 2 = 12 − 2 = 10. Areas grow by a factor of 10.

This is the maths behind 2D and 3D graphics, and machine learning stores its data as vectors and matrices.

The usual mistake: assuming the order of multiplication does not matter. It does. With A = [[1, 1], [0, 1]] and B = [[1, 0], [1, 1]], the product AB is [[2, 1], [1, 1]] but BA is [[1, 1], [1, 2]].

16. Probability and statistics

Probability is a number from 0 to 1: favourable outcomes divided by all outcomes.

How to do it

1. List the equally likely outcomes and count them.
2. Count the ones you want. Probability is wanted ÷ total.
3. Independent events: multiply. "And" means multiply.
4. "At least one": work out the chance of none, and subtract it from 1.
5. Without replacement: the total shrinks after each pick.

Example 1: rolling an even number on a die

• Even: 2, 4, 6 is 3 outcomes out of 6. Probability 1/2.

Example 2: two dice add up to 7

1. Total outcomes: 6 × 6 = 36.
2. Ways to make 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) is 6.
3. Probability: 6/36 = 1/6.

Example 3: two heads in a row

• 1/2 × 1/2 = 1/4. Three heads: 1/2 × 1/2 × 1/2 = 1/8.

Example 4: at least one head in two flips

1. The chance of no heads (two tails) is 1/4.
2. At least one head: 1 − 1/4 = 3/4.

Example 5: without replacement. A bag holds 3 red and 2 blue balls. You draw two. What is the chance both are red?

1. First red: 3/5.
2. One red is gone, 4 balls left, 2 red. Second red: 2/4.
3. Both: 3/5 × 2/4 = 6/20 = 3/10.

Expected value is the average result over many tries: each outcome times its probability, added up.

Example 6: a die. (1 + 2 + 3 + 4 + 5 + 6) ÷ 6 = 3.5, a value you can never roll but which the average approaches.

Example 7: a game. You win 6 if you roll a 6 and nothing otherwise. Expected value: 1/6 × 6 + 5/6 × 0 = 1. Paying 2 to play loses money on average.

The standard deviation measures how spread out values are around the mean.

How to do it

1. Find the mean.
2. Subtract the mean from each value and square each result.
3. Average those squares. This is the variance.
4. Take the square root. This is the standard deviation.

Example 8: the data 2, 4, 4, 4, 5, 5, 7, 9

1. Mean: 40 ÷ 8 = 5.
2. Differences squared: 9, 1, 1, 1, 0, 0, 4, 16. Sum: 32.
3. Variance: 32 ÷ 8 = 4.
4. Standard deviation: the root of 4 is 2.

For data shaped like a bell curve, about 68% of values fall within one standard deviation of the mean and about 95% within two.

Probability often surprises people. Try the classic example.

The usual mistake: thinking a streak changes the next outcome. A fair coin that has landed heads five times still lands heads with probability 1/2 next time.

17. Counting

Counting methods tell you how many arrangements exist without listing them.

How to do it

1. Independent steps: multiply the number of choices at each step.
2. Arrangements of n different things: n! = n × (n − 1) × ... × 1.
3. For picking k things out of n, ask whether order matters:
   • Permutations (order matters, like gold, silver and bronze): n! ÷ (n − k)!
   • Combinations (order does not matter, like a team): n! ÷ (k! × (n − k)!)

Example 1: an outfit. 3 shirts, 4 trousers, 2 pairs of shoes.

• 3 × 4 × 2 = 24 outfits.

Example 2: a plate with 3 letters then 3 digits

• 26 × 26 × 26 × 10 × 10 × 10 = 17,576,000.

Example 3: a 4-digit PIN

• 10 × 10 × 10 × 10 = 10,000.

Example 4: lining up 4 people

• 4 × 3 × 2 × 1 = 4! = 24.

Example 5: first, second and third prize among 5 people. Order matters.

• 5 × 4 × 3 = 60. By the formula: 5! ÷ 2! = 120 ÷ 2 = 60.

Example 6: a team of 2 from 6 people. Order does not matter.

1. Permutations first: 6 × 5 = 30.
2. Each pair was counted twice (A then B, B then A), so divide by 2! = 2.
3. 30 ÷ 2 = 15.

Example 7: the lottery. Choose 6 numbers from 49.

• C(49, 6) = 49! ÷ (6! × 43!) = 13,983,816. One ticket wins with probability 1 in about 14 million.

Choosing a team of 3 from 10 people has 120 combinations. Giving out three ranked prizes among the same 10 has 720 permutations. The same idea explains passwords: 8 characters from the 62 letters and digits give 62⁸, about 2.2 × 10¹⁴ possibilities.

The usual mistake: using permutations when order does not matter, which overcounts.

Reference tables

Keep these open while you work. Everything here is checked against a calculation, not copied from memory.

Times table, 1 to 12

Squares and cubes

Prime numbers up to 100

There are 25 of them: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.

2 is the only even prime. To test whether a number up to 100 is prime, check whether 2, 3, 5 or 7 divides it. If none does, it is prime.

Divisibility rules

Powers of 2

Degrees and radians

The wheel in the trigonometry section shows these on a circle. Multiply degrees by π ÷ 180 to get radians, and radians by 180 ÷ π to get degrees.

Fractions, decimals and percentages

Common unit conversions

The mistakes in one table

Formula sheet

Every formula from the post in one place, grouped by topic, with a small example for each so you can see what the letters stand for. The examples were checked with a calculation.

Fractions, percentages and ratios

The average speed row is a trap: the answer is not the mean of 60 and 40, because you spend longer at the slower speed.

Exponents and logarithms

Algebra, quadratics and sequences

Geometry: shapes

Geometry: solids

Coordinates and lines

Trigonometry

The cosine rule with C = 90° loses its last term, because cos 90° = 0, and becomes Pythagoras.

Calculus

Vectors and matrices

An inverse exists only when the determinant is not 0.

Probability, statistics and counting

A worked case for Bayes' rule. Take assumed numbers: 1% of people have a condition, a test finds it 90% of the time when it is present, and gives a false alarm 5% of the time when it is absent. You test positive. How likely is it that you have the condition?

1. Chance of a positive from someone who has it: 0.01 × 0.90 = 0.009.
2. Chance of a positive from someone who does not: 0.99 × 0.05 = 0.0495.
3. Chance of a positive overall: 0.009 + 0.0495 = 0.0585.
4. Chance you have it, given the positive: 0.009 ÷ 0.0585, about 0.154, or 15%.

Most positives come from the large healthy group, which is why a positive result on a rare condition is less certain than it feels.

Where to go next

Sources and further reading

• Khan Academy, free courses from arithmetic through calculus and statistics.
• 3Blue1Brown, the video series "Essence of calculus" and "Essence of linear algebra": the best visual introductions to those two topics.
• Gilbert Strang, Introduction to Linear Algebra.
• James Stewart, Calculus, a standard textbook.
• Graham, Knuth and Patashnik, Concrete Mathematics, for counting and discrete mathematics.
• George Pólya, How to Solve It (1945), on how to approach a problem you cannot see through yet.