Mathematics

A digit's value is set by its place, and zero holds an empty place. Places continue below the units as decimal fractions: 0.25 = 2 × 10⁻¹ + 5 × 10⁻².

Use: Long multiplication and division on paper, and so every ledger, survey and engineering calculation (arithmetic for everyone).

First: Brahmagupta calculated with zero and negative numbers in 628; Fibonacci's Liber abaci, 1202, carried the Hindu–Arabic numerals into Europe; Stevin set out decimal fractions in 1585 (MacTutor, 1100–1300, 1500–1600).

The largest number dividing both a and b is unchanged when the larger is replaced by its remainder on division by the smaller. gcd(3009, 884): remainders 357, 170, 17, 0, so 17.

Use: Reducing fractions, ratios, gear trains and calendar cycles to lowest terms.

First: Euclid, Elements, Book VII, Proposition 2, about 300 BC (Joyce's edition).

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

Use: A rope knotted into 3, 4 and 5 equal lengths makes a right angle (9 + 16 = 25), which is how foundations are set square; and any distance across a rectangle, a roof or a map grid.

First: Known in Babylon by about 1850 BC; proved as Euclid's Elements I.47, about 300 BC (MacTutor; Joyce).

π ≈ 3.14159 is the ratio of every circle's circumference to its diameter; C is the circumference and A the area of a circle of radius r, V the volume of a sphere.

Use: Sizing wheels, pipes, tanks, domes and silos. 22/7 is within 0.04 % of π, close enough for most building.

First: Archimedes, about 250 BC, gave the sphere's volume and bounded π between 223/71 and 22/7 with polygons of 96 sides (MacTutor; Pi through the ages).

Triangles with the same angles have their sides in the same ratio: here a post of height h casting shadow s, and a tower of height H casting S at the same moment.

Use: Heights from shadows, the width of a river from one bank, and every scale drawing and map.

First: Thales, about 575 BC, used geometry to find the height of pyramids and the distance of ships from shore (MacTutor).

The ratios of a right triangle's sides depend only on its angle θ. The law of cosines, last line, holds in any triangle, where C is the angle opposite side c, and reduces to Pythagoras when C is a right angle.

Use: An angle and one measured length give any other length: surveying by triangulation, heights, the range of a landmark, and position at sea.

First: Hipparchus, about 127 BC, used an early form of trigonometry in his astronomy (MacTutor).

Every point is a pair of numbers (x, y), so lines and curves become equations: y = mx + c is a straight line of slope m.

Use: Maps, surveys, orbits and machine drawings are calculated rather than drawn; the distance between two surveyed points comes straight from their coordinates.

First: Descartes, La Géométrie, 1637, applied algebra to geometry (MacTutor).

The two solutions of ax² + bx + c = 0. If b² − 4ac is negative there is no real solution.

Use: Any problem in which a quantity multiplies itself: a field's sides from its area and perimeter, the time a falling or thrown object takes to land.

First: Brahmagupta, 628, gave methods for quadratic equations; al-Khwarizmi's book of about 810 gave algebra its name, from al-jabr (MacTutor).

n equations in n unknowns, written as a matrix A of coefficients. Elimination subtracts multiples of one equation from the others until each unknown can be read off.

Use: Mixing rations, feeds and alloys to a specification, the forces in every member of a frame, the adjustment of a survey network, and nearly every computer model.

First: The Chinese Nine Chapters on the Mathematical Art, possibly from about 200 BC, solve up to six equations in six unknowns by this elimination (MacTutor); Cayley defined the matrix in 1858 (MacTutor).

A logarithm turns multiplying into adding, and raising to a power into multiplying.

Use: With a table or a slide rule, products, roots and powers become additions (calculating by hand). pH, decibels and earthquake magnitudes are logarithmic scales.

First: Napier, 1614; Briggs's tables to base 10, 1617 (MacTutor).

Anything that grows or shrinks in proportion to its own size changes exponentially; with e as the base, the rate k appears directly in the exponent.

Use: Compound interest, populations, epidemics in their early days, cooling and radioactive decay. Doubling time, or half-life, is ln 2 / k ≈ 0.693 / k.

First: Jacob Bernoulli met e in 1683 as the limit of interest compounded ever more often; Euler named it e in a letter of 1731 and set out its theory in 1748 (MacTutor).

The rate at which a quantity changes at an instant: the slope of its graph at that point. At a maximum or minimum it is zero.

Use: Speed from distance, acceleration from speed, and the best design: the strongest beam from a log, the tank with least material for its volume.

First: Newton began the differential calculus in 1665; Leibniz published his in 1684 (MacTutor, 1675–1700).

Adding up a rate of change across an interval gives the total change, so integration undoes differentiation.

Use: Areas and volumes of any shape, the work done by a varying force, distance from a speed record, the load of water on a dam.

First: Barrow's work on tangents, 1670, was Newton's starting point; Leibniz first wrote ∫ in 1675; Cauchy put analysis on a formal footing in 1821 (MacTutor, 1675–1700, 1820–1830).

Near a point a, a smooth function equals a sum of powers whose coefficients are its derivatives there. Each extra term adds accuracy.

Use: How tables of sines, logarithms and exponentials are computed by hand; the first term alone gives working approximations, such as sin x ≈ x for small angles in radians.

First: Brook Taylor, Methodus incrementorum, 1715; Gregory, Newton, Leibniz and others had found special cases (MacTutor).

Each step replaces a guess by the point where the tangent at the guess crosses zero. Near a simple root, the number of correct digits roughly doubles each step.

Use: Solving equations no formula solves. For a square root of a, it becomes xₙ₊₁ = ½(xₙ + a/xₙ): from 1.5, two steps give √2 = 1.41421…

First: Newton described it in his Method of Fluxions, written in 1671 and unpublished until 1736; Raphson published it in 1690 (MacTutor).

A complex number a + ib is a point in a plane; multiplying it by the right-hand side turns it through the angle θ.

Use: Alternating current, vibration and every kind of wave are calculated as turning complex numbers, which turns their differential equations into algebra.

First: Bombelli gave the rules for calculating with complex numbers in 1572; the identity giving e raised to the power πi as −1 first appears in Euler's Introductio, 1748 (MacTutor, 1740–1760).

A repeating signal, even one with corners such as a square wave, is a sum of sine and cosine waves at whole multiples of its frequency.

Use: Heat flow, the notes in a sound, tides and the vibration of machines are analysed one frequency at a time.

First: Fourier, 1807, in a paper on the propagation of heat (MacTutor).

Turns a signal in time into its spectrum of frequencies, and back again. Books differ in where they put the 2π and the sign of i.

Use: Radio, filtering noise, spectroscopy, medical imaging and image compression. The fast Fourier transform does it for N samples in about N log N steps instead of N².

First: Fourier's Théorie analytique de la chaleur, 1822; Cooley and Tukey's fast Fourier transform, 1965 (MacTutor, 1960–1970).

Probabilities of independent events multiply; the probability that something does not happen is one minus the probability that it does.

Use: Reliability and insurance: if each of 10 independent parts works 99 % of the time, all ten work 0.99¹⁰ ≈ 90 % of the time.

First: Pascal and Fermat worked out the laws of chance in letters of 1654; Huygens published the first book on probability in 1657 (MacTutor).

How far evidence B should change the probability of A, starting from how common A was before.

Use: Reading a test: one that is 99 % accurate for a condition 1 person in 1,000 has gives about 10 false positives for every true one, so a positive result is right only about 1 time in 11.

First: Bayes's essay on the doctrine of chances, published in 1764 (MacTutor).

The bell curve that sums of many small independent errors follow; μ is the mean and σ the spread. About 95 % of values lie within 1.96 σ of the mean.

Use: Measurement error, manufacturing tolerances, and how many samples a result needs: the uncertainty of an average of n measurements is σ/√n (knowing whether a result is real).

First: De Moivre first described the curve in 1733 (MacTutor).

The straight line y = a + bx that makes the sum of the squared misses (yᵢ − a − bxᵢ)² as small as possible; x̄ and ȳ are the means.

Use: Fitting a line or a calibration curve to measured points, and adjusting surveys and orbits to many imperfect observations.

First: Legendre, 1806; Gauss used it to find the orbits of celestial bodies in 1809 (MacTutor).

Place value with two digits, 0 and 1, so every number is a row of on-off states.

Use: Switches, relays and transistors have two states; in binary they can store and compute any number.

First: Leibniz perfected binary arithmetic by 1679 and published it in 1701 (MacTutor).

Logic as algebra on true and false (1 and 0), with AND (∧), OR (∨) and NOT (¬).

Use: Designing and simplifying circuits of switches; every digital computer is built of such circuits (computing).

First: Boole, The Mathematical Analysis of Logic, 1847, and The Laws of Thought, 1854; Shannon's master's thesis applied it to relay and switching circuits (MacTutor, Shannon).

The average information per symbol from a source whose symbols have probabilities pᵢ, in bits: the fewest binary digits per symbol any code can use.

Use: Compressing data, and adding just enough check bits to a message to correct the errors a noisy line makes.

First: Shannon, A Mathematical Theory of Communication, Bell System Technical Journal, 1948, which introduced the word "bit" (MacTutor).