Algebra: finding the unknown

A line such as 2x + 3 = 35 is an equation: a claim that two amounts are the same number, one on each side of the equals sign.

Picture old-fashioned scales, one side of the equation in each pan, hanging level. Take 3 from the left pan alone and it rises: the claim is now false. Take 3 from both pans and they stay level. So you may add, subtract, multiply, or divide by any number but zero, as long as the same is done to the other side. Each step leaves a simpler equation, still true. Take 3 from both sides of 2x + 3 = 35 and you have 2x = 32. Halve both sides and you have x = 16. A value that makes an equation true is called its solution: with 16 in place of x, the left side comes to 35 as well.

Which step comes first? Look at what was done to x and undo it in reverse order. In 3x + 5 = 41, x was multiplied by 3, and then 5 was added. The adding was done last, so it is undone first: subtract 5 from both sides, leaving 3x = 36. Then divide both sides by 3, leaving x = 12.

Subtracting undoes adding, and dividing undoes multiplying. Operations that undo each other like this are called inverse operations, and solving a simple equation is a matter of applying them, last step first, the way shoes come off before socks. One step remains: the check. Put 12 back into the first equation. 3 × 12 + 5 is 41, so the answer is right.

The method was written up as a subject of its own in ninth-century Baghdad. Muḥammad ibn Mūsā al-Khwārizmī wrote a short book on it, dedicated to the caliph al-Maʾmūn, who ruled from 813 to 833. His preface says it covers what people constantly need “in cases of inheritance, legacies, partition, law-suits, and trade”, and in measuring land. The longest of its three parts works through problems of Islamic inheritance law, sharing an estate among the heirs.

There are no symbols in the book, not even numerals: every problem is written out in words. The unknown is shayʾ, “thing”, and its square is māl, “wealth”. Problems with an unknown were far older, and Greek and Indian mathematicians had gone further in places. What was new was a handbook that taught the method for its own sake.

Al-Khwārizmī brought every problem into a standard form with two moves, and they gave his book its title. The first, al-jabr, means restoring, the word Arabic also uses for setting a broken bone. It clears an amount being taken away on one side by adding it to both sides. His own example, in modern letters, is x² = 40x − 4x². Add 4x² to both sides and nothing is taken away any more: 5x² = 40x.

The second, al-muqābala, means balancing. When the same kind of amount stands on both sides, take the smaller from both. In 50 + x² = 29 + 10x there are plain numbers on both sides; take 29 from each and it becomes 21 + x² = 10x. Both moves are the scales of an equation at work. Through Latin translations made in the twelfth century, al-jabr became the English word algebra.

Many of al-Khwārizmī’s problems contain the unknown multiplied by itself, which we write x². An equation whose highest power is x² is called quadratic, from the Latin for “square”. The simplest kind, such as x² = 49, is solved by undoing the squaring: the square root of 49 is 7, since 7 × 7 = 49. Today minus 7 counts as well, since a negative times a negative is positive.

Quadratics hold a surprise. Take al-Khwārizmī’s own example, 21 + x² = 10x. Try 3: 21 + 9 is 30, and 10 × 3 is 30. Try 7: 21 + 49 is 70, and 10 × 7 is 70. Both make the sides equal, so both are answers, and his rule finds the pair. He also warned that for some numbers an equation of this kind has no answer at all, and called such a case impossible.

For “a square and ten roots are equal to thirty-nine”, x² + 10x = 39, al-Khwārizmī gives a recipe: halve the 10 to get 5, multiply it by itself to get 25, add 39 to make 64, take its square root, 8, and subtract the 5. The answer is 3. A fixed recipe like this, reaching its answer every time, is called an algorithm, a word grown from the Latin spelling of his name.

He proved it with drawings. In one, x² is a square of side x, and 10x lies along two of its sides as two strips, each 5 by x. One corner is left empty, a square 5 by 5. Add its area, 25, to both sides, and the left becomes a whole square of side x + 5 and area 64. So x + 5 is 8. Filling the gap is called completing the square.

In 1637 René Descartes published La Géométrie, which showed that an equation can be drawn as a curve, and a curve studied through its equation. Pierre de Fermat reached it on his own. Today it is done on a grid where every point is named by two numbers, how far across and how far up: the coordinate plane.

An equation in x and y picks out every point that makes it true. y = 2x draws a straight line. y = x² draws a U-shaped curve called a parabola, and so does every quadratic drawn this way. The drawing shows answers too. Gather 21 + x² = 10x on one side as y = x² − 10x + 21 and draw it. The U dips below zero, and each place where the curve crosses zero is an answer: x = 3 and x = 7, al-Khwārizmī’s pair.

Al-Khwārizmī stopped at the square. One step up is x³, three x’s multiplied together, which is the volume of a box whose every side is x. An equation whose highest power is x³ is called cubic. In Samarkand, around 1070, the Persian mathematician ʿUmar al-Khayyām (Omar Khayyam) set out to sort and solve them all. Like al-Khwārizmī he wrote only in words and allowed only positive numbers, so he counted 25 kinds of equation up to the third power.

Eleven could be handled with the methods of Euclid’s geometry. For the other 14 he drew two curves, such as a circle and a parabola, and showed that the length he wanted could be measured where they cross. He found no rule by calculation alone, and hoped someone after him would. One was published in Italy in 1545, by Gerolamo Cardano.

For centuries after al-Khwārizmī, algebra was written in words. ʿUmar al-Khayyām wrote that way, and so did the Latin translations that carried the subject into Europe. Symbols came slowly. In 1557 the Welsh mathematician Robert Recorde, tired of writing out “is equal to”, drew two parallel lines of one length instead, “bicause noe .2. thynges, can be moare equalle”. That was the equals sign.

In 1591 François Viète wrote letters for the known numbers in a problem as well as for the unknown, so that one line could stand for a whole family of problems. In 1637 Descartes fixed the habit still used today: the last letters of the alphabet, x, y and z, for unknowns, and the first, a, b and c, for known amounts.

Letters obey the same rules as numbers, and two of those rules do most of the everyday work. The first: 3x + 2x is 5x, three of something and two more of the same thing. But 3x + 2y cannot be squeezed into one term, because x and y may stand for different numbers, like apples and oranges. Only like terms, with the same letter raised to the same power, combine.

The second is the distributive law. A number outside a bracket multiplies everything inside it: 2(x + 3) is 2x + 6. Mental arithmetic uses it all the time, since 2 × 13 is 2 × 10 plus 2 × 3, which is 26. Together the two rules let a long expression shrink to a short one, so that 3(x + 2) + 4x becomes 7x + 6.

In 2x + 3 = 35 the letter x is an unknown: one fixed number, waiting to be found. Letters have a second job. A bus ticket costs 5, so n tickets cost 5n. Here n is not hiding a single number. It can be 1, 2 or 20, and the cost changes with it. A letter that is free to take many values is a variable.

A rule linking variables can be drawn. Plot the number of tickets across and the total cost up, and the points fall on a straight line, rising by 5 for every extra ticket. A straight line rising from zero is the mark of this kind of relationship, where one amount grows in step with another: double the tickets and you double the cost.

A formula is an equation that links several quantities, so that knowing all but one gives the last. Distance = speed × time is one. It can be rearranged by the same rules of balance: divide both sides by speed and you get time = distance ÷ speed. One line now answers three questions. A 600-kilometre drive at a steady 50 kilometres an hour takes 600 ÷ 50, which is 12 hours.

Rearranged, a formula also shows how quantities pull on each other. Keep the distance fixed and raise the speed, and the time falls: double the speed and the trip takes half as long, so 100 kilometres an hour takes 6 hours. Plotted, time against speed is a curve that falls steeply and then flattens.

Why bother with letters when numbers will do? Because a letter can stand for every number at once. Add two odd numbers: 3 + 5 is 8, 7 + 9 is 16, 11 + 13 is 24, all even. Three examples, or three thousand, still leave the next case unchecked.

Now use letters. Any odd number is one more than an even number, so it can be written 2a + 1, where a is a whole number. Two odd numbers are 2a + 1 and 2b + 1. Their sum is 2a + 2b + 2, which is 2(a + b + 1): two times a whole number, so even. Because a and b could be any whole numbers, this one short argument covers all cases at once. An argument that shows a claim must hold is a proof, and letters are what let a proof speak about every number.