The history of equation solving

Equations in our sense are expressions of the form

A = B where A and B are polynomials in one variable x.

• Linear equations: 3x = 5, 3x+2 = 4x-5

• Quadratic equations: x2 = 9, x2–3x = 2x+4

• cubic, quartic, higher order equations (with higher powers of x) Linear equations

• As we have seen, solving linear equations (and even simple systems of linear equations) go back to at least the Babylonian era. General systems of two equations in two variables could be solved by Chinese mathematicians around 200 BCE.

• An example problem: “I found a stone, but did not weigh it; after I added one-seventh and then one-eleventh [of the total], it weighed 60 gin. What was the original weight of the stone?” (Yale Babylonian Collection #4652)

• Thus, (x+x/7) + (x+x/7)/11 = 60.

• We would solve: (1+1/7+1/11+1/77)x = 60 => 96/77x=60 => x = 48 1/8

• But this requires equation notation and factoring out coefficients of x, which was not a method known then. They would use the “method of surplus and deficit” or “false position”:

• Guess that x+x/7 = 11 (because that’s nicely divisible by 11)

• But x+x/7 + (x+x/7)/11 = 12, not 60, so increase the guess by the factor 60/12 = 5, that is, x+x/7 = 55.

• To solve this, repeat the process and guess x=7. Then x+x/7 = 8, not 55, so multiply by 55/8 to give x=48 1/8. Quadratic equations

• The Babylonians also solved quadratic equations by geometric means (by cutting and pasting rectangles).

• For square roots, they had extensive tables. They set √2 = 1 + 25/60 = 1.41666…

• They could solve x2+bx = c in general but had different methods depending on b>0 or b<0 (“completing the square”)

• Additional complications arose because negative solutions did not make sense to the Babylonians.

• So this was a deep and involved problem to them. Quadratic equations in Diophantus’s Arithmetica (after 150 CE)

• Greek mathematics was overwhelmingly geometric and had no room for quadratic equations. An exception is Diophantus, who for instance was interested in how to write a square as a sum of two squares (such as 32+42=52).

• He solved an example (x2+y2=16) by what we would call an ad-hoc method (assuming y=2x-4).

• Yet this text inspired future readers, in particular Fermat. Al-Khwarizmi’s Al-jabr (ca 825 CE)

• Records of equation solving in Arabia stop around 200 BCE and only reappear in an important treatise: “Al-kitab almukhtasar fi hisab al-jabr w’al- muqabala (The book on restoration and balancing)”

• The methods are not all that different (completing squares) but presented more abstractly and more generally, pointing out the possibility of no solutions, one positive solution, or two positive solutions, and degenerate cases. The language also now contained words such as “add”, “subtract”, “divide”. Negative numbers and manipulation of equations were still not in use, so al-Khwarizmi classified 6 types of quadratic equations:

1. squares equal to roots (x2 = bx) 2. squares equal to numbers (x2 = c) 3. roots equal to numbers (bx = c) 4. squares and roots equal to numbers (x2+bx = c) 5. squares and numbers equal to roots (x2+c = bx) 6. roots and numbers equal to squares (x2 = bx+c) And now for something quite different

When the cube with the things next after Quando chel cubo con le cose apresso Together equal some number apart Se aguaglia à qualche numero discreto Find two others that by this differ Trovan dui altri differenti in esso And this you will then keep as a rule Dapoi terrai questo per consueto

That their product will always be equal Ch’el lor’ produtto sempre fia equale To a third cubed of the number of things Al terzo cubo delle cose neto The difference then in general between El residuo poi suo generale The sides of the cubes subtracted well Delli lor lati cubi ben sottratti Will be your principal thing. Varra la tua cosa principale.

N. Tartaglia to G. Cardano, 1539 (“Ars magna”) Higher order equations

• Cardano’s student Ferrari solved the general fourth-order equation. While Cardano introduced negative numbers into European mathematics, Rafael Bombelli, some 20 years later, introduced complex numbers to deal with lacking solutions of cubic equations.

• Fifth-order (quintic) and higher equations could not be solved, and the problem endured another 350 years until Niels Henrik Abel and Évariste Galois (ca 1820-30), who showed that then general quintic is unsolvable with radicals resp. which kinds of equations can be solved with radicals.