Sep 11, 2017 | Pflugerville

For those students in high school, and even some younger, we are familiar with the quadratic formula, "the opposite of b, plus or minus the square root of b squared minus 4ac, all divided by 2a". This formula allows you to find the root of quadratic equations of the form: ax^{2} + bx + c = 0.

Where did this formula come from? Why did older civilizations need to solve equations of this form in the first place? The following article, taken from h2g2, explores the origins of this famous formula.

Original article: https://h2g2.com/approved_entry/A2982567

This is the quadratic formula, as it is taught to most of us in school:

*x*_{1,2}=(*-b*/2*a*) ± (1/2*a*)(*b*^{2}-4*ac*)^{1/2}

gives the solution to a generic quadratic equation of the form:

*ax*^{2} + *bx* + *c* = 0

The development, or derivation, of a mathematical idea is usually as logical, deducible and rectilinear as possible. This brings about the common notion that its historical development is similarly as continuous, logical and rectilinear: one mathematician picking up an idea where another mathematician left it.

Using the quadratic formula as an example, it will be shown that the historical development of mathematics is not at all rectilinear. Instead, parallel developments, interconnections and confluences can be found, which - to complicate this stuff even further - are also interrelated with social, cultural, political and religious matters.

The so-called quadratic formula has been derived in the course of a few millennia to its current form, which is taught to most of us in school. This Entry will strictly concentrate on the historical development of the quadratic formula. Some mathematical background may be of use to fully understand the described development, however the maths used in this Entry will be kept at a necessary minimum.

Egyptian, Chinese and Babylonian engineers were really smart people - they knew how the area of a square scales with the length of its side. They knew that it's possible to store nine times more bales of hay if the side of the square loft is tripled. They also found out how to calculate the area of more complex designs like rectangles and T-shapes and so on. However, they didn't know how to calculate the sides of the shapes - the length of the sides, starting from a given area - which was often what their clients really needed. And so, this is the original problem: a certain shape^{1} must be scaled with a total area, and in the end what's needed is lengths of the sides, or walls to make a working floor plan.

The first aspect that finally led to the quadratic equation was the recognition that it is connected to a very pragmatic problem, which in its turn demanded a 'quick and dirty' solution. We have to note, in this context, that Egyptian mathematics did not know equations and numbers like we do nowadays; it is instead descriptive, rhetorical and sometimes very hard to follow. It is known that the Egyptian wisemen (engineers, scribes and priests) were aware of this shortcoming - but they came up with a way to circumvent this problem: instead of learning an operation, or a formula that could calculate the sides from the area, they calculated the area for all possible sides and shapes of squares and rectangles and made a look-up table. This method works much like we learn the multiplication tables by heart in school instead of doing the operation proper.

So, if someone wanted a loft with a certain shape and a certain capacity to store bales of papyrus, the engineer would go to his table and find the most fitting design. The engineers did not have time to calculate all shapes and sides to make their own table. Instead, the table they used was a reproduction of a master look-up table. The copyists did not know if the stuff they were copying made sense or not as they didn't know anything about maths. So, obviously, sometimes errors crept in, and copies of the copies were known to be less trustworthy^{2}. These tables still exist, and it is possible to see where errors crept in during the copying of the documents.

The Egyptian method worked fine, but a more general solution - without the need for tables - seemed desirable. That's where the Babylonian geeks come into play. Babylonian maths had a big advantage over the one used in Egypt, namely they used a number-system that is pretty much like the one we use today, albeit on a hexagesimal basis, or base-60. Addition and multiplication were a lot easier to perform with this system, so the engineers around 1000 BC could always double-check the values in their tables. By 400 BC they found a more general method called 'completing the square' to solve generic problems involving areas. There are no indications that these people used a specific mathematical procedure to find out the solutions, so probably some educated guessing was involved. Around the same time, or a bit later, this method also appears in Chinese documents. The Chinese, like the Egyptians, also did not use a numeric system, but a double checking of simple mathematical operations was made astonishingly easy by the widespread use of the abacus.

The first attempts to find a more general formula to solve quadratic equations can be tracked back to geometry (and trigonometry) top-bananas Pythagoras (500 BC in Croton, Italy) and Euclid (300 BC in Alexandria, Egypt), who used a strictly geometric approach, and found a general procedure to solve the quadratic equation. Pythagoras noted that the ratios between the area of a square and the respective length of the side - the square root - were not always integer, but he refused to allow for proportions other than rational. Euclid went even further and found out that this proportion might also not be rational. He concluded that irrational numbersexist.

Euclid's opus *Elements* covered more or less all the mathematics needed for technical applications from a theoretical point of view. However, it didn't use the same notation with formulas and numbers like we use nowadays. For that reason it was not possible to calculate the square root of any number by hand, in order to obtain a good approximation for the exact value of the root, which is what the architects and engineers were after. Because all (theoretically relevant at least) maths seemed to be complete^{3} but otherwise useless, the many wars occurring in Europe, and also the early Middle Ages turned the mathematical world in Europe silent until the 13th Century. In this period mathematics also suffered a big shift, going from a pragmatic science to a more mystical, philosophical discipline.

Hindu mathematics has used the decimal system (the one we use) at least since 600AD. One of the most important influences on Hindu mathematics was that it was widely used in commerce. The average Hindu merchant was pretty fast in simple maths. If someone had a debt the numbers would be negative, if someone had a credit the numbers would be positive. Also, if someone had neither credit, nor debt, the numbers would add up to zero. Zero is an important number in the history of mathematics, and its relatively late appearance is due to the fact that many cultures had difficulty of conceiving 'nothing'. The concept of 'nothing', like in 'shunya', the void, or the concept of 'equilibrium', was already anchored in Hindu culture.

Around 700AD the general solution for the quadratic equation, this time using numbers, was devised by a Hindu mathematician called Brahmagupta, who, among other things, used irrational numbers; he also recognised two roots in the solution. The final, complete solution as we know it today came around 1100AD, by another Hindu mathematician called Baskhara^{4}. Baskhara was the first to recognise that any positive number has two square roots.

Around 820AD, near Baghdad, Mohammad bin Musa Al-Khwarismi, a famous Islamic mathematician^{5} who knew Hindu mathematics, also derived the quadratic equation. The algebra used by him was entirely rhetorical, and he rejected negative solutions. This particular derivation of the quadratic formula was brought to Europe by Jewish mathematician/astronomer Abraham bar Hiyya (whose Latinised name is Savasorda) who lived in Barcelona around 1100.

With the Renaissance in Europe, academic attention came back to original mathematical problems. By 1545 Girolamo Cardano, who was a typical Renaissance scientist (ie, interested in alchemy, occultism and suchlike), and one of the best algebraists of his time, compiled the works related to the quadratic equations - that is, he blended Al-Khwarismi's solution with the Euclidean geometry. He was possibly not the first or only one, but the most famous. In his (mainly rhetorical) works he allows for the existence of complex, or imaginary numbers - that is, roots of negative numbers. At the end of the 16th Century the mathematical notation and symbolism was introduced by amateur-mathematician François Viète, in France. In 1637, when René Descartes published *La Géométrie*, modern Mathematics was born, and the quadratic formula has adopted the form we know today.

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