Squares and numbers equal to roots, e. Roots and numbers equal to squares, e. Al-Khwarizmi gives the rule for solving each type of equation, essentially the familiar quadratic formula given for a numerical example in each case, and then a proof for each example which is a geometrical completing the square.
A new phase of mathematics began in Italy around It was written by Luca Pacioli although it is quite hard to find the author's name on the book, Fra Luca appearing in small print but not on the title page. In many ways the book is more a summary of knowledge at the time and makes no major advances.
The notation and setting out of calculations is almost modern in style: 6. References show. It is based on al Khwarizmi's technique of completing the square. Share this post. Al Khwarizmi is often considered the father of algebra, due to an influential text he wrote, and his name is the origin of the term algorithm.
The usual quadratic formula is a consequence. Want to keep learning? Having already solved Q1, we rewrite this equation as. These are the two solutions. Once we know how to solve quadratics, Descartes theorem allows us a simpler way to factor a quadratic expression: just find its zeroes first, and then each of these will correspond to a linear factor!
You are now in a position to systematically solve all quadratic factorisation problems, without any guess work. Thank you al Khwarizmi! In such case, the quadratic formula needs to be used.
The quadratic formula can be used when the equation of a quadratic function is given in the standard form :. The quadratic formula is a rather useful and straightforward tool for solving quadratic equations. However, it is sometimes frustrating when you set up the formula and evaluate half of it only to find out that it cannot be evaluated further. The equation of the function has no solutions — the parabola has no x-intercepts see the diagram above.
This is when the Discriminant — the expression under the square root of the quadratic formula — comes to the rescue, offering more efficiency and clarity in terms of how many, if any, solutions to expect.
Real numbers are numbers that we can evaluate and determine knowing methods known to us and they produce quantities that correspond to what we experience in real life. 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 Addition and multiplication were a lot easier to perform with this system, so the engineers around BC could always double-check the values in their tables.
By 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 BC in Croton, Italy and Euclid 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 numbers exist. 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 AD. 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 AD 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 AD, by another Hindu mathematician called Baskhara 4. Baskhara was the first to recognise that any positive number has two square roots. The algebra used by him was entirely rhetorical, and he rejected negative solutions.
With the Renaissance in Europe, academic attention came back to original mathematical problems. By 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. Get Directions. Math Enrichment and Accelerated Programs in Pflugerville. Did you know Mathnasium offers enrichment programs for students?
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