Quadratic Equations Starting with the Chinese in 2000 BC

What they knew The area of a square is the square of the lengths of its sides. If they multiplied the lengths of the sides of a square hay loft by 3 the area would be multiplied by 9. How to work out the area of compound shapes such As rectangles and T shapes.

What they didn’t know How to work out the lengths of the sides of a square if they knew its area So if somebody went to the builder knowing what area he needed to store his hay the builder couldn’t work out how long to make the sides of the loft

The Egyptians 1500 BC The Egyptians did not have a formula for working out the lengths of the sides if they knew its area. They made a table which showed the area for all the possible outcomes. If someone wanted a space with a certain area, they would look in their table

What the Egyptian area table might have looked like:

THE BABYLONIANS The Babylonians used a base 60 number system which meant that they could check their calculations and produce more accurate tables than the Egyptians. By 400BC they had found a method called ‘completing the square’ to solve problems involving areas.

Mediterranean information Pythagoras (500bc in Italy ) and Euclid(300bc in Egypt) used geometry to solve the Quadratic equations. Pythagoras noted that the ratio of the area of a square and the length of its side (the square root) was not always a whole number but he refused to accept any that were not rational. Euclid however, accepted the existence of irrational numbers

The architects, builders and engineers of the day were still looking for a way to find the square root of any number so that they could make their buildings the right size. Euclid’s big work called Elements gave the theory but he didn’t use the same notation as we do today and it still wasn’t possible to find a square root.

Mediterranean's and Quadratic equations 300bc

Indian maths Hindu maths has used the decimal system since 600AD. Hindu maths was strongly influenced by the commercial world and the average Hindu merchant was quite fast at simple maths. The numbers would be negative if people had debts and the positive if someone had credits. Zero is an important number in mathematics and the Hindus were amongst the first to accept its existence.

Indian maths continued Around 700AD the general solution for the quadratic equation (using numbers) was devised, by a Hindu mathematician called Brahmagupta, who used irrational numbers. He also recognised 2 roots in the solution. The final complete solution, that we know, came around 1100AD by another Hindu mathematician called Baskara. He was the first to recognise that any positive number has 2 square roots.

Persian Mathematics This part of our presentation is to inform of the exciting history of Persian quadratics and maths.

The History of the Persian Empire Persia is now known as Iran and is situated next to Iraq and Afghanistan. At the peak of its time, its empire stretched from Greece to Egypt and India.

Persian Mathematicians Around 820AD, near Baghdad, Mohammad bin Musa Al-Khwarismi, a famous Islamic mathematician and ‘the father of algebra’ also derived the quadratic equation. But he rejected negative solutions. This derivation of the quadratic equation was brought to Europe by a Jewish mathematician called Abraham bar Hiyya. He lived in Barcelona in around 1100AD.

1500AD The renaissance in Europe By 1545 Girolamo Cardano blended Al-Khwarismi’s solution with Euclidean Geometry to come up with a solution for a quadratic equation. He allowed for the existence of complex or imaginary numbers which involve the square root of negative numbers. In 1637 when Rene Descartes published La Geometrie, modern mathematics was born and the quadratic formula appeared in the form that we know today!