An Algebraic Proof for the Quadratic Equation - also a bit about Pythagoras, Fermat's Last Theorem, Archimede's Eureka, and Trigonometry: SOH CAH TOA.
The Quadratic Equation has many uses in Engineering and Mathematics. It can be used to represent Complex Numbers which are based on the Square Root of -1 :
Ö-1 which has no meaning in reality as both 1 x 1 and -1 x -1 yield 1! It can however be used to represent a two dimensional Cartesian number with the real component being the X-axis, and the imaginary component Ö-1 being the Y-Axis. The Square Root of -1 is known a j by Engineers and i by Mathematicians. With j x j being -1 by definition. One use for Complex Numbers is Robotic Servo-motor theory. Cartesian geometry is named after the French Philosopher and Mathematician René Descartes (1596-1650). He also deduced the Philosophical Axiom: 'I think there for I am' (Latin: Cogito ergo sum; French: Je pense, donc je suis).
-b ± Öb² - 4ac
x =2a Multiply both sides by 2a giving:
2ax = -b ± Öb² - 4ac Add b to both sides giving:
2ax + b = ± Öb² - 4ac Square both sides, remembering (+x)² and (-x)² both
give x², thus giving:
(2ax + b)² = b² - 4ac Multiplying out the Algebraic square gives:
(2ax + b)(2ax + b) = b² - 4ac
4a²x² + 2.2axb + b² = b² - 4ac
Note . is short for Multiply. Next subtract b² from both sides (note
the b² on each side cancels the other out) to give:
4a²x² + 4axb = - 4ac Divide both sides by 4a giving:
ax² + xb = - c Add c to both sides giving:
ax² + bx + c = 0 Which gives the final proof!
Of course the original Quadratic Equation had to be solved in the reverse order to this. So some mathematician had to think to multiply by 4a and add b² to both sides. This isn't as difficult as it might appear as this mathematician would know a Algebraic square would be needed to isolate x, remember 2x2 = 4 and bxb = b². If you ever have to remember the Quadratic Equation for an exam knowing how it is derived will help jog your memory and allow you to check if you remembered it correctly.
The simplest form of Algebra was originally developed by an Arabic scholar in Baghdad in the early 800's. He was called Al-Khowarizmi, the word algorithm is derived from his name. The word algebra is derived from the first words of his most well known book Al Jabr Wa'l Muqabalab. His work used earlier concepts such as Hindu symbols, Mesopotamian mathematics and Euclid's geometry. '0' and the highly important decimal system were developed in India. Try multiplying two Roman Numerals together!
Every school child is taught Pythagoras's (582?-500? B.C.) triangle where by the sum of the squares adjacent and opposite side of a triangle equal the square of the hypotenuse. All in a time before the decimal system. Pythagoras actually didn't invent this theorem it probably goes back to the ancient Egyptians (perhaps used in their calculations of making a Pyramid
exactly square) or the Babylonians (modern day Iraq who used a number system based on 60 hence: 60 seconds, 60 minutes and 6 x 60 degrees. The commonly used integer examples being 3˛ + 4˛ = 5˛ and 5˛ + 12˛ = 13˛. A common formulae for this is [(n + 1)˛ - n˛]˛ + n˛ = (n + 1)˛ of course this may not cover all examples. I also worked out a non-integer example myself 1˛ + [SQRT (24)]˛ = 5˛.
Fermat's Last Theorem
Going on naturally from this Mathematicians had to consider whether there were any solutions for the next power up of the equation: a³ + b³ = c³ for integer values only. No solution was readily found and many Mathematicians spent years trying to solve this apparently simple puzzle. Some discovering new mathematical theories as a result of their research.
"The 17th-century mathematician Pierre de Fermat wrote about this in 1637 in his copy of Claude-Gaspar Bachet's translation of the famous Arithmetica of Diophantus: 'I have a truly marvelous proof of this proposition which this margin is too narrow to contain.' (Original Latin: 'Cuius rei demonstrationem mirabilem sane detexi. Hanc marginis exiguitas non caperet.') However, no correct proof was found for 357 years, until it was finally proven using very deep methods by Andrew Wiles in 1995."
Pythagoras actually belonged to a religion called the cult of numbers as did Archimedes of Eureka fame ('I've found it'). He proved a gold smith had swindled the King of Syracuse, a powerful Greek Colony in the Island of Sciliy, by submerging his newly mad crown in water with the displaced water giving it's volume. An idea he got from having a bath. Gold of course weighs more than Silver so he was able to determine the weight was wrong from the volume. He also built siege engines to hold of the Romans for 2 years. And magnifying glass to burn the sails of their Trireme Ships. He discovered the volume of a sphere (also before the decimal System): 4/3πr3. Which he had engraved on his gravestone. Their religion outlawed irrational numbers like Ö2 because they had no end!
SOH CAH TOA
Sin ø = Opposite / Hypotenuse
Cos ø = Adjacent / Hypotenuse
Tan ø = Opposite / Hypotenuse
Note you can work out all other relationships using Algebra:
e.g. Sin ø = Opposite / Hypotenuse -> Hypotenuse x Sin ø = Opposite OR
Hypotenuse = Opposite / Sin ø