PYTHAGORAS, IRRATIONAL NUMBER, AND PYTHAGORAS THEOREM

A. PYTHAGORAS

Pythagoras is a Greek mathematician at the same time ancient a philosopher for the 6th century. He is very influence for science especially in mathematics. One of his famous omission is Pythagoras Theorem which almost people have ever heard it. Pythagoras Theorem said that hypotenuse of right triangle is sum of square of 2nd other side from the right triangle. Because of his omissions in mathematics, he also called as “The Father of Number”.

One of the pupils which so called Hippasus said that √2 which is hypotenuse of isosceles triangle which having length each feet is 1 is the irrational number. However, Hippasus then murdered because Pythagoras cannot argue evidence raised by Hippasus.

Hippasus is a pupil of Pythagoras coming from Metapontum. He also a mathematician at the same time ancient Greek philosopher about the 6th century. He considered to be inventor of irrational number, especially prove that square root of 2 √2 is irrational number. Ironically, the invention exactly cause the death. Pythagoras argue existence of irrational number. Pythagoras and the other pupils assumed that all nmber are rational number and there is no irrational number. Hippasus prove this theorem by using reductio ad absurdum (prove by contradiction) proving number that it is irrational number. Pythagoras cannot argue this statement and assume that Hippasus is errant teaching follower so that he set mind on to engulf Hippasus.

B. IRRATIONAL NUMBER

Irrational number is a real number which cannot divided (result for its have never desisted). In this case, irrational number cannot expressed as a/b, with a and b as integer, and b unlike null. So irrational number is not rational number. The example for irrational number such as π, √2, and e number. Phi number (π) which during the time we recognizing, actually imprecise 3,14 but 3,1415926535897932…. Also √2 number which if we formulate becoming 1,41421356237309504880….And e number that is 2,71828182….

The irrational number can be proved with using reductio ad absurdum or in english called proof by contradiction. It is logic argument started with an assumption, then from the assumption found an absurd result, illogical, or contradictive, so the conclusion of the assumption will to be wrong valuable and the deny will become correct valuable. A mathematical statement sometime be proved by reductio ad absurdum that is by assuming the deny (negation) from the statement which will be proved, then from the assumption degraded a contradiction. When contradiction reachable logically, then the assumption have proven fault, so that the statement is correct.

Proved by contradiction or reductio ad absurdum is not wrong argument, but if done truly will be valid argument. If prove by contradiction yield a mistake, the mistake lay at the process degradation of contradiction, not at may of the prove.

The classical example for the prove by contradiction at ancient Greek era is prove that square root of two is irrational number (cannot expressed as comparison of integer). This statement is provable by the way of assuming on the contrary that 2 is rational number, so that can expressed as comparison of integer a/b in simplest fraction. But if a/b = √2, so a2 = 2b2.It means a2 is even number. Because square from odd number is not possibly even, then a is even number. Because a/b is simplest fraction, b surely anomalous (because fraction of even/even number still can be made moderate). But because a is even number (assume 2r = a, mean a2 = 4r2) is fold number of 4, and b2 is fold number of 2 (even). This mean b is also even number, and this is contradiction to conclusion before all that b surely anomalous. Because assumption early that 2 is rational number result the contradiction, the assumption surely wrong, and the deny (that 2 is irrational) is correct statement.

C. PYTHAGORAS THEOREM

One of the Pythagoras omission that very popular is Pythagoras Theorem. The theorem called as the ancient Greek mathematician and philosopher, he is Pythagoras. Pythagoras is not the inventor of the theorem but he is the first people who proved the truth of the theorem so he given appreciation with give name the theorem like his name.

This theorem express that summing up wide squares at foots a right triangles equal broadly squares in hypotenuses. Right triangle is triangle having a right angle (90o0); the foots are two sides which the bevels angular shapes, and hypotenuse is third side dealing with the right angle. The formula of this theorem is a2+ b2 = c2, where a and b is the sides of the right triangle, and c is the hypotenuse.