Numbers: Irrational Numbers/ Greek Mathematics

Then the class looked at irrational, rational, and real numbers.

Rational numbers are ratios of integers. Irrational numbers are defined as all real numbers that are not rational. Real numbers, all occurring distances and their negatives, have 

Rational numbers have terminating or periodic representation, irrational numbers have (non-periodic) infinite representations. For example the number: not rational because it is neither terminating nor periodic. The number 0.1234567891011121314151617181920212223... is irrational for the same reason.

The class then moved on to discuss the Pythagoreans, who believed that “Everything is numbers” (everything is rational).

The class then learned about Hippasos (a member of the group) who in around 450 B.C. discovered that not everything is rational and disproved the Pythagoreans because he found that the square root of two is not rational.

Proof that sqrt(2) is irrational

We began the class with a discussion of how different mathematical statements could be shown. To prove that a number, say 1.7, is rational, all you have to give is its representation as a fraction, say 17/10. Proving that a number, say sqrt(2) is not rational requires usually much more effort.

This proof was discussed. It uses the so-called "Reductio ad Absurdum" format to assume the opposite. Assume that the square root of 2 is rational. We know that there are integers a and b such that the sqrt(2)= a/b. We also assume that this ratio is reduced to lowest terms, meaning that a and b cannot both be even.

Squaring both sides of the equation we get 2 = a2/b2. We multiply both sides of the equation by b2 to obtain 2b2 = a2. Then a2 must be an even number. Using the fact that squares of even numbers are even and squares of odd numbers are odd (something that could be shown rather easily, but wasn't done in class). a too must then be even. By definition of the word "even" that means that a=2c for some natural number c. Substituting this into the equation (2b2=a2) above, we obtain 2b2=4c2, and therefore b2=2c2. So b2 must be even, making b even as well (by the same fact used above). Then both a and b are even. This is a contradiction.

Since the assumption (that sqrt(2) is rational) ends in a contradiction, it must be false. Therefore sqrt(2) is irrational.

We also discussed the difference between a theorem and a proof. A theorem is a statement. A proof is something to convince us that the statement is true.

Hippasos' Proof of the irrationality of the golden ratio

This geometrical proof of the irrationality of the golden ratio F = (1+sqrt(5))/2 was sketched only briefly. Taking a regular pentagon of side length 1 , the diagonals within have length F . Obviously this side also has length 1, therefore if we subtract it from F we get F-1. . If we subtract F-1 from 1 we get 1-(F-1) =2-F . Since we applied the geometrical version of Euclid's algorithm, every common unit of the two starting red and blue lines 1 and F is also a common unit of the smaller red and blue lines 2-F and F-1. But these are side and diagonal of the smaller, inscribed pentagon . Proceeding we get smaller and smaller red and blue lines, but since the process is not ending, we don't have a common unit.

A modern, but geometrical proof of the irrationality of the square root of 2.

see the link here

Greek Mathematics

Egyptians had some, and Babylonians (Mesopotamia, nowadays Iraq) had an amazing knowledge of mathematics already between 2000BC and 600BC. For instance, Babylonians had some insight into solving quadratic equations by completing the square (although they lacked the notion of an equation and they didn't have negative numbers either) and also knew what is nowadays called the Theorem of Pythagoras. What neither of these cultures established was a solid foundation of their "truths". Even though the creator of these formulas may had derivations that convinced themselves, they did no effort to convince others why these formulas are true. In other words, they lacked the concept of a proof. Greeks... deductive instead of incductive reasoning.

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