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

- terminating, like 12.453,
- periodic, like 1/3 = 0.3, 15/56 = 0.2678571428, or
- (non-periodic) infinite, like 1.1414213562 ... with unpredictable "..."

Rational numbers have terminating or periodic representation, irrational numbers have (non-periodic) infinite representations. For example the number:

0.101001000100001000001000000100000001000000001...is 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.

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 = a 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.

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.

- Thales of Miletus, around 600BC
- Hippocrates of Chios gives the first proof. (about 440BC)
- Theodorus of Cyrene shows the irrationality of the square roots of 3, 5, 7, 11, 13, 17.
- Somewhere between somewhere discovered the irrationality of the square root of 2.
- Theaetetus and the five (now called Platonic) solids.
- Hippasus of Metapontum (about 450BC)
- Plato
- Aristotle (384-322BC)
- Euclid
- Archimedes

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