4. 2010 LC Paper 2

8. ENTER Deductive Reasoning

24. Incommensurable Magnitudes (Irrational Numbers) 1 1 √ 2 The whole of Pythagorean mathematics and philosophy was based on the fact that any quantity or magnitude could always be expressed as a whole number or the ratio of whole numbers. Unit Square The discovery that the diagonal of a unit square could not be expressed in this way is reputed to have thrown the school into crisis, since it undermined some of their earlier theorems. Story has it that the member of the school who made the discovery was taken out to sea and drowned in an attempt to keep the bad news from other members of the school. He had discovered the first example of what we know today as irrational numbers .

25. 1 1 √ 2 1 It is possible to draw a whole series of lengths that are irrational by following the pattern in the diagram below and using Pythagoras’ Theorem. Continue the diagram to produce lengths of  √ 3, √ 5, √ 6, √ 7 , etc. See how many you can draw. You should get an interesting shape.

26. 1 1 √ 2 1 √ 3 1 √ 4 1 √ 5 1 √ 6 1 √ 7 1 √ 8 1 √ 9 1 √ 10 1 √ 11 1 √ 12 1 √ 13 1 √ 14 1 √ 15 1 √ 16 1 √ 17 1 √ 18

31. Check for understanding!

39. How to use method.

40. Example of use.