Rational and Irrational Numbers © 2004 All rights reserved 5 7 2 1

Rational and Irrational Numbers A rational number is any number that can be expressed as the ratio of two integers. All terminating and repeating decimals can be expressed in this way so they are rational numbers. Examples 4 5 2 3 = 8 6 = 1 -3 = 3 1 - 2.7 = 27 10 0.7 = 7 10 0.625 = 5 8 34.56 = 3456 100 0.3 = 1 3 0.27 = 3 11 0.142857 = 1 7

Rational and Irrational Numbers An irrational number is any number that cannot be expressed as the ratio of two integers. 1 Pythagoras The history of irrational numbers begins with a discovery by the Pythagorean School in ancient Greece. A member of the school discovered that the diagonal of a unit square could not be expressed as the ratio of any two whole numbers. The motto of the school was “All is Number” (by which they meant whole numbers). Pythagoras believed in the absoluteness of whole numbers and could not accept the discovery. The member of the group that made it was Hippasus and he was sentenced to death by drowning. (See slide 19/20 for more history)

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Rational and Irrational Numbers An irrational number is any number that cannot be expressed as the ratio of two integers. Intuition alone may convince you that all points on the “Real Number” line can be constructed from just the infinite set of rational numbers, after all between any two rational numbers we can always find another. It took mathematicians hundreds of years to show that the majority of Real Numbers are in fact irrational. The rationals and irrationals are needed together in order to complete the continuum that is the set of “Real Numbers”. 1 Pythagoras

Rational and Irrational Numbers Example questions Show that is rational a rational Show that is rational b rational

Rational and Irrational Numbers Questions State whether each of the following are rational or irrational. a b c d irrational rational rational irrational e f g h irrational rational rational rational

Rational and Irrational Numbers Combining Rationals and Irrationals Addition and subtraction of an integer to an irrational number gives another irrational number, as does multiplication and division. Examples of irrationals

Rational and Irrational Numbers Combining Rationals and Irrationals Multiplication and division of an irrational number by another irrational can often lead to a rational number. Examples of Rationals 21 26 8 1 -13

Rational and Irrational Numbers Combining Rationals and Irrationals Determine whether the following are rational or irrational. (a) 0.73 (b) (c) 0.666…. (d) 3.142 (e) rational irrational rational rational irrational (f) (g) (h) (i) (j) irrational irrational rational rational irrational (j) (k) (l) irrational rational rational

History The Pythagoreans Pentagram Pythagoras was a semi-mystical figure who was born on the Island of Samos in the Eastern Aegean in about 570 B.C. He travelled extensively throughout Egypt, Mesopotamia and India absorbing much mathematics and mysticism. He eventually settled in the Greek town of Crotona in southern Italy. He founded a secretive and scholarly society there that become known as the “Pythagorean Brotherhood”. It was a mystical almost religious society devoted to the study of Philosophy, Science and Mathematics. Their work was based on the belief that all natural phenomena could be explained by reference to whole numbers or ratios of whole numbers. Their motto became “All is Number”. They were successful in understanding the mathematical principals behind music. By examining the vibrations of a single string they discovered that harmonious tones only occurred when the string was fixed at points along its length that were ratios of whole numbers. For instance when a string is fixed 1/2 way along its length and plucked, a tone is produced that is 1 octave higher and in harmony with the original. Harmonious tones are produced when the string is fixed at distances such as 1/3, 1/4, 1/5, 2/3 and 3/4 of the way along its length. By fixing the string at points along its length that were not a simple fraction, a note is produced that is not in harmony with the other tones. Pentagram Pythagoras Spirit Water Air Earth Fire History

Proof The Square Root of 2 is Irrational 1 This is a “reductio-ad-absurdum” proof. To prove that is irrational Assume the contrary: 2 is rational That is, there exist integers p and q with no common factors such that: (Since 2q2 is even, p2 is even so p even) (odd2 = odd) So p = 2k for some k. (Since p is even is even, q2 is even so q is even) So q = 2m for some m. This contradicts the original assumption. Proof is irrational. QED