week18 - Constructible Numbers, Fields and Surds Original...

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Constructible Numbers, Fields and Surds Original Notes adopted from February 5, 2002 (Week 18) © P. Rosenthal , MAT246Y1, University of Toronto, Department of Mathematics typed by A. Ku Ong Constructible Numbers If a,b,c are constructible & > 0, if b<c c = x b a x = ac/b if b> c b = a c x x = ac/b So can construct ac/b for a,b,c positive constructed numbers. In particular, take b =1, shows can construct the product of any two constructible positive numbers. Take c =1, show can construct quotient of any two constructible positive numbers. Let C = set of all constructible numbers. If x C, -x C.
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0,1 C If a,b, C, so is a +b. Definition: A subset F of R is a number field if 1) 0,1 F 2) If x,y F, so are x + y & x * y. 3) If x F, so is –x. 4) If x F & x 0, then 1/x F. Above we showed: C is a number field. C Q Eg. R,Q are number fields Q( 2) is defined to be {a + b 2: a,b Q} Obviously Properties 1,3,4 hold Property 2: (a + b ¥ 2 )(c + d ¥ 2 ) = ac + 2bd + (bc + ad) ¥ 2 Q( ¥ 2) Property 4: 1 * a-b ¥ 2 = a-b ¥ 2 = a + -b ¥ 2 a +b ¥ 2 a-b ¥ 2 a 2 – 2b 2 a 2 – 2b 2 a 2 – 2b 2 If a 2 – 2b 2 = 0 a 2 – 2b 2 (a/b) 2 = 2 ¥ 2 rational, contradiction. If a,b not both 0, 1/a+b
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week18 - Constructible Numbers, Fields and Surds Original...

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