Question

(3) (a) Prove that, between any two rational numbers, there is an irrational number (b) Prove that, between any two irrational numbers, there is a rational number

Answer 1

0 Between any two unequal rational hu. there is an inational, Suppose beq are two rational hot with p<q,, we want to find an inational no blw them. , het n be an integer large enough so that - n(9-b) >2 , we can always find such NEN Simply choose "n=10 kt I, where k= 1 it 9-63, 1 t otherwise choose k to be larger than the zero numbers af zero after the Desimal point before the first hon Zeno no. so Then, we note between nq & np. there must be furg Integer at least call them atb a cb. Now consider the no. z=53-1+ a. so His no z we between afb, This also between 1. nqt np., Now, we just need to show he is inational. Subhore / is rational then By Definition of nations of Z Z = (53 -1 +a)= e By Definition of rational Now for some integer efk.

53-1+0 = nel » $3 = m +(2-a) = netkoka y so ti rectional as nolikta as netk-kace 9 netkoka see ke which it contradicetion , as we knew tect 13 is an inational so our assumption was Wrong - 7 is sweational. so ) zie irational es nella 2) npa zanq e zie inational e 7 is inational = p< Z < q peq for rational so any ; } an irational

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