MathDB

Suppose that

a

is an irrational number.
(a) If there is a real number

b

such that both

(a+b)

and

ab

are rational numbers, show that

a

is a quadratic surd. (

a

is a quadratic surd if it is of the form

r+\sqrt{s}

r-\sqrt{s}

for some rationals

r

and

s

, where

s

is not the square of a rational number).
(b) Show that there are two real numbers

b_1

and

b_2

such that
i)

a+b_1

is rational but

ab_1

is irrational.
ii)

a+b_2

is irrational but

ab_2

is rational. (Hint: Consider the two cases, where

a

is a quadratic surd and

a

is not a quadratic surd, separately).

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