Question

If p is a rational # and q is an irrrational #

Is the number p/q rational or irrational and prove your answer.

Answer #1

If p is a rational number and q is an irrrational number
Is the number p/4 + q/5 rational or irrational and prove your
answer.

Prove that if p is a positive rational number, then √p + √2 is
irrational.

Prove the following: (By contradiction)
If p,q are rational numbers, with p<q, then there exists a
rational number x with p<x<q.

Let p and q be irrational numbers. Is p−q rational or
irrational?
Show all proof.

10. (a) Prove by contradiction that the sum of an irrational
number and a rational number must be irrational. (b) Prove that if
x is irrational, then −x is irrational. (c) Disprove: The sum of
any two positive irrational numbers is irrational

Prove by contradiction that 5√ 2 is an irrational number. (Hint:
Dividing a rational number by another rational number yields a
rational number.)

: Prove by contradiction that 5√ 2 is an irrational number.
(Hint: Dividing a rational number by another rational number yields
a rational number.)

1. Prove that the sum of any rational number with an irrational
number must be irrational.
2. Prove or disprove: If a,b, and c are integers such that
a|(bc), then a|b or a|c.

Irrational Numbers
(a) Prove that for every rational number µ > 0, there exists
an irrational number λ > 0 satisfying λ < µ.
(b) Prove that between every two distinct rational numbers there
is at least one irrational number. (Hint: You may find (a)
useful)

Prove that between any two rational numbers there is an
irrational number.

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