Question

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)

Answer #1

Prove, that between any rational numbers there exists
an irrational number.

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

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.

Find two rational numbers and two irrational numbers between
1.41 and (square root of 2), Clearly identify which are rational
and which are irrational and explain in detail. Minimum of 1
paragraph..

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.

(1) Let x be a rational number and y be an irrational. Prove
that 2(y-x) is irrational
a) Briefly explain which proof method may be most appropriate to
prove this statement. For example either contradiction,
contraposition or direct proof
b) State how to start the proof and then complete the proof

The definition of a rational number is a number that can be
written with the form a/b with the fraction a/b being in lowest
form. Prove that √27 is an irrational number using a proof by
contradiction. You MUST use the approach described in class (and on
the supplemental material on cuLearn) and your solution MUST
include a lemma demonstrating that if ? 2 is divisible by 3 then ?
is divisible by 3. Hint: reduce √27 to the product...

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