Question

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

Answer #1

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

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)

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

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 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.)

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

Consider a number line. How many rational numbers are there?
How many irrational numbers? How are they separated? This is a very
abstract concept, but will help when considering the graph.
What is the definition of a rational number? Of an irrational
number?

consider the statement "The quotient of any two
rational numbers is a rational number."
A. Determine whether the statement is true or false if
true prove the statement directly fron the definitions and if false
provide a counter example
B.In case the statement is false, determined whether a
small change would make it true. if so, make the change and prove
the new statement
C. Follow the directions for writing proofs on pg
154-6

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