Question

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

Answer #1

Lets prove this:

We have 2 ratios: m/n and a/b are rational if m,a∈︎ℤ , n,b∈︎ℕ\{0}.

Lets Assume both are postive rationals or the quotient of naturals where the denomiantor isn’t zero rather than integers.

2mb/2nb = m/n, 2an/2nb = a/b. But now we have common denominator and guaranteed even number in both numerator and denominator.

WLG let m/n < a/b

m/n = 2mb/2nb

< (2mb+1)/2nb ≤︎ (2an-1)/2nb <

2an/2nb = a/b

(since 2mb < 2an and both numbers are even, there is at least one odd number between them,

if exactly one odd exists, then 2mb+1 = 2an-1.)

Since the d/dx √︎(x²+1) < d/dx x for all x≥︎1,

2mb+1 > √︎((2mb)²+1) > 2mb (second part trivial)

So I assert that:

m/n < √︎(4m²b²+1)/2nb < a/b

and the middle term as the square root of a perfect square plus 1, (all divided by a rational), is clearly irrational.

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.

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.

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.

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.

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?

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..

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