irrational and rational numbers anyone please!?

Question

ok here goes..

does anyone have any idea how to show that 5/sqrt2 is irrational??

or to show that 0.451111 (with 1 infinitely repeating) is a rational number??

any help would be much appreciated!

thanks

Answer 1

Proof by contradiction:

Assume that 5/sqrt(2) is a rational number. {We want to prove that this cannot happen by proving a contradiction.} Then 5/sqrt(2) can be expressed as a quotient of integers. That is,

5/sqrt(2) = m/n, for m and n having no common factors {If they did have a common factor, we can reduce the fraction, so let's assume this fraction is in its reduced form.}

Squaring both sides,

[5/sqrt(2)]^2 = (m^2)/(n^2)

25/2 = (m^2)/(n^2)

Multiply both sides by 2n^2,

25n^2 = 2m^2

This implies n^2 is even, which means n is even. Therefore,
n can be expressed in the form 2k, for some integer k. That is

n = 2k. Therefore,

25[2k]^2 = 2m^2

25[4k^2] = 2m^2
100k^2 = 2m^2. Dividing both sides by 2,

50k^2 = m^2. Factoring a 2 out on the left hand side,

2(25k^2) = m^2.
Since m^2 can be expressed as 2 times something, it follows that m^2 is an even number, which means m is an even number.

Therefore, m and n are both even numbers. This is a contradiction (since we stated at the start that m and n have no common factors).

Therefore, 5/sqrt(2) is an irrational number.

Answer 2

The set of rational numbers is usually denoted by Mathematicians as script Q and is defined as the set of all numbers m/n where m is an integer and n is a nonzero integer.

some infinitely repeating numerals are not rational. others, like 1/3, 1.333333333... are; if they are rational, there will always be a repeating pattern resulting from the conversion of the divisor m (from n/m) to base 10.

I calculated your second example to rational expression. It's 406/900. Try it.

Here's how it's converted:

let m/n = 0.4511111...
m = .4511111... * n
100 m = 45.11111... *n
100 m = (45 + .111111...)*n
100m = 45n + n/9 # (1/9 = .11111...)
m = (45n + n/9)/100

use the value n(1) =9 to get a fractional m = 4.06;
multiply 4.06/9 by 100/100 to get the rational denotation.

you'll find the proof of the square root of 2 being non-rational here:
http://en.wikipedia.org/wiki/Irrational_number

from which it can be easily shown that 5/sqrt2 is also non-rational.

Answer 3

The proof that 5/sqrt2 is irrational would be similar to the proof that sqrt2 itself is irrational. This is proof by contradiction. You start by assuming that it is rational and equal to p/q in lowest terms. Then show that both p and q must be even numbers which contradicts previous statement. The only way to resolve this is to say that original assumption was incorrect.

Answer 4

sqrt2 is irrational since it is a non-terminating, non repeating decimal. you can not express it as a ration of two numbers. So any operation involving sqrt2 will give you result (if not zero) an irrational number.

0.451111... is a rational number.

you can express it as a ratio of two numbers:

ignore 0.45 in the calculation

a1 = 0.001
r=0.1

S = a1/(1-r)

S = 0.001/(1-0.1)

S = 0.001/0.9

S = 1/900

therefore, 0.451111...=45/100 + 1/900

= 406/900 ...a ratio

Answer 5

5 / √2
= (√25) / (√2)
= √12.5

Since 12.5 is not a perfect square, so 5/√2 is not a rational number.

0.45111111111111111111111111111
= 203/450

Since it can be expressed as a fraction, it is a rational number.

Answer 6

any number with a denominator having a square root sign is irrational numbers.. all others are not. :)

Edited:

here's some more info from google;

http://www.homeschoolmath.net/teaching/proof_square_root_2_irrational.php

Answer 7

x = .45111111111111111111111111111...
10x = 4.5111111111111111111111111111...

10x - x = 4.51 - .45
9x = 4.06
x = 203/450

Answer 8

x = .4511111111111111111111111111111111111
10x = 4.51111111111111111111111111111

10x - x = 4.51 - .45
9x = 4.06
x = 203/450