What is the difference between rational and irrational numbers?

Jan 2020
2
0
US
Yesterday, I was reading a post about 'Difference between rational and irrational numbers' and I was able to understand that
1. All the perfect squares are rational numbers
2. All the surds are irrational numbers
3. All the terminating decimals are rational numbers
4. All the repeating decimals are rational numbers
5. All the non-repeating decimals are irrational numbers
Can you guys provide some further information about the difference between rational and irrational numbers?
 

skeeter

Math Team
Jul 2011
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A rational number can be written in the form $\dfrac{a}{b}$, where $a$ and $b$ are integer values ... an irrational number cannot be represented by a ratio of integers.
 
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romsek

Math Team
Sep 2015
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A rational number can be written in the form $\dfrac{a}{b}$, where $a$ and $b$ are integer values ... an irrational number cannot be represented by a ratio of integers.
one small addition, $b \neq 0$
 
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Nov 2018
24
1
Iran
A rational number can be written in the form \(\displaystyle \frac{a}{b}\), where $a$ and $b$ are integer values ... an irrational number cannot be represented by a ratio of integers.
Do you have any idea about $$\frac {\sqrt{2}}{\sqrt{2}}=1 $$?
I've seen they say the numerator and the denominator must be an integer but in this special example
it seems to be a little incomplete maybe adding the condition that if $a=b$ then it's $$ a,b \in R$$
(just an idea).
 

romsek

Math Team
Sep 2015
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A rational number can be written as the ratio of two integers, the denominator being non-zero.
It doesn't have to be. $\dfrac{\pi}{\pi} =1$ too and $\pi$ is transcendental!
 
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Aug 2012
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Do you have any idea about $$\frac {\sqrt{2}}{\sqrt{2}}=1 $$
I've seen they say the numerator and the denominator must be an integer but in this special example
it seems to be little incomplete maybe adding the condition that if$a=b$ then it's$$ a,b \in R$$
(just an idea)
This wouldn't work because $\frac{2 \pi}{\pi} = 2$ and you'd have no end of special cases.

A number is rational if it can be written as the ratio of two integers. That doesn't mean that it can't be written as the ratio of two irrationals. In fact any rational can be so written. For example if $a$ and $b$ are integers then $\frac{a \pi}{b \pi} = \frac{a}{b}$.
 
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Feb 2020
62
1
St Louis
One can't make its mind up. :p


The definition of rationality is: The quality of being based on or in accordance with reason or logic.
therefore, irrationality is given without accordance to reason or logic.

That would imply that a number is irrational when there is no reasoning conducted or assessed according to strict principles of validity.

The system or set of principles underlying the arrangements of its elements is incomplete.
 

romsek

Math Team
Sep 2015
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One can't make its mind up. :p


The definition of rationality is: The quality of being based on or in accordance with reason or logic.
therefore, irrationality is given without accordance to reason or logic.

That would imply that a number is irrational when there is no reasoning conducted or assessed according to strict principles of validity.

The system or set of principles underlying the arrangements of its elements is incomplete.
you're an idiot
 
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romsek

Math Team
Sep 2015
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No, but I can ignore you. Bye.