What is the difference between rational and irrational numbers?

mathman

Forum Staff
May 2007
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Rational and irrational number definitions have nothing to do with the more general definitions of these words.
 
Feb 2020
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St Louis
Rational and irrational number definitions have nothing to do with the more general definitions of these words.
Could you explain why that is? How else would they have got their names?
 
Aug 2012
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Could you explain why that is? How else would they have got their names?
Good question. The irRATIOnal numbers are numbers that are not the RATIO of two integers. It's a double meaning. They're irrational, "not a ratio." Nothing to do with irrational meaning crazy or unreasonable.
 
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Feb 2020
62
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St Louis
Good question. The irRATIOnal numbers are numbers that are not the RATIO of two integers. It's a double meaning. They're irrational, "not a ratio." Nothing to do with irrational meaning crazy or unreasonable.
Ratio: The relative magnitudes of two quantities (usually expressed as a quotient). The relativity is the quality or state of being dependent or determined in value by relation to something else.

Rationality: the quality or state of being agreeable to reason.
this was first used in 1630 and then adopted by mathematics only 20 years later with good reason.

The OP had asked:
"Can you provide some further information about the difference between rational and irrational numbers?"
My answer was that relatively speaking, an irrational number is missing an element that would make it logical. agreeable to reason. reason: the power of the mind to think, understand, and form judgments by a process of logic. one cannot understand a number that does not end nor form any valid logical absolute judgements from it.
By adding another element you can. That is definitive.

To estimate an irrational, imperfect square to an approximate decimal or fractional value, we need to look at the rational, perfect squares around it. A perfect square is the product of a whole number that is multiplied by itself. Every imperfect square has a value between two perfect squares. Think of it as a square root number line.


It's all there but often overlooked.
 
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topsquark

Math Team
May 2013
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The OP had asked:
"Can you provide some further information about the difference between rational and irrational numbers. "
my answer was that relatively speaking, an irrational number is missing an element that would make it logical. agreeable to reason. reason: the power of the mind to think, understand, and form judgments by a process of logic. one cannot understand a number that does not end nor form any valid logical absolute judgements from it.
By adding another element you can. That is definitive.
This has nothing to do with Math.

To estimate an irrational, imperfect square to an approximate decimal or fractional value, we need to look at the rational, perfect squares around it. A perfect square is the product of a whole number that is multiplied by itself. Every imperfect square has a value between two perfect squares. Think of it as a square root number line.
Good idea, but your argument is improved if, instead of using rational perfect squares, you simply used rational numbers. No "perfect" squares of rational numbers are needed.

As a famous example, \(\displaystyle \pi \approx \dfrac{22}{7}\).

-Dan
 
Feb 2020
62
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St Louis
This has nothing to do with Math.


Good idea, but your argument is improved if, instead of using rational perfect squares, you simply used rational numbers. No "perfect" squares of rational numbers are needed.

As a famous example, \(\displaystyle \pi \approx \dfrac{22}{7}\).

-Dan
It is my belief that what I said has a lot to do with math.


My reasoning for using Imperfect squares is that although irrational square roots cannot be written as fractions, we can still write them exactly.

For example, suppose you need to find √18. You know there is no whole number squared that equals 18, so √18 is an irrational number. The value is between √16=4 and √25=5. However, we need to find the exact value of √18

We begin by writing the prime factorization of the square root of 18.. √18=√9×2=√9×√2.
√9=3 but √2 does not have a whole number value. Therefore, the exact value of √18 is 3√2. A value that represents itself in its entirety. That is different from ≈ "approximately equal to"
 

topsquark

Math Team
May 2013
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an irrational number is missing an element that would make it logical.
It is my belief that what I said has a lot to do with math.
Numbers aren't either logical or illogical. They just are. So no, there are no missing elements.

Unless you have a precisely defined definition of the word "logical" that has meaning in terms of Mathematics. If so, please share.

-Dan
 

skipjack

Forum Staff
Dec 2006
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The definition of rationality is: The quality of being based on or in accordance with reason or logic.
Like many English words, it has more than one meaning, so your use of "the definition", as distinct from "a definition", was inappropriate. In some contexts, "rationality" means the quality of being commensurable with natural numbers, and this meaning doesn't have the implication for irrational numbers that you gave earlier.
 
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