# What is the difference between rational and irrational numbers?

#### SDK

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"
You know we also know how to represent $\pi$ exactly right?
\begin{align*}
& \downarrow\\
\rightarrow & \ \pi \leftarrow \\
& \uparrow
\end{align*}

BAM!

Greens

You know we also know how to represent $\pi$ exactly right?
\begin{align*}
& \downarrow\\
\rightarrow & \ \pi \leftarrow \\
& \uparrow
\end{align*}

BAM!
lol.. best i've seen so far.

#### mathman

Forum Staff
Could you explain why that is? How else would they have got their names?
Lain "ratio" was used in the sense of calculation as well as reasoning.

Lain "ratio" was used in the sense of calculation as well as reasoning.
you are correct.
as I had stated. it was first used around 1630 and adopted (with good reason) by mathematicians only 20 years later.

they did not arbitrarily decide to use this word and give it a different definition. the definition applied to something that the were having to describe.

#### topsquark

Math Team
you are correct.
as I had stated. it was first used around 1630 and adopted (with good reason) by mathematicians only 20 years later.

they did not arbitrarily decide to use this word and give it a different definition. the definition applied to something that the were having to describe.
I imagine that there was something like that when the term "rational" was coined for a specific kind of number. (Perhaps they were called that because they "behaved" nicely.) But once Mathematicians started using it the definition had to be something more strict and unambiguous than the language. A rational number is a number that can be written in terms of two integers: a/b (where b is not 0) and / is the usual division operation on the real numbers. That's it. There is nothing left to compare with the language any more. A rational number has nothing to do with any "rational" logic structure.

As a better indication of how language can be "massaged" by Mathematicians I give you a joke:
Q: How does a set differ from a door?
A: A set can be open, closed, both, or neither.

The terms open and closed have radically different definitions in Mathematics than in their dictionary equivalents.

-Dan

I imagine that there was something like that when the term "rational" was coined for a specific kind of number. (Perhaps they were called that because they "behaved" nicely.) But once Mathematicians started using it the definition had to be something more strict and unambiguous than the language. A rational number is a number that can be written in terms of two integers: a/b (where b is not 0) and / is the usual division operation on the real numbers. That's it. There is nothing left to compare with the language any more. A rational number has nothing to do with any "rational" logic structure.

As a better indication of how language can be "massaged" by Mathematicians I give you a joke:
Q: How does a set differ from a door?
A: A set can be open, closed, both, or neither.

The terms open and closed have radically different definitions in Mathematics than in their dictionary equivalents.

-Dan
lol... thats a good one.
sounds almost like quantum mechanics. I myself prefer cookies over quanta. they "behave" better

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