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## Irrational numbers

I love the English term for this type of number because it can be interpreted in two ways.

On the one hand, it sounds like it cannot be expressed by a ratio. It's irrational. There's no ratio. On the other, it sounds like it's weird, illogical, ridiculous. It's irrational.

When I first heard the term, I thought they were named so because they're weird and illogical. Then, I realized it's more about the ratio rather than calling it names.

How did you react when you first heard the term irrational number? What did you think was the reason why it's named like that?

2. ## Re: Irrational numbers

Is this really a language question, Glizdka? I think a moderator would have good reason to transfer it to the discussions forum.

I don't think many of us understood that those numbers were 'weird and illogical', but I suppose that in a sense they are. Personally, I don't like to think of Pi as a number at all—it's just a ratio. Mathematicians like to think of everything as numbers, for operational rather than philosophical reasons. They even think of zero and infinity as numbers when it suits them—which I think seems silly to many non-mathematicians.

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## Re: Irrational numbers

I'm only interested in how native speakers feel about the word irrational in irrational numbers. Preferably, people without any deeper knowledge in math or understanding what makes a number irrational because I'm interested in language, not math. I think it qualifies as a language question.

My question aside, I don't agree that mathematicians think of infinity as a number. Maybe bad mathematicians do—there's lots of them around—but the idea that infinity is a number seems just as silly to a mathematician as it does to a non-mathematician, but that's a story for another time and not for this forum.
Last edited by Glizdka; 14-Apr-2021 at 19:44.

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## Re: Irrational numbers

Originally Posted by Glizdka
I love the English term for this type of number because it can be interpreted in two ways.
When I first heard the term, I thought they were named so because they're weird and illogical. Then, I realized it's more about the ratio rather than calling it names.
The following might interest you:

3) The word ‘ratio’ next appears in 1660 in Isaac Barrow’s work as described above. So what we have here is the fact that in the mathematical sense, ‘rational’ actually developed as a back-formation of ‘irrational,’ and ‘ratio’ as a back-formation of ‘rational’– where a back-formation is the process by which a new word develops from the dropping of an affix from an earlier word as in ‘swindle’ from ‘swindler’ and ‘typewrite’ from ‘typewriter.’
http://www.wordwizard.com/phpbb3/viewtopic.php?t=6491

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## Re: Irrational numbers

It's a technical term. I wouldn't think it meant the numbers were crazy or behaved in ways that were inexplicable.

It literally means that the number can not be expressed (with complete accuracy) by a ratio.

6. ## Re: Irrational numbers

Did you also think that prime numbers were particularly fine specimens of integers, Glizdka?

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## Re: Irrational numbers

Originally Posted by jutfrank
Did you also think that prime numbers were particularly fine specimens of integers, Glizdka?
It has never occurred to me. I love it!

I think it's cute my five has an imaginary friend, but their relationship seems rather complex.

8. ## Re: Irrational numbers

Actually, to mathematicians there isn't just one infinity. The first and "smallest" infinity is called aleph-null and corresponds to the number of integers. The second is called C and corresponds to the number of real numbers. I'm not a mathematician but I think there must also be more and "larger" infinities because my intuition says there must be fewer real numbers than there are complex numbers.

You can stop reading now if you are satisfied, but here is an easy mathematical proof for those who are interested. First, the number of primes is infinite. If not, there must be a largest prime. Call it n. But then consider (n!+1). It must be prime because when divided by every integer up to n it leaves a remainder of 1. But n!+1>n. Therefore n is not the largest prime and the number of primes is infinite. But the primes are only a subset of the positive integers. Therefore the number of positive integers is infinite.
Last edited by probus; 16-Apr-2021 at 00:50. Reason: Typo

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## Re: Irrational numbers

Thinking about the number of numbers is kind of like speaking about how we speak. Mathematicians and linguists aren't that far apart after all.

Infinite bottles of beer on the wall, infinite bottles of beer.
Take one down, pass it around, infinite bottles of beer on the wall.

Now that I think about it, surreal numbers and transcendental numbers sound beautiful.

Anyway, to answer the question. It's just me and maybe a few others, but irrational numbers is generally just a term and nobody really gives it much more thought, right?
Last edited by Glizdka; 16-Apr-2021 at 01:29.

10. ## Re: Irrational numbers

Originally Posted by probus
I'm not a mathematician but I think there must also be more and "larger" infinities because my intuition says there must be fewer real numbers than there are complex numbers.
Well, that is one way of looking at it, but I don't really see where that gets you. What this view does is to take a concept that I think is more naturally seen as incomplete and apply a notion of completeness to it. What I think you're doing is imagining numbers on a line. You have in mind two number lines side by side, and see that one is longer than the other, and so you conclude that one infinity is bigger than the other because each line has a finite length. In my opinion, all you're saying with this is something about how the human mind likes to make sense of abstract ideas. This handy 'trick' of viewing infinity as complete is also evident when mathematicians talk of 'infinite sets' and when they attempt to express infinity as a number. Philosophers, being much more sensible types, tend to think of infinity as incomplete (not least because it's a much more difficult and interesting view!). With this view, there are no 'bigger' or 'smaller' infinities because, by definition, there is no concept of limit to be had.

George Lakoff has some very interesting work that demonstrates how these metaphors that we use to make sense of infinity are the same ones we use to makes sense of events in time when we use grammatical aspect in language.

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