Irrational numbers

Status
Not open for further replies.

Glizdka

Key Member
Joined
Apr 13, 2019
Member Type
Other
Native Language
Polish
Home Country
Poland
Current Location
Poland
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?
 

jutfrank

VIP Member
Joined
Mar 5, 2014
Member Type
English Teacher
Native Language
English
Home Country
England
Current Location
England
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.
 

Glizdka

Key Member
Joined
Apr 13, 2019
Member Type
Other
Native Language
Polish
Home Country
Poland
Current Location
Poland
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:

YAMATO2201

Senior Member
Joined
Dec 29, 2016
Member Type
Student or Learner
Native Language
Japanese
Home Country
Japan
Current Location
Japan
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
 

SoothingDave

VIP Member
Joined
Apr 17, 2009
Member Type
Interested in Language
Native Language
American English
Home Country
United States
Current Location
United States
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.
 

jutfrank

VIP Member
Joined
Mar 5, 2014
Member Type
English Teacher
Native Language
English
Home Country
England
Current Location
England
Did you also think that prime numbers were particularly fine specimens of integers, Glizdka?
 

Glizdka

Key Member
Joined
Apr 13, 2019
Member Type
Other
Native Language
Polish
Home Country
Poland
Current Location
Poland
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.
 

probus

Moderator
Staff member
Joined
Jan 7, 2011
Member Type
Retired English Teacher
Native Language
English
Home Country
Canada
Current Location
Canada
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:

Glizdka

Key Member
Joined
Apr 13, 2019
Member Type
Other
Native Language
Polish
Home Country
Poland
Current Location
Poland
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:

jutfrank

VIP Member
Joined
Mar 5, 2014
Member Type
English Teacher
Native Language
English
Home Country
England
Current Location
England
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.
 

probus

Moderator
Staff member
Joined
Jan 7, 2011
Member Type
Retired English Teacher
Native Language
English
Home Country
Canada
Current Location
Canada
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

No. Although I put in the disclaimer that I am not a mathematician, I did take a lot of math courses as an undergraduate, and worked for a time as an actuary. I know what I'm talking about.

The real numbers can indeed be represented by points on a line, and mathematicians do that. The technical term in mathematics for such correspondences is isomorphism. The real numbers and the points on a line can be regarded as identical for many purposes

The "second line" lies not parallel to the first but across it at right angles. That is intuitively why the complex numbers are more "numerous" than the real numbers (C > aleph-null) in a sense. Just as mathematicians speak of the real number line, they also speak of the complex number plane.

Since we are supposed to be discussing English, not mathematics, I am closing this thread.
 
Last edited by a moderator:
Status
Not open for further replies.
Top