# What are hyperreal numbers?

My first encounter with hyperreal numbers was two months ago. I read a lot of articles about them, but I did not understand what and which they are, because of this:

On the french wikipedia page I can read: Un nombre hyperréel x est dit infinitésimal, si |x| est strictement inférieur à tout réel positif. Translation : x is an infinitesimal hyperreal number if |x| < a with a > 0 ...

So if a = 1000 then -1000 < x < 1000, all x between -1000 and 1000 are infinitesimals?

I can not believe that this is true...

Thank you!

Yann

• They mean for all $a>0$, $|x|<a$. An infinitesimal number is one whose magnitude is smaller than any positive number. – Seth Oct 10 '14 at 20:06
• The word "tout" is important – Hagen von Eitzen Oct 10 '14 at 20:09

The hyperreal numbers are numbers which extend the usual real numbers. The idea is that in the real number, if $x>0$ then for some $n\in\Bbb N$ we have that $\frac1n<x$.

In the hyperreal numbers we don't have that. There are numbers which are smaller than any $\frac1n$, but are larger than $0$. Note that these numbers are not real numbers, but that's okay since the hyperreal numbers extend those.

However, the hyperreal numbers and the real numbers share many properties, both are fields and basic statements we can write down in the language of fields (namely the existence of roots of certain polynomials) are true in the real numbers if and only if they are true in the hyperreal numbers. These similarities can be extended, but that requires some additional knowledge in mathematics.

Two important differences (except the existence of infinitesimals) are these:

1. The hyperreal numbers are not necessarily unique. Namely, there is exactly one field (up to isomorphism) which is $\Bbb Q$, and there is exactly one field which is $\Bbb R$ (again, up to isomorphism). However the statement that every two hyperreal fields are isomorphic requires additional assumptions on the mathematical universe.

2. The hyperreal numbers do not make a complete field. Namely, in $\Bbb R$ is a set is bounded, it has a least upper bound. In the hyperreal numbers this is not true anymore. Consider $\Bbb N$, then since there is $x>0$ such that $x<\frac1n$ for all $n\in\Bbb N$, it follows that $\frac1x$ is an upper bound for $\Bbb N$, but in that case $\frac1x-1$ is a smaller upper bound. So there is no least upper bound.

• OK thank you I see, and if I take a such as |a-0| < ε with ε > 0 can I find some hyperreal numbers for witch |x| < a ? :) – ParaH2 Oct 10 '14 at 20:28
• Note that (1) you didn't specify if $\varepsilon$ is a real number or a hyperreal number; and (2) you didn't specify if $\varepsilon$ is fixed or not. So I can't quite answer that. – Asaf Karagila Oct 10 '14 at 20:33
• Well can you answer 4 the for cases? ^^ Is ε is real and fixed, real and not fixed, hyperreal and fixed, hyperreal and not fixed ^^ – ParaH2 Oct 10 '14 at 20:35
• That also depends on whether or not $a$ is real or not. Too many variables, but you can analyze these cases yourself based on this one: If you allow $\varepsilon$ to be hyperreal, and it's not fixed then the answer is negative for the same reason it is negative in the real numbers with real $\varepsilon$, but if it's fixed it will be true for the hyperreals (and the reals if $\varepsilon$ is real); on the other hand if $\varepsilon$ is real and unfixed then the answer is positive in the hyperreals. This means exactly that $|a-x|$ is infinitesimal. – Asaf Karagila Oct 10 '14 at 20:45
• Well thank you guy/girl !! It's very nice :) but i don't understand all of your answer, i will speak about it with a friend and if i don't understand i'll come back ! Have a nice day, bye ! :) – ParaH2 Oct 10 '14 at 20:52

A possible answer tailored to a non-mathematician: you are probably familiar with the idea that number systems can be extended according to the needs of the applications one has in mind. Thus the natural numbers $\mathbb N$ are extended to the integers $\mathbb Z$ to accomodate easier solution of the simplest kind of linear equation. These are further extended to the rationals $\mathbb{Q}$ to be able to solve any linear equation. To solve geometric problems of even the simplest kind like expressing the diagonal of a unit square or the area of a unit circle, one needs irrational numbers, resulting in a further extension denoted $\mathbb R$ (the real numbers).

If all one were interested in were for example Euclidean geometry, this would be more than enough (in fact a much smaller number system would suffice). For the purposes of infinitesimal calculus, it is convenient to extend the number system further to include infinitesimal numbers. A number $\epsilon>0$ is called infinitesimal if it is smaller than every positive real number $r\in \mathbb R$.

The hyperreals are a particularly useful extension of $\mathbb R$ that incorporates infinitesimals. More details on the hyperreals can be found in a number of posts here; see tag .