Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

It can't be $5$. And it can't be $4.\overline{9}$ because that equals $5$. It looks like there is no solution... but surely there must be?

share|cite|improve this question
Why must there be one? – Tobias Kildetoft May 20 '13 at 20:09
@Tom Not all subsets of the real numbers have a maximum, the set $\{x\in \Bbb R: x<5\}$ is one such instance. – Git Gud May 20 '13 at 20:10
Who says $x$ is a real number. I say $x$ is a number which is less than or equal to $4$ hence the greatest such $x$ is 4. Problem solved ;) – James S. Cook May 20 '13 at 20:11
If $x<5$ then $x<\frac{x+5}{2}<5$. So there is no greatest such $x$. – Thomas Andrews May 20 '13 at 20:13
What do you mean "must there still be a way to describe it?" There are lots of ways to describe things that don't exist. I can describe a moon made of green cheese, but that doesn't mean it exists. – Thomas Andrews May 20 '13 at 20:14
up vote 39 down vote accepted

There isn't one. Suppose there were; let's call it $y$, where $y<5$.

Let $\epsilon = 5 -y$, the difference between $y$ and 5. $\epsilon$ is positive, and so $0 < \frac\epsilon2 < \epsilon$, and then $y < y+\frac\epsilon2 < y+\epsilon = 5$, which shows that $y+\frac\epsilon2$ is even closer to 5 than $y$ was.

So there is no number that is closest to 5. Whatever $y$ you pick, however close it is, there is another number that is even closer.

Consider the analogous question: “$x < \infty$; what is the greatest value of $x$?” There is no such $x$.

share|cite|improve this answer
To answer your analogous question, there is no such $x$ as a value, but $x = \infty-\frac1\infty$ would approach the limit quite nicely. – Gary S. Weaver May 21 '13 at 1:00
I brought that up only to illustrate the point that one can make up a set of properties that are not satisfied by any object. – MJD May 21 '13 at 1:12
Actually, if you wanted a simple way to describe this number, couldn't you just say $x = 5 - \frac1\infty$ ? – Darrel Hoffman May 21 '13 at 1:30
You can say whatever you like, but hardly anyone will understand what you mean. If you want a simple way to describe this particular collection of properties, then Tom's original description, the greatest $x$ that is less than 5, is perfectly clear. Your suggestion just obfuscates that. – MJD May 21 '13 at 1:38
@GaryS.Weaver What about $\infty - \frac1{2\infty}$? If that doesn't work, so many things are broken. – PyRulez May 30 '15 at 14:58

The answer is $4$, assuming the domain of $x$ is $\Bbb Z$.

share|cite|improve this answer
The original question mentioned 4.99999, so the intention should have been clear. – MJD May 20 '13 at 21:22
He said $4.999...$ which is in fact an integer. – PyRulez May 20 '13 at 21:23
It doesn't answer OP's question. – srijan May 20 '13 at 23:50
+1 because I always approve of picking an interpretation of a problem that renders it trivial. – Kyle Strand May 21 '13 at 3:56
The answer is 3, assuming the domain of x is {3}. – Plutor May 21 '13 at 11:48

If $x<5$ then $2x<x+5$ so $x<\frac{x+5}{2}$. Similarly, $x<5$ means $x+5<5+5=10$ or $\frac{x+5}{2}<5$. So if $x<5$ we have $x<\frac{x+5}{2}<5$, and therefore there is a larger number, $\frac{x+5}{2}$ less than $5$.

Basically, the average of two different numbers must be strictly between those two numbers.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.