# In what base does “100” equal “4 in base 10”

I was reading this answer to an amusing comic related question: http://math.stackexchange.com/a/166891/35132 and I understand that in the linked answer, the examples of how four may be expressed used base (expressed in decimal!!) is 10, 4, 3 for 4, 10, 11.

What I can't figure out is what base would need to be used for his last example (100) to equal 4 in decimal?

P.S. Are maths questions like this always this hard to put in words?!

-
For the PS: Yes. People who don't use very specific types of math on a regular basis don't have any need to distinguish between a number and a semantic representation of a number. They are capable of using the concept of "place value" and a base ten system to perform operations on numbers but don't tend to philosophize about it. The language for talking about math developed when this distinction was not well-understood and is primarily used by people who (for legitimate reasons) don't care about it. As a result, it's hard to ask questions like yours in natural language. – user29743 Jul 5 '12 at 15:29
@countinghaus Interesting. So is there a better "language" or way of talking about maths? – Grezzo Jul 5 '12 at 16:34
I should have said "syntactic" and not "semantic" in that comment. And, hmm, I guess no, except you strive to be really precise. If there's any risk that you'll be confusing when you're talking about multiple bases (and there usually is!), you should say things like "the number denoted one zero zero in base seven" or "the number which is called fourteen in english." – user29743 Jul 5 '12 at 18:14
Note that the answer holds for "base 10" in all bases N>4. So if by "base 10" you mean base S(S(S(S(S(0))))) (where S is the successor operator ("plus one")), you're okay. You don't have to mean ten. – Rex Kerr Jul 5 '12 at 18:18

## 2 Answers

Let the required base be $b$. Then, $$1 \cdot b^2 + 0 \cdot b^1 + 0 \cdot b^0 = 4 \implies b = 2.$$

-

$$100_{(b)}=b^2+0\cdot b + 0=b^2 \,.$$

So 100 in base $b$ is just the number $b^2$.... Can you find $b$ now?

-
What does "in base 10" mean here? – Henning Makholm Jul 5 '12 at 15:34
Yup I should had said the number $b^2$... – N. S. Jul 5 '12 at 15:36
Like this? ${}{}$ – Henning Makholm Jul 5 '12 at 15:41
Thankyou, upvoted for your help, but I marked pritram as the answer because he was first and he actually gave me the answer with workings, not just the means to get to the answer myself – Grezzo Jul 5 '12 at 16:31