Base 2\sqrt{2} and other peculiar bases

You're probably familiar with the different integer bases: for example, 4545 is 2314231_4, or 231 base 4231\text{ base }4. Why? Because 45=2×42+3×41+1×4045=2\times 4^2+3\times 4^1+1\times 4^0. This should be old news to you. If you need a refresher on bases and base conversion, read through this note: Number Base Conversion

But do bases have to be integers? For example, is there any meaning to something like base "2\sqrt{2}"?

Well, let's try it. Any number (dndn1d1d0)2=d0×20+d1×21++dn×2n(\overline{d_nd_{n-1}\cdots d_1d_0})_{\sqrt{2}}=d_0\times \sqrt{2}^0+d_1\times \sqrt{2}^1+\cdots +d_n\times \sqrt{2}^n

So what would 1102110_{\sqrt{2}} be in base 1010?

It would be 0×20+1×21+1×22=2+20\times \sqrt{2}^0+1\times \sqrt{2}^1+1\times \sqrt{2}^2=2+\sqrt{2}. Interesting.

But how would we convert 33 to base 2\sqrt{2}? Well, 3=2+1=22+20=10123=2+1=\sqrt{2}^2+\sqrt{2}^0=101_{\sqrt{2}}

Everything more or less is going as expected right now. We have that 1102>1012110_{\sqrt{2}} > 101_{\sqrt{2}}, and 2+2>32+\sqrt{2} > 3.

But things in 2\sqrt{2} aren't always normal. Let's compare the numbers 11211_{\sqrt{2}} with 1002100_{\sqrt{2}}:

112=2+111_{\sqrt{2}}=\sqrt{2}+1 1002=2100_{\sqrt{2}}=2

Wait, what? We should have 1002>112100_{\sqrt{2}} > 11_{\sqrt{2}}, because that's just common sense. But clearly 2<2+12 < \sqrt{2}+1. Where did we go wrong?


Rest assured, nothing went wrong. Base 2\sqrt{2} is just a peculiar base that has these sort of strange properties.

A question is then brought into mind: For what bases does intuition fail us, that a kk digit numeral is larger than a k+1k+1 digit numeral? Think about it yourself before reading below.


To solve this problem, we compare the largest kk digit numeral with the smallest k+1k+1 digit numeral in base BB. First off, we know that B<2B < 2 because when B=2B=2, the normal intuitive results hold (it's just binary). Also, B>1B > 1 or else there cannot be any digit except 00. Thus, the smallest k+1k+1 digit numeral is (10000k zeroes)B=Bk(1\underbrace{00\cdots 00}_{k\text{ zeroes}})_B=B^k

and the largest kk digit numeral is clearly (1111k ones)B=i=0k1Bi=Bk1B1(\underbrace{11\cdots 11}_{k\text{ ones}})_B=\sum_{i=0}^{k-1}B^{i}=\dfrac{B^k-1}{B-1}

Thus we want Bk1B1>Bk\dfrac{B^k-1}{B-1} > B^k

This means Bk1>Bk+1BkB^k-1 > B^{k+1}-B^k

or Bk+12Bk+1<0B^{k+1}-2B^k+1 < 0

When k=1k=1, which means 1B>10B1_B > 10_B, then B<1B < 1 which is not possible.

When k=2k=2, which means 11B>100B11_B > 100_B, then it turns out 1<B<φ1 < B < \varphi where φ=1+52\varphi=\dfrac{1+\sqrt{5}}{2} is the golden ratio. This brings up a good point: 11φ=100φ11_{\varphi}=100_{\varphi} because of the identity φ2=φ+1\varphi^2=\varphi + 1.

When k=3k=3, which means 111B>1000B111_B > 1000_B, then 1<B1.8391 < B \lesssim 1.839 where the upper limit is the only real constant that satisfies the identity x3=x2+x+1x^3=x^2+x+1.

As kk increases, the upper bound tends to 22. Thus, for sufficiently large kk, all bases under 22 and greater than 11 exhibit the strange property of a kk digit numeral being larger than a k+1k+1 digit numeral.


I hope you found this little tidbit of non-integral bases and their strange behavior pretty interesting. Thanks for reading!

As an exercise, see if you can:

somehow relate base 2\sqrt{2} to base 22

prove or disprove that every integer and real number of the form a+b2a+b\sqrt{2} can be uniquely represented in base 2\sqrt{2}.

Note by Daniel Liu
5 years, 5 months ago

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I think this note was a little bit disorganized and "jumpy". I just noticed this intriguing fact, and wanted to share it to everyone in note-form. Enjoy!

Daniel Liu - 5 years, 5 months ago

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Loved it!!!

Siddharth Brahmbhatt - 5 years, 5 months ago

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You should have called the note "When 11 is greater than 100"

Nathan Ramesh - 5 years, 5 months ago

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Eh I like to keep my titles (for notes) saying what I'm about to discuss

Daniel Liu - 5 years, 5 months ago

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Nice discussion. Loved the way of its peculiarity.

Sharky Kesa - 5 years, 5 months ago

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@Daniel Liu Can you edit this note into Fractional NUmber Bases Wiki? Thanks!

Calvin Lin Staff - 4 years, 11 months ago

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This is awesome. It would be interesting to do a table of addition and multiplication. So 1 + 1 = 100. This can also be justified by the fact that the digits 0 and 1 don't divide the square root of 2 into equal parts.

Adrian Neacșu - 5 years, 5 months ago

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Can anyone explain me what is a|b means..???

Sudipta Biswas - 5 years, 5 months ago

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Like the one in this question

See the problem

Sudipta Biswas - 5 years, 5 months ago

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It means that a divides b. For example, 24 2|4 , 36 3|6 , 278278278|278 , etc

Siddhartha Srivastava - 5 years, 5 months ago

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Thanks.!!

Sudipta Biswas - 5 years, 5 months ago

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It means 'a' divides 'b'.

Aabhas Mathur - 5 years, 5 months ago

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I AM STUNNED!!

A Former Brilliant Member - 5 years, 3 months ago

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