You're probably familiar with the different integer bases: for example, \(45\) is \(231_4\), or \(231\text{ base }4\). Why? Because \(45=2\times 4^2+3\times 4^1+1\times 4^0\). 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 "\(\sqrt{2}\)"?

Well, let's try it. Any number \[(\overline{d_nd_{n-1}\cdots d_1d_0})_{\sqrt{2}}=d_0\times \sqrt{2}^0+d_1\times \sqrt{2}^1+\cdots +d_n\times \sqrt{2}^n\]

So what would \(110_{\sqrt{2}}\) be in base \(10\)?

It would be \(0\times \sqrt{2}^0+1\times \sqrt{2}^1+1\times \sqrt{2}^2=2+\sqrt{2}\). Interesting.

But how would we convert \(3\) to base \(\sqrt{2}\)? Well, \[3=2+1=\sqrt{2}^2+\sqrt{2}^0=101_{\sqrt{2}}\]

Everything more or less is going as expected right now. We have that \(110_{\sqrt{2}} > 101_{\sqrt{2}}\), and \(2+\sqrt{2} > 3\).

But things in \(\sqrt{2}\) aren't always normal. Let's compare the numbers \(11_{\sqrt{2}}\) with \(100_{\sqrt{2}}\):

\[11_{\sqrt{2}}=\sqrt{2}+1\] \[100_{\sqrt{2}}=2\]

Wait, what? We should have \(100_{\sqrt{2}} > 11_{\sqrt{2}}\), because that's just common sense. But clearly \(2 < \sqrt{2}+1\). Where did we go wrong?

Rest assured, nothing went wrong. Base \(\sqrt{2}\) 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 \(k\) digit numeral is larger than a \(k+1\) digit numeral? Think about it yourself before reading below.

To solve this problem, we compare the largest \(k\) digit numeral with the smallest \(k+1\) digit numeral in base \(B\). First off, we know that \(B < 2\) because when \(B=2\), the normal intuitive results hold (it's just binary). Also, \(B > 1\) or else there cannot be any digit except \(0\). Thus, the smallest \(k+1\) digit numeral is \[(1\underbrace{00\cdots 00}_{k\text{ zeroes}})_B=B^k\]

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

Thus we want \[\dfrac{B^k-1}{B-1} > B^k\]

This means \[B^k-1 > B^{k+1}-B^k\]

or \[B^{k+1}-2B^k+1 < 0\]

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

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

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

As \(k\) increases, the upper bound tends to \(2\). Thus, for sufficiently large \(k\), all bases under \(2\) and greater than \(1\) exhibit the strange property of a \(k\) digit numeral being larger than a \(k+1\) 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 \(\sqrt{2}\) to base \(2\)

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

No vote yet

1 vote

×

Problem Loading...

Note Loading...

Set Loading...

Easy Math Editor

`*italics*`

or`_italics_`

italics`**bold**`

or`__bold__`

boldNote: you must add a full line of space before and after lists for them to show up correctlyparagraph 1

paragraph 2

`[example link](https://brilliant.org)`

`> This is a quote`

Remember to wrap math in \( ... \) or \[ ... \] to ensure proper formatting.`2 \times 3`

`2^{34}`

`a_{i-1}`

`\frac{2}{3}`

`\sqrt{2}`

`\sum_{i=1}^3`

`\sin \theta`

`\boxed{123}`

## Comments

Sort by:

TopNewestI 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!

Log in to reply

Loved it!!!

Log in to reply

Nice discussion. Loved the way of its peculiarity.

Log in to reply

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

Log in to reply

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

Log in to reply

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

Log in to reply

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

Log in to reply

It means 'a' divides 'b'.

Log in to reply

It means that a divides b. For example, \( 2|4 \), \( 3|6 \) , \(278|278 \), etc

Log in to reply

Thanks.!!

Log in to reply

Like the one in this question

See the problem

Log in to reply

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.

Log in to reply

I AM STUNNED!!

Log in to reply