Ask Uncle Colin: A Binary Fraction

Dear Uncle Colin,

How would you write $\frac{1}{10}$ in binary?

Binary Is Totally Stupid

Hi, BITS, and thanks for your message!

I have two ways to deal with this: the standard, long-division sort of method, and a much nicer geometric series approach.

Long division-esque

While I can do the long division method, I can’t easily typeset it. So I’m going to tackle it by the double-and-remainder method.

Let $x = \frac{1}{10}$.

Then I find:

• $2x = \frac{1}{5}$
• $4x = \frac{2}{5}$
• $8x = \frac{4}{5}$
• $16x = 1 + \frac{3}{5}$
• $32x = 3 + \frac{1}{5}$

Now, I know what 3 is in binary - it’s $11_2$. If you’ll forgive me mixing decimal and binary representations, I know that that $2^5 x = 11_2 + 2x$.

Dividing both sides by $2^5$ gives $x = 0.00011_2 + 2^{-4} x$, which gives the recursive result $x = 0.0001100110011…_2$.

Geometric series

It’s always been a source of delight for me that, in base 10, $0.\dot{9} = 1$ exactly. This is a result that can be shown using the sum of a geometric sequence: $S =\frac{a}{1-r}$, where $a$ is the first term and $r$ is the common ratio (as long as it’s between -1 and 1). (In that example, $0.\dot{9} = \frac{9}{10} + \frac{9}{100} + \dots$; we have $a = \frac{9}{10}$, $r =\frac{1}{10}$ and the result follows.)

Now, can we engineer a similar result for $\frac{1}{10}$ using “nice” binary numbers? Let’s have a think. We want $\frac{a}{1-r} = \frac{1}{10}$, or $10a = 1-r$, where $a$ and $r$ are numbers easy to express in binary.

One possibility is $r = -\frac{1}{4}$ and $a = \frac{1}{8}$, but this would give me a bit of a headache with negative numbers.

Suppose $a$ and $r$ have the same denominator, $2^k$, and numerators $A$ and $R$. Then we can rewrite our target expression as $10A = 2^k - R$ - we’re looking for a “nice” multiple of 10 that’s a nice distance away from a power of 2. For example, with $k=5$, we get $A=3$ and $R=2$ - which gives us $a = \frac{3}{32}$ and $r = \frac{1}{16}$.

A geometric series with first term $\frac{3}{32}$ and common ratio $\frac{1}{16}$ would look like: $0.0001100110011…_2$, as we had before!

Hope that helps,

- Uncle Colin

Colin

Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.

Share

This site uses Akismet to reduce spam. Learn how your comment data is processed.

Sign up for the Sum Comfort newsletter and get a free e-book of mathematical quotations.

No spam ever, obviously.

Where do you teach?

I teach in my home in Abbotsbury Road, Weymouth.

It's a 15-minute walk from Weymouth station, and it's on bus routes 3, 8 and X53. On-road parking is available nearby.