Base conversions

courteous · 2009-01-02 23:14:38

How do you convert, let's say 6.47 (base-10), into both binary and ternary system?

mathsyperson · 2009-01-03 05:19:35

In base 10, you have thousands, hundreds, tens and units. But this continues after the decimal point and you have tenths, hundredths and so on.

Similarly, base 2 has eights, fours, twos, units, but then it continues with halves, quarters, etc.
To convert a number from base 10 to base 2, write it in terms of powers of two.

eg.
3.25 = 2 + 1 + 1/4, so 3.25 (base 10) = 11.01 (base 2)

One important point: in the same way that you can get endless decimals in base 10 (1/3 = 0.333...), you might get some endless decimals when converting between bases.

So for 6.47, you'd try to write it in powers of 2:

4
4+2 = 6
6+1/4 = 6.25
6.25 + 1/8 = 6.375
6.375 + 1/16 = 6.4375
6.4375 + 1/32 = 6.46875
...

You can get very close to what you're trying to convert, but it will take an infinite amount of fractions to actually get there, so the base 2 decimal form will have to be non-terminating.

I'd say your best plan is to round it. To five decimal places, 6.47 (base 10) = 110.01111 (base 2).

Base 3 conversion is very similar. Write the original number in terms of nines, threes, ones, thirds, ninths and so on, then write the base 3 form using that.

Last edited by mathsyperson (2009-01-03 10:08:10)

JaneFairfax · 2009-01-03 09:59:00

mathsyperson wrote:

4
4+2 = 6
6+1/4 = 6.25
6.25 + 1/8 = 6.375
6.25 + 1/16 = 6.4375
6.25 + 1/32 = 6.46875
...

Arent you missing some terms?

mathsyperson · 2009-01-03 10:09:25

Aah, sorry. Edited now.
I knew what I was doing, I just didn't type it.

JaneFairfax · 2009-01-03 10:41:17

I was trying find a geometric series involving powers of to express complicated rational numbers in base 2. Some are straightforward, e.g.

Unfortunately I couldnt get

. I got

but I couldnt do the division.

mathsyperson · 2009-01-03 11:33:48

This way works:

So now we can convert fractions into base 2 if some multiple of their denominator is one less than a power of 2.

Edit: In fact, by also using Jane's method for 1/6, we can convert a fraction if it can be written as

I'm fairly sure all fractions can do this (based on the fact that they can if the 2's are replaced by 10's).

courteous · 2009-01-05 08:27:05

JaneFairfax wrote:

This can't be legal multiplication?

Last edited by courteous (2009-01-05 08:28:27)

mathsyperson · 2009-01-05 09:29:56

No, that bit's fine. You can see why if you think of it terms of the fractions:

0.1 x 0.010101...

0.0010101...

Math Is Fun Forum

#1 2009-01-02 23:14:38

Base conversions

#2 2009-01-03 05:19:35

Re: Base conversions

#3 2009-01-03 09:59:00

Re: Base conversions

#4 2009-01-03 10:09:25

Re: Base conversions

#5 2009-01-03 10:41:17

Re: Base conversions

#6 2009-01-03 11:33:48

Re: Base conversions

#7 2009-01-05 08:27:05

Re: Base conversions

#8 2009-01-05 09:29:56

Re: Base conversions

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