# Thread: Converting decimal to bicimal and other things

1. ## Re: Converting decimal to bicimal and other things

Ok thanks. I've never seen that method before.
My question is, is there another way?
Given the preceding narrative on expressing decimals as quotients of two rationals I thought it may be possible to do so and convert both numerator and denominator to binary and go from there.
I don't see the preceding text as being of any use here. (see EDIT below)

The method I would use is repeated subtraction of powers of 2 (incl neg powers for after the "decimal" point ie 0.5, 0.25. 0.125, 0.0625, 0.03125, 0.015625 etc).
eg

0.703125 - 0.50 = 0.203125 ….. .1 (put a one if you can subtract)

can't subtract 0.25 ……..………… .10 (put a zero if you can't)

0.203125 - 0.125 = 0.078125 …. .101

0.078125 - 0.0625 = 0.015625 …. .1011

can't subtract 0.03125 ……………….. .10110

0.015625 - 0.015625 =0 …………… .101101

which is basically what you would do with an integer (using positive powers of 2).

The "repeated multiplication" way seems more efficient.

EDIT: Ok had another thought:

You could eg write 0.703125 as 703125/1000000 which cancels down to 45/64 which in binary is 101101/1000000.

This is an easy division to do and get 0.101101 as before. (Easy division because denominator 64 is a power of 2). Nice method after all!

2. ## Re: Converting decimal to bicimal and other things Originally Posted by Debsta Ok thanks. I've never seen that method before.

I don't see the preceding text as being of any use here.

The method I would use is repeated subtraction of powers of 2 (incl neg powers for after the "decimal" point ie 0.5, 0.25. 0.125, 0.0625, 0.03125, 0.015625 etc).
eg

0.703125 - 0.50 = 0.203125 .. .1 (put a one if you can subtract)

can't subtract 0.25 .. .10 (put a zero if you can't)

0.203125 - 0.125 = 0.078125 . .101

0.078125 - 0.0625 = 0.015625 . .1011

can't subtract 0.03125 .. .10110

0.015625 - 0.015625 =0  .101101

which is basically what you would do with an integer (using positive powers of 2).

The "repeated multiplication" way seems more efficient.
Interesting! Thanks very much Debsta. I wonder why both methods work!?

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3. ## Re: Converting decimal to bicimal and other things

I edited my previous post, but must have been after you read it. Have another look.

4. ## Re: Converting decimal to bicimal and other things Originally Posted by Debsta I edited my previous post, but must have been after you read it. Have another look.
Thanks! Can I trouble you with another question. Ex.16 below. I have answered it like this:  Sent from my iPhone using Tapatalk

5. ## Re: Converting decimal to bicimal and other things

Ok. I think that proves it. You need a concluding statement though.

My approach would be:

Suppose y>x and let y= x+k where k >0.

$\displaystyle \frac{x+y}{2}=\frac{x+x+k}{2}=\frac{2x+k}{2} = x +\frac{k}{2} >x$ . (if k>0, then k/2>0)

If y =x+k then x =y-k

$\displaystyle \frac{x+y}{2} =$ ...you can fil this in $\displaystyle <y$.

Therefore ...