# Converting decimal to bicimal and other things

#### Monoxdifly

My Math Forum. Are you also a member there?

#### Debsta

MHF Helper
My Math Forum. Are you also a member there?
Thanks. Don't think so. Certainly not an active one. I'll check it out.

#### flashylightsmeow

Can you please show how you approach Ex 13 for 3.75? The example above Ex 13 refers back to Ex 1 which you have not included.

"Regarding Ex. 13. I understand that the answers can be found by multiplying successively by 2 and taking the whole unit at each stage as 1 and 0 where there is none."
I don't understand what that is saying.

The way I would approach it is 3.75 = 2 + 1 + 0.5 + 0.25 = 1x2^1 +1x2^0 + 1x2^-1 +1x2^-2 = 11.11 (bicimal)
Thanks Debsta. Please see the page below.
I used the same method to convert 3.75, but when it comes to 0.703125 it's not as straightforward.

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#### Debsta

MHF Helper
You say: "Regarding Ex. 13. I understand that the answers can be found by multiplying successively by 2 and taking the whole unit at each stage as 1 and 0 where there is none."

Can you please show me how you do this for 3.75 (ie using the hint given) and then I can try the next one. I'm still unsure how this process works.

#### flashylightsmeow

You say: "Regarding Ex. 13. I understand that the answers can be found by multiplying successively by 2 and taking the whole unit at each stage as 1 and 0 where there is none."

Can you please show me how you do this for 3.75 (ie using the hint given) and then I can try the next one. I'm still unsure how this process works.
Sure

Here we go

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#### Debsta

MHF Helper
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.

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!

Last edited:

#### flashylightsmeow

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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#### Debsta

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

#### flashylightsmeow

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:

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#### Debsta

MHF Helper
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 …………...