# Does any real number has a unique binary decimal representation...?

• Mar 15th 2013, 07:16 AM
jojo7777777
Does any real number has a unique binary decimal representation...?
Any real number in [0,1] has a unique binary decimal representation o.b1b2b3..., where each bi is either 0 or1. Numerically, o.b1b2b3...=from1 to ∞ Σ(bn/2n)
where the infinite series converge to a number in[0,1].
The question is- how does it possible for this series "to converge to a number in[0,1]" for example, how 1/3 can be expressed using this series? Is there a proof for that statement?
• Mar 15th 2013, 09:14 AM
HallsofIvy
Re: Does any real number has a unique binary decimal representation...?
Quote:

Originally Posted by jojo7777777
Any real number in [0,1] has a unique binary decimal representation o.b1b2b3..., where each bi is either 0 or1. Numerically, o.b1b2b3...=from1 to ∞ Σ(bn/2n)
where the infinite series converge to a number in[0,1].
The question is- how does it possible for this series "to converge to a number in[0,1]" for example, how 1/3 can be expressed using this series? Is there a proof for that statement?

Since bi has to be either 1 or 0, that sum is less than or equal to $\displaystyle \sum_{i= 1}^\infty \frac{1}{2^i}$ which is a geometric series converging to 1.
To find the binary decimal for 1/3, note that 1/3< 1/2 so the first decimal place is 0. It is larger than 1/4 so the first two places are 0.01. Now, 1/3- 1/4= (4- 3)/12= 1/12. That is less than 1/8 but larger than 1/16 so the first four places are 0.0101. Now, 1/12- 1/16= 4/48- 3/48= 1/48. That is less than 1/32 but larger than 1/64 so the first six places are 0.010101. You might be able to guess, now, how that continues!

Quote:

• Mar 15th 2013, 12:22 PM
emakarov
Re: Does any real number has a unique binary decimal representation...?
Quote:

Originally Posted by jojo7777777
Any real number in [0,1] has a unique binary decimal representation o.b1b2b3..., where each bi is either 0 or1.

It is not true that every number has a unique representation. For example, 0.100000... = 0.011111...
• Mar 15th 2013, 10:32 PM
jojo7777777
Re: Does any real number has a unique binary decimal representation...?
Thank you very much...!!!
Does there is a proof for 0 and 1 to alternate like this forever- concerning the binary decimal representation of 1/3?
• Apr 1st 2013, 05:33 PM
johng
Re: Does any real number has a unique binary decimal representation...?
Hi,
This may be overkill in answer to your last question. But here it is. Just like in base 10, the binary expansion of any rational eventually is repeating.

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