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

Any real number in [0,1] has a unique binary decimal representation o.b_{1}b_{2}b_{3}..., where each b_{i} is either 0 or1. Numerically, o.b_{1}b_{2}b_{3}...=from1 to ∞ Σ(b_{n}/2^{n})

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?

Thanks in advance

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.b_{1}b_{2}b_{3}..., where each b_{i} is either 0 or1. Numerically, o.b_{1}b_{2}b_{3}...=from1 to ∞ Σ(b_{n}/2^{n})

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 b_{i} 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!

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.b_{1}b_{2}b_{3}..., where each b_{i} is either 0 or1.

It is not true that every number has a unique representation. For example, 0.100000... = 0.011111...

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?

2 Attachment(s)

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.

Attachment 27760

Attachment 27761