Irrational numbers in binary?

[Daecon]

Is there a way to convert irrational numbers from decimal to binary? Has it already been done?

[Shadow]

Sure...while I'm not absolutely positive, I think it's the same as converting any number with a fractional part. Just keep fitting in powers of two. When you get to [math]2^0[/math], you just go on with [math]2^{-1}, 2^{-2}[/math] etc...but since it's irrational you'll just keep going on for ever. Here are the first few digits of Pi in binary (source: Wiki)

 

11.00100100001111110110...

 

Here's [math]\sqrt {2}[/math](wiki again):

 

1.0110101000001001111...

 

An algorithm for converting fractional numbers into binary is described in the wiki article on the binary numeral system. You should also have a look at representing real numbers.

 

Hope this helps,

 

Gabe

[Severian]

Is there a way to convert irrational numbers from decimal to binary? Has it already been done?

 

Surely an irrational number can't be written down in decimal in the first place?

[Shadow]

I'm sure Transdecimal knows what irrational numbers are. And in any case, he didn't say "write down with a finite number of digits", he said "convert from decimal to binary". Which is very possible, unless there's a catch in that sentence I'm missing.

[Daecon]

Well take Pi for example, which begins with 3 and then 1/10ths, 4/100ths, 1/1'000ths, 5/10'000ths, 9/100'000ths... and so on.

 

How would you write those fractions in binary and then string them together?

[Shadow]

Using the algorithm on wikipedia, you just have to take Pi in decimal to the number of digits you want, and then convert to binary. That, or fit in powers of two. Eg 3, 14... -> [math]2^1 + 2 ^ 0[/math], a you get left with 0.14... in decimal and 11 in binary. Now you just go on:

 

[math]0.1415926... = 0 \cdot 2^{-1} + 0 \cdot 2^{-2} + 1 \cdot 2^{-3} +....[/math]

 

Hope this helps,

 

G

[Severian]

I'm sure Transdecimal knows what irrational numbers are. And in any case, he didn't say "write down with a finite number of digits", he said "convert from decimal to binary". Which is very possible, unless there's a catch in that sentence I'm missing.

 

As soon as you have converted it to decimal, it ceases to be an irrational number. So it is impossible to "convert and irrational number from decimal to binary".

[Daecon]

Maybe I should have said "represent an irrational number in binary", then.

 

Instead of using decimal with it's 1/10ths, 1/100ths, 1/1000ths, etc. and by using binary with it's 1/2ths, 1/4ths, 1/8ths, 1/16ths, etc...

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Aha. I understand now.

 

The Wikipedia entry was no help whatsoever as I couldn't understand a word of it, but I googled and found http://cs.furman.edu/digitaldomain/more/ch6/dec_frac_to_bin.htm which explained it much better.

 

I see where I was getting confused.