## Decimal like Concepts in Other Bases?

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liberonscien
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### Decimal like Concepts in Other Bases?

Are there decimal like concepts in other bases?
If so what are they called and how are they expressed?
For example, I was attempting to determine what the binary equivalent of 0.75base 10 is and if it exists. I thought it might be something like 0.1001011base 2, but I wasn't certain if that was correct or valid.

Soupspoon
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### Re: Decimal like Concepts in Other Bases?

As more leftwards digits of decimal than the base0 place are base+1, base+2, etc (i.e. units, 10s, 100s, whatever base 1<some zeros> is recorded in), to the right of the basic units (and the 'decimal' point, even if it isn't in decimal any more) are base-1, base-2, etc, which are 10ths,100ths, and onwards, whether thats 10ths as in 1/10dec for decimal or 1/10bin (i.e. 1/2dec) for binary.

0.75dec is actually quite simple. 0.1bin is ½dec, 0.01 is ¼dec. And thus 0.11bin is what you need to represent 0.75dec.

Does that make sense?

Xenomortis
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### Re: Decimal like Concepts in Other Bases?

Digits in a decimal expansion merely represent multiples of a power of 10.
So for instance:
17.32 is
1 * 101 + 7 * 100 + 3 * 10-1 + 2 * 10-2
That is "One 10, seven 1's, three 10ths, two 100ths".

The idea is the same for other number bases, but instead of powers of 10, it's powers of the other base.
So for base 2, the digit expansion "ab.cd" actually represents
"a * 21 + b * 20 + c * 2-1 + d * 2-2"

So, since 0.75 = 0.5 + 0.25 = (1/2 + 1/4), it's simply 0.11

(Ninja'd by faster typing skills)
Last edited by Xenomortis on Mon Aug 15, 2016 5:01 pm UTC, edited 2 times in total. Soupspoon
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### Re: Decimal like Concepts in Other Bases?

Xenomortis wrote:(Ninja'd by faster typing skills)
(Actually, I like your explanation of the same thing better. Looks cleaner, probably even less confusing.)

liberonscien
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### Re: Decimal like Concepts in Other Bases?

Xenomortis wrote:Digits in a decimal expansion merely represent multiples of a power of 10.
So for instance:
17.32 is
1 * 101 + 7 * 100 + 3 * 10-1 + 2 * 10-2
That is "One 10, seven 1's, three 10ths, two 100ths".

The idea is the same for other number bases, but instead of powers of 10, it's powers of the other base.
So for base 2, the digit expansion "ab.cd" actually represents
"a * 21 + b * 20 + c * 2-1 + d * 2-2"

So, since 0.75 = 0.5 + 0.25 = (1/2 + 1/4), it's simply 0.11

(Ninja'd by faster typing skills)

What would 17.32 be in binary?

Xenomortis
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### Re: Decimal like Concepts in Other Bases?

17 is 16 + 1 (24 + 20), so that's 10001.
0.32 is a little more tricky - you need to work out 32 / 100 in binary (specifically, 1000002 / 11001002) - you can do the long division if you want, the technique is the same as it is in base 10.
It won't be terminating - easy decimal expansions seldom terminate in base 2.
The first four digits are .0101 - see if you can see why. So we have an approximation of 10001.0101... PsiCubed
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### Re: Decimal like Concepts in Other Bases?

Pure repeated decimals in base 10 are numbers which can written as [something]/999...999.

For example:

377/999 = 0.377 377 377 377.

1/41 = 2439/99999 = 0.02439 02439 02439 ...

Note that the number of repeated digits is equal to the number of repeated 9's in the denominator.

And the same is true for base 2 numbers. The only difference is that you need to replace 999...999 with a binary 111...111, which is 1 less than a power of two.

Well, 32/100 = 8/25.

The smallest power of two equal to 1 (mod 25) is 220. So we have:

220-1 = 1048575 = 25x41943.

And:

8/25 = (8x41943)/(25x41943) = 335544/1048575 = 335544/(220-1).

The denominator would be 111111111111111111112 (that's 20 1's). This means that 8/25 will have 20 repeated digits.

Converting 335544 to binary we get 10100011110101110002. So:

8/25 = 335544/(220-1) = 10100011110101110002/111111111111111111112.

Now because "1010001111010111000" has only 19 digits, we need to add a leading zero to get a period of 20. So the final answer is:

0.32 = 8/25 = 0.01010001111010111000 01010001111010111000 01010001111010111000 ...

And

17.32 = 10001.01010001111010111000 01010001111010111000 01010001111010111000 ...

P.S. Xenomortis, nice to see a fellow Eulerean here. Last edited by gmalivuk on Mon Aug 15, 2016 7:18 pm UTC, edited 1 time in total.
Reason: changed an errant 'sup' tag to a 'sub' tag EmelinaVollmering
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### Re: Decimal like Concepts in Other Bases?

Nope, the methods are same and almost look alike. Let's take a look inside.

Base 10
7 + 1 = 8

Base 8
7 + 1 = 10

This is the only rule, while performing operation. First, if n is the base, then 0 to n-1 are the valid numbers. Second, if the 'n-1 + n+1' It will not be 2n. It will not be 2n, it will be 20.

Only clear rule is n = 10. After that everything start over again. No matter what the operation is. 0,1,....n-1,10,11. And so on.