Quite a bit.

>>17799
>Quite a bit.
Chortle-chortle!

But seriously, is there a theoretical way to calculate it? Because it seems like a binary number will always require more digits than the decimal number it represents.

Of course you could like "cheat" and use half-bytes to represent every decimal number. For instance "10" would be written

0001 0000

Don't know if it's possible to use clever bit operations that way. Hmm... Allow me to think out loudly.

If you move the 1's one step to the left, you are doubling the number. So let's assume that we double the number 19 in a binary way:

00010011 = 16 + 3 + 1 = 19

00100110 = 32 + 4 + 2 = 38

Is it possible to do the same thing with half-bytes and get the same result?

0001 1100 = 1 & 4 + 2 = 1 & 6 = 16

0011 1000 = 2 + 1 & 8 = 3 & 4 = 34

No, it doesn't work.

>it seems like a binary number will always require more digits than the decimal number it represents.
It will.
Each digit in a binary number has two states
Each digit in a decimal number has ten states

The closest we have is hexadecimal (16 states) which is a conversion of 4 bits = 1 byte = 16 states. But hex isn't really a numbering system, just shorthand.

>>17801
I see.

The number of bits to represent an n-digit decimal number is the smallest integer that's >= n/log10(2). log10(2) is approximately 0.30103.

>>17806
I see.

So how many logs for a 200-million digit number? Will it break my calculator to calculate this?

>>17807
A scientific calculator should handle it just fine. To represent 10^200,000,000 you'd need 664,385,619 bits, or about 79.2 MiB.

>>17808
>A scientific calculator should handle it just fine. To represent 10^200,000,000 you'd need 664,385,619 bits, or about 79.2 MiB.
Oh wow! You just made an insane huge number manageable for me. I thought I would need like 1 TB of RAM to handle it! XD

>>17809
I can also tell you how long the number that you could store in 1 TiB is.

1 TiB
1024^4 bytes
1,099,511,627,776 bytes
8,796,093,022,208 bits
about 2.6 trillion decimal digits

>>17813
>I can also tell you how long the number that you could store in 1 TiB is.

>1 TiB
>1024^4 bytes
>1,099,511,627,776 bytes
>8,796,093,022,208 bits
>about 2.6 trillion decimal digits

Is there any limits to your powers? XD

>>17808
>>A scientific calculator should handle it just fine. To represent 10^200,000,000 you'd need 664,385,619 bits, or about 79.2 MiB.

A 664,385,619-bit number is too much. But according to my calculations it's 83,048,202 bytes. Would it be possible to string together a bunch of bytes to create a "virtual" 664,385,619-bit variable? Because a 32-bit variable is just 4 bytes.

>>17818
Note that when I said that a scientific calculator should be able to handle it, I was referring to the calculation of 200M divided by 0.30103. Obviously most calculators don't have over 70 megs of memory.

>Would it be possible to string together a bunch of bytes to create a "virtual" 664,385,619-bit variable?
Of course. The technique is called "arbitrary precision arithmetic". There are programming libraries that implement efficient algorithms to operate on such large values. For example: https://gmplib.org/
664 million bits is pretty big, but it fits comfortably in a PC's memory nowadays. I don't think there's any application for such a huge number, though. Even in cryptography the largest numbers commonly used are only 4096-bit, and there are no primes longer that 83 million bits.
https://www.mersenne.org/primes/

File 163302965397.png - (753.10KB , 1683x2432 , 1491634557010.png )

>>17819
Thanks a lot! :-)

>>17819
Oops. LOL. No KNOWN primes longer than 83 million bits.

A little off-topic, but if you guys like extremely huge (finite) numbers, check out this youtube playlist:
https://www.youtube.com/playlist?list=PLUZ0A4xAf7nkaYHtnqVDbHnrXzVAOxYYC

>>17834
Thanks! :-)