ELI5: The difference between base 10 and other bases, and how to count in some.
Examples: base 5 or base 3 compared to base 10.
Posted May 21, 2012, 7:03 AM UTC
010
Base 5 has 5 digits. So, 0,1,2,3,4. Then, the next number is 10 which converts to 5 in base 10. Then 11, 12, 13, 14, 15, 20, 21 etc...
Base 3 is the same but 0, 1, 2, 10, 11, 12, 20, 21...
3
Each base has different number of symbols that can be used to represents numbers.
The counting works exactly the same way as in decimal(base 10). You start from 0 and go up to n-1 where is n is your base. When you reach n-1 you increment the next position and set the other one to zero and start over.
In the case of decimal you got from 00 to 09(n-1) then go to 10. Then from 10 to 19 then 20 etc...
In base 3 you choose your symbols to be 0, 1 and 2. These are arbitrary. And you count like so:
00, 01, 02, 10, 11, 12, 20, 21, 22, etc...
22 is the largest number you can represent with double digits just like 99 in decimal.
You can't have a base-0 counting system. Base-1 is possible and to represent a number N you just repeat the symbol N times.
2
In each base, the columns represent powers of that base number, so in base ten the columns represent ones, tens, hundreds and so on. We have ten digits (0 to 9) and when we run out of digits we spill over to the next column and start there with 1, resetting the first column with zero (so the symbol 10 means one ten and zero ones).
Taking this idea to another base, let's look at base 5. Here we have 5 digits (0 to 4). When we run out of digits we do the same as we did in base ten, which is to spill over into the next column. In base 5 the columns represent ones, fives, twenty-fives instead of ones, tens, hundreds, because we're looking at powers of five instead of powers of ten. So "10" written in base five means "one five and zero ones".
Exactly the same thing happens in all the other bases. If we count in a base system that needs more than ten digits, we borrow letters of the alphabet to represent the other numbers. In hexadecimal, which is base 16, the symbols used are 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F, and they represent the numbers zero to fifteen.
2
I see. Thank you! What would be the functional purpose of this?
1
I don't know about the other bases, but binary (base 2) and hexadecimal (base 16) are very common when describing how computers work and what they're doing internally.
Everything in a computer is a little switch, which is either on or off. We can use 1 to represent "on" and 0 for "off". This straight away gives you a binary system. Computers store numbers in groups of 8 switches (each switch is a "bit" and 8 of them is a "byte"). So the letter "A" would be stored inside the computer as 01000001. It gets tedious writing out all the zeros and ones all the time, so hexadecimal is used as a shortcut for binary: 1001 is nine in binary, which is 9 in hex. 1101 is thirteen in binary, but D in hex. By moving from binary to hex, we're reducing the number of characters we need to represent each number. The number 54321 in decimal would be 1101010000110001 in binary, but we'd write it as D431 in hexadecimal.
Have you ever seen a MAC address? If you ever needed to add your phone or games console to your home router to give it internet access you might have seen one. It's something like FD:30:01:A4:C2:DE - and all it is, is a unique identifying number written in hexadecimal.
2
Jesus, that's awesome. D: Thank you again!
1
More info: when writing numbers in different bases it's common practice to put the base as a subscript after the number. 1234 means one thing in decimal and something completely different in hexadecimal. The subscript might be the base number, or it might be a "d" for decimal or "h" for hex, different people use different notations. But it is important to specify what base you're writing the numbers in.
2
edit: added base I as suggested by Poddster.
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