Bases

The numbers you are most probably used to seeing are decimal numbers, those in base 10 (unless you're a life long computer programmer and can only work in hexadecimal and binary).
 
Technically, the base of a number indicates how many different digits it takes to get from the beginning to the next level, that is the next power (see the section on powers and roots).

For example, in base 10 you need 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 which is 10 digits.

So if our base was 2 (binary) we would need 2 digits, 0 and 1.

In base 10 (decimal) the different levels are powers of 10 (see also powers and roots, and scientific notation):

10⁰, 10¹, 10², 10³, 10⁴ etc (anything to the power zero is 1)

and

10^-1, 10^-2, 10^-3 etc

So when we write 100 what we mean is:

(1 x 10²) + (0 x 10¹) + (0 x 10⁰)

and

456 = (4 x 10²) + (5 x 10¹) + (6 x 10⁰)

In base 16 (hexadecimal)

456 = (4 x 16²) + (5 x 16¹) + (6 x 10⁰)

certainly a different quantity. So make sure there's no confusion about bases.

Binary and hexadecimal are the ones you are most likely to encounter, other than decimal. Even then, that's only true if you are studying computing or communications in any depth.

How do you get between bases (and we're not talking rounders or softball here).  

From binary (or any other base) to decimal is fairly straightforward. Write the number as its full sum of powers of 2 (or whatever) and then calculate the result of the expression. For example:

101101 = (1x2⁵) + (0x2⁴) + (1x2³) + (1x2²) + (0x2¹) + (1x2⁰)

= (1x32) + (0x16) + (1x8) + (1x4) + (0x2) + (1x1)

= 32 + 8 + 4 + 1

= 45 (base 10)

Going from decimal to binary (or any other base) is slightly more intricate. Start with the greatest power of 2 that is less than (or equal to) the number you have. Let's say that number is n. Write down 1x2ⁿ. Subtract that from your first number. Now start the process again with that new number and continue until you are left with no remainders. Now fill the gaps between the powers with 0x2^{missing power}s, if you haven't already. That expression will then give you the sequence of 1s and 0s for your number in base 2.
 
Here's an example: Write 89 (base 10) in binary.

The biggest power of 2 less than 89 is 64 which is 1x2⁶

89-64 = 25

The next lowest power is 32, but that is bigger than 25 so we have 0x2⁵  

The next lowest power is 16, which is less than our number, so we have 1x2⁴ leaving

25-16 = 9

8 is in there too, so we have 1x2³ to add to our expression and leaves us with

9 -8 = 1

2² isn't, nor is 2¹ but 2⁰ (=1) is, and that leaves nothing, so we've finished. Writing down everything from above we have:

(1x2⁶) + (0x2⁵) + (1x2⁴) + (1x2³) + (0x2²) + (0x2¹) + (1x2⁰)

and so 1011001 (base 2) = 89 (base 10).  

The same process will convert decimal to any other base, but clearly in other bases you may have to have a number other than 1 or 0 multiplying your power of the base number. For example 32 in base 10 is 2x16¹, so:  

32 (base 10) = 20 (base 16)

Note also that in bases greater than 10, you'll need more symbols for digits than you are used to. There is already a convention for this, and in base 16, for example, the 16 digits required for each power of 16 are 0-9,A-F. So, F (base 16) = 15 (base 10), and AC (base 16) = 172 (base 10).

Have a go at the Quiz!