Binary Representation vs. Decimal

When we write the number 2345 we imply (without any further clarification) that what we actually mean is:

But this is only an agreement: If we do not specify the base, then it is assumed to be 10.

Now suppose that our agreement was that the base is 7. Then the number 2345 would have been interpreted as:

Which would have resulted to the decimal number: 2 * 343 +3 * 49 + 4 * 7 + 5 = 866.

We can be more specific if we write  2345₁₀  and  2345₇  where the subscript denotes the base used.

Note that if the base is 7 then we only need the digits 0,1,2,3,4,5,6 to represent any number. If we have 7, then we write 10₇  and, for 8 we write 11₇,  for 9 we write 12₇, etc..

This brings us to the binary representation. What should the base be if only digits 0 and 1 are to be used? The answer of course is 2.

If we write 100011001₂ , then what we mean, in decimal representation, is:

For numbers involving parts which are less than 1, the same rules apply.

For example, 12.438 stands for 1 * 10¹ + 2 * 10⁰ + 4 * 10^-1 + 3 * 10^-2 + 8 * 10^-3 and 1011.0111₂ stand for 2³ + 2¹ + 2⁰ + 2^-2 + 2^-3 + 2^-4 which in decimal representation gives:

Similarly we can convert a decimal number into binary. For example if we have the decimal number 47 we can write:

47

= 1 + 2 (23)

= 1 + 2 ( 1 + 2 ( 11 ) )

= 1 + 2 ( 1 + 2 ( 1 + 2 ( 5 ) ) )

= 1 + 2 ( 1 + 2 ( 1 + 2 ( 1 + 2 ( 2 ) ) ) )

= 1 + 2 ( 1 + 2 ( 1 + 2 ( 1 + 2 ( 0 + 2 ( 1 ) ) ) ) )

which results in:

101111 ₂