Chapter 2. Binary and Number Representation

Binary is a number system which builds numbers from elements called bits. Each bit can be represented by any two mutually exclusive states. Generally, when we write it down or code bits, we represent them with 1 and 0. We also talk about them being true and false, and the computer internally represents bits with high and low voltages.

We build binary numbers the same way we build numbers in our traditional base 10 system. However, instead of a one's column, a 10's column, a 100's column (and so on) we have a one's column, a two's columns, a four's column, an eight's column, and so on, as illustrated below.

For example, to represent the number 203 in base 10, we know we place a 3 in the 1's column, a 0 in the 10's column and a 2 in the 100's column. This is expressed with exponents in the table below.

Or, in other words, 2 × 10² + 3 × 10⁰ = 200 + 3 = 203. To represent the same thing in binary, we would have the following table.

That equates to 2⁷ + 2⁶ + 2³+2¹ + 2⁰ = 128 + 64 + 8 + 2 + 1 = 203.

To represent all the letters of the alphabet we would need at least enough different combinations to represent all the lower case letters, the upper case letters, numbers and punctuation, plus a few extras. Adding this up means we need probably around 80 different combinations.

If we have two bits, we can represent four possible unique combinations (00 01 10 11). If we have three bits, we can represent 8 different combinations.

8 bits gives us 28 = 256 unique representations, more than enough for our alphabet combinations. We call a group of 8 bits a byte. Guess what size a C char variable type is? One byte.

Hexadecimal refers to a base 16 number system. We use this in computer science for only one reason, it makes it easy for humans to think about binary numbers. Computers only ever deal in binary and hexadecimal is simply a shortcut for us humans trying to work with the computer.

So why base 16? Well, the most natural choice is base 10, since we are used to thinking in base 10 from our every day number system. But base 10 does not work well with binary -- to represent 10 different elements in binary, we need four bits. Four bits, however, gives us sixteen possible combinations. So we can either take the very tricky road of trying to convert between base 10 and binary, or take the easy road and make up a base 16 number system -- hexadecimal!

Hexadecimal uses the standard base 10 numerals, but adds A B C D E F which refer to 10 11 12 13 14 15 (note we start from zero).

Traditionally, any time you see a number prefixed by 0x this will denote a hexadecimal number.

As mentioned, to represent 16 different patterns in binary, we would need exactly four bits. Therefore, each hexadecimal numeral represents exactly four bits. You should consider it an exercise to learn the following table off by heart.

Of course there is no reason not to continue the pattern (say, assign G to the value 16), but 16 values is an excellent trade off between the vagaries of human memory and the number of bits used by a computer (occasionally you will also see base 8 used, for example for file permissions under UNIX). We simply represent larger numbers of bits with more numerals. For example, a sixteen bit variable can be represented by 0xAB12, and to find it in binary simply take each individual numeral, convert it as per the table and join them all together (so 0xAB12 ends up as the 16-bit binary number 1010101100010010). We can use the reverse to convert from binary back to hexadecimal.

We can also use the same repeated division scheme to change the base of a number. For example, to find 203 in hexadecimal

Hence 203 in hexadecimal is 0xCB.