How Hexidecimal Works

Below is a post I wrote a while back somewhere else about how Hexidecimal works. This will become important for when I post the next part of my NES emulator series.

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Decimals to Hex and Binary

The numbers we use everyday work on a base ten system. That is we count from 0 to 9, then add a 1 in front of the 9 then return the 9 back to 0 ( 09 becomes 10 ). At 99, a 1 gets added in front of the 99, and the 99 loops back to 00 ( 099 becomes 100 ). This way of looking at it is very important to understand other base systems.

Now imagine a system that works on base sixteen instead of base ten. You would essentially be counting up to 15, then adding a 1 in front and setting 15 to zero. So, like 8, 9, 10, 11, 12… you would have (0)(13), (0)(14), (0)(15), (1)(0), (1)(1)…. and so on. ( (0)(15) becomes (1)(0) )

I used brackets to separate each 'digit' but we need a way to represent 10-15 as a one digit character. Lets replace 10, 11, 12, 13, 14 and 15 with A,B,C,D,E,F. So 9 in decimal would still be 9, but 10 in decimal would be A. 15 in decimal would be F, and 16 in decimal would be 10, because if you go one higher, you set F to zero and add 1 at the beginning (0F becomes 10). Congratulations, you have learned the essentials of Hexadecimal.

Converting Dec to Hex

An easy way to convert to Hex is by looking at the number of digits and the values each represent. As an example, we will look a couple of examples in decimal. The number 300. It has three digits, and can be rewritten as 3x10^2. The number 50 can be rewritten as 5x10^1. Finally, 8 can be rewritten as 8x10^0. Because 300 + 50 + 8 = 358, you could say that (3x10^2)+(5x10^1)+(8x10^0) = 358.

That in mind, lets do the same for Hex. We will decode, or find the decimal value that is associated with the hex value 0x166. The "0x" will now be added to distinguish Hex from decimal. We can rewrite this number as (1x16^2)+(6x16^1)+(6x16^0) to give us 358.

Finally, lets encode a Hex value given a decimal number. Starting small, this number will be 39.
- To start, we want to see how many powers of 16 the number has. So grab a calculator, unless you are good with exponents. See how many times you can multiply 16 by its self without going over your decimal number. In this example, we can only go as far as 16^1, because 16^2 is 256, way over.
- Now we need to see how close we can get to our number by multiplying 16, again without going over. The most we can go is 2x16^1 = 32.
- We need to find what is left after this multiplication, so subtracting 32 from 39 leaves us with 7.
- Because this is inside the base 16, we leave this at 7.
- Therefore, the hex value for 39 is 0x27

Converting Hex to Binary

Binary is simply another system of numbering. Remember that while decimal is base-10 (0-9), and hexidecimal is base-16 (0-F), binary is base-2 (0-1)! 

To count up in binary, you start at zero. Then you count up until the number reaches 1. Then, on the next count, you set that digit back to zero and add a 1 in front of it, like so:

0 -> 1 -> 10 -> 11 -> 100 -> 101 -> 110 -> 111-> 1000 -> 1001 -> 1010 -> 1011 -> 1100 -> 1101 -> 1110 -> 1111

Now, here is where it gets interesting. Check this table out:

 

1111 corresponds to F... So, the last one digit hexidecimal number fits in four digits of binary. In computer science, each 1 or a 0 of binary is called a bit. Because one digit of Hex is 4 bits, you can fit 2 digits of Hex in a byte (which is 8 bits).

11111111 is equal to 0xFF, and equal to 255. In one byte, you can store a number up to 0xFF (255 in decimal).

Infographic

Here is the infographic that really helped me learn this. Credits to the original owner... I just found it on Google Images :)

Count Like a Computer Infographic