Binairy numbers - only 1 and 0

Computer Myth

It is often said the computer's internal language consists of combinations of zeros and ones. That is not true. The computer would not know what to do with a 1 or a 0. These values are a representation for our convenience. All a computer understands is electronic signals.

An electronic signal is on or off, which translates to 1 or 0 for us. Each pattern of on and off signals represents a command or a value. Which combination represents what is determined by standards that have been set.

Extremely simplified you could compare it with a house shared by different people. Next to the doorbell there is a note, ring once for Joe, ring twice for Jane and so on. It is not a common rule, but it is what they agreed on.

Decimal Numbers

We have grown so accustomed to using our decimal system we don't need to think of the system behind it to be able use it.

When we see a number like 386, we see 386 as a whole number.
Taking it apart would result in:
3 x 100 plus,
8 x 10 plus,
6 x 1

Representing the same number in boxes would result in:

Going from right to left, the value of each box is 10 times higher than the former one.
We have all been moving a full box of 1s to the box on the left and starting over with an empty box of 1s, a full box of 10s to the box on the left and starting over with an empty box of 10s and so on, in school to grasp the logic of counting.

Why do we use 10? Because, going from 0 to 9, we have 10 different digits.

Binairy Numbers

Luckily for us, the binairy system, and more generally every system, works the same way.
The binairy system only has two digits, 0 and 1, so instead of using 10 we use 2.
A binairy number could be 1011.

Representing this in boxes results in:

Now it is easy to figure out what the decimal value of the binairy value 1011 is:
Going from right to left
1 x 1 plus,
1 x 2 plus,
0 x 4 plus,
1 x 8

The result is 11

The other way around is just as easy.
Let's say you need to convert 123 into its binairy value.
First find the box that has the highest number that is equal or lower than the number you need to convert.

In this case that is box 64, so we write a 1 in box 64

Now we have 123 - 64 = 59 left
The next one is box 32, we write a 1 in box 32

Now we have 59 - 32 = 27 left
The next one is box 16, we write a 1 in box 16

Now we have 27 - 16 = 11 left
The next one is box is 8, we write a 1 in box 8

Now we have 11 - 8 = 3 left
The next one is box 4, this doesn't fit, we only have 3, we write a 0 in box 4

The next one is box 2, we write a 1 in box 2

Now we have 3 - 2 = 1 left
The last one is box 1, we write a 1 in box 1

The result is
1111011

Calculations are performed the same way in every system. When a box is full we move it to the next block on the left and start over with an empty box.

Hexadecimal Numbers

With the information above you should be able to handle hexadecimal numbers as well. The hexadecimal system has 16 different digits. Since we only have 10 different numbers, they added characters to represent digits.

The available digits in a Hexadecimal system are:
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F

Looking at the boxes, the decimal system has base 10, the binairy system has base 2, so the hexadecimal system has base 16. All the systems work the same, it is just the value of the boxes that changes.

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