1 Numbers
Numbers are a way of expressing a quantity of things. While we could write these quantities in full words, like They have five apples,
we can also use numerals, like They have 5 apples.
Wether you should use full words or numerals is a matter of preference. In math and computer science however, we have generally agreed on always using numerals.
While most numeric representations use the digits 0 to 9, there are some exceptions. But before we get to those exceptions, lets first look at the numeric representation we already know.
2 Decimal numbers, our normal
numbers
When we write a numeral, we generally use one of the following symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. These are called digits.
There are numbers that only contain one digit: the natural numbers below
When we are increasing a number, we use the next digit in the list of digits for the new number.
But now we have a problem, since there is no digit after 9!
Technically speaking, every number has an infinite amount of zeroes to the left.
Knowing this, we can increase the digit to the left by one, and restart the digit we were already working with with the lowest digit!
By following this pattern, we can keep counting forever with just 10 digits!
3 Hexadecimal numbers
Hexadecimal numbers follow the same rules as decimal numbers, they just have more digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E and F.
Yep! You read that right. We just use the symbols we normally use for letters as digits here.
We also prepend hexadecimal numbers with a $-sign, so we cannot mistake them for decimal numbers.
So what happens if we have a hexadecimal number
The same thing as with decimal numbers! When we go from digit F back to 0, we increase the value of the next digit.
Keep in mind that
Some sources do not prefix hexadecimal numbers with a $-sign, but with 0x
, 0h
, 16r
or h
.
4 Binary numbers
We saw that decimal numbers and hexadecimal numbers follow the same rules. So do binary numbers, but again with a different amount of digits. Binary numbers only have access to the digits 0 and 1.
Binary numbers are prefixed with a %-sign. Keep in mind there is a big different between
Counting with binary numbers looks as follows:
Some sources do not prefix binary numbers with a %-sign, but with 0b
or bin
. In URI's you might see some number being prefixed by a %-sign as well, but that is a hexadecimal number, to make things confusing.
5 Converting between decimal, hexadecimal and binary numbers
Most OSses have a calculator that supports converting between numbers. Windows has calc.exe
and Ubuntu has Calculator
.
For both calculators you have to go to the settings and set the mode to programming-mode.
We can also do it by hand. Let's see how we accomplish that!
5.1 Converting decimal to hexadecimal
When converting from and to hexadecimal, it is important to know the powers of 2. We get these by repeatedly multiplying by 2.
| 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 | Result | |
|---|---|---|---|---|---|---|---|---|---|
| Binary | 1 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | %10001001 |
| Hexadecimal | 8 | 9 | $89 | ||||||
While this trick is handy for small decimal numbers, it is difficult to do this for numbers like
The steps to find it are as follows:
- Take a decimal number;
- Execute a long division on the decimal number until you get a remainder less than
- If the quotient is bigger than
- Take all the remainders from right to left to get the hexadecimal number.
This is how the steps would look for
If we take all the remainders, we get our digits for the hexadecimal number. Don't forget to convert the numbers above 9 to hexadecimal digits! One of our remainders is
The digits are as follows: 6, B, 1, 3, 2.
If we swap the order around, we get our hexadecimal number:
5.2 Converting hexadecimal to decimal
This one is a lot easier if we first look at decimal numbers. Lets take
We can write that number in the following ways:
What is great about this way of showing a number is that it applies to all number systems. We can do the same for the hexadecimal number
Which means that