Bases

When we normally count, in what we call decimal, is actually base-10, which means we use 10 numbers: from 0 to 9

So if you count:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9...

And then what? 10 of course, but why is that: you reached the end of your base, thus increment the second digit, and loop back to beginning for the first digit, you could use this notation to see it better:

08, 09, 10...

Another way to think of it is to use the calculation behind it, we use base 10, so each digit is number * base ^ rank, with the rank starting out at 0. Lets use 42:

(4 * 10¹) + (2 * 10⁰)
(4 * 10) + (2 * 1)

Any bases are possible, as long as you define what needs to be used to represent it, of course, some are more common than others, especially in computer science.

Binary

Given the previously explained on/off paradigm in electronic, it was chosen to use binary, or base-2 in computers. We so have 2 numbers to count: 0 and 1

If you apply the same as earlier:

0, 1

And that’s it, you reached the end of the base, so increment next digit:

0, 1, 10, 11, 100, 101, 110, 111...
0  1   2   3    4    5    6    7...

And as of 42:

101010

0 * 2⁰ + 1 * 2¹ + 0 * 2² + 1 * 2³ + 0 * 2⁴ + 1 * 2⁵
0      + 2      + 0      + 8      + 0      + 32
= 42

In computer, the number of bits you may have heard about, is the number of binary digit used to represent the number. The byte which is used to compute volume of data, is 8 bits, which allows for values up to 255.

To identify we sometimes use prefix 0b:

0b101010

Hexadecimal

To avoid the need to read binary all the time, it is actually easier given the data is grouped in bytes in computers to represent values in hexadecimal rather than binary or decimal. Hexadecimal is base-16, we use numbers from 0 to 9, and letters form A to F, and the prefix 0x:

0x1, 0x2, 0x3, 0x4, 0x5, 0x6, 0x7, 0x8, 0x9, 0xa, 0xb, 0xc, 0xd, 0xe, 0xf

Applying the same technique as before:

0x0e, 0x0f, 0x10, 0x11

And for 42:

0x2a

a * 16⁰ + 2 * 16¹
10 * 1 + 2 * 16
= 42

Ascii

Aka American Standard Code for Information Interchange, is the basic representation of characters used in computers, using numbers up to 128 to encode characters representation. It was later extended in various different versions to suite more languages.

For example A is 0x41 (65), and following capital letters are following in order, and a is 0x61 (97) and so on.

In binary, this means:

0x41 -> 0b01000001
0x61 -> 0b01100001

You may notice that the left hand bit is 0 in both case, as said, ascii is limited to 128, thus never using the bit of rank 7. The various extended versions make use of it, but have to be a different version for different languages as this still is quite limited.

Later, the unicode, and UTF variations where created, but this is a vast topic, which I will not talk about in here.