## 1. Octal encoding

Speaking of encodings... As you know, in everyday life we use decimal notation: all our numbers are represented using 10 symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. There are 10 numerals, so the system is called decimal.

But programmers are big-time inventors. They immediately came up with encodings that use a different number of symbols. For example, 16, 8 and 2.

The alternative encoding that uses 8 symbols is the easiest: just drop the 8 and 9 and you get an octal encoding (the octal numeral system).

And, yes, you can use the octal system to specify numeric literals. If, of course, you really need to. It's easier than it sounds. You just need to write the number 0 before the number.

In other words, Java treats any integer literal starting with 0 as an octal value.

Examples:

Code Notes
``int x = 015; ``
x is 13: 1*8+5
``int x = 025; ``
x is 21: 2*8+5
``int x = 0123; ``
x is 83: 1*64+2*8+3 == 1*82+2*81+3*80
``int x = 078;``
This will not compile: 8 is not one of the symbols used in the octal encoding.

It's unlikely that you will need to write octal numbers in your code, but you should know what they are. After all, you will have to read code written by others. And as mentioned above, programmers are big inventors.

Well, remember that you can't simply write 0 in front of every number.

## 2. Binary encoding

Binary encoding is even more interesting. If octal has only the digits 0-7, then binary has only 0 and 1. Why is this encoding necessary?

This has everything to do with the internal structure of a computer. Everything in a computer runs on electricity, and as it happens, the most efficient way to store and transmit something using electricity is to use two states: either there is no electricity in the wire (zero) and there is electricity (one).

This is the origin of the popularity of the binary numeral system.

In principle, it is not used very often in Java: Java is considered a high-level language, completely abstracted from the hardware it runs on. Indeed, do you really care what format is used to store and process data inside a computer?

But over the past decades, programmers have come to love the binary encoding (and other encodings based on it). As a result, Java has operators that take binary numbers as inputs. And the accuracy of floating-point numbers depends on their binary representation.

And as was the case with octal encoding, Java has a way to encode literals using the binary system. That is, literals comprised only of 0s and 1s. In order for the Java compiler to understand that the code contains a numeric literal encoded in binary rather than simply a decimal number consisting of zeros and ones, all binary literals are identified using the prefix 0b (the 'b' comes from the word binary).

Examples:

Code Notes
``int x = 0b100; ``
x is 4: 1*4+0*2+0
``int x = 0b1111; ``
x is 15: 1*8+1*4+1*2+1
``int x = 0b1111000111; ``
x is 967: 1*29+1*28+1*27+1*26+0*25+0*24+0*23+1*22+1*2+1;
``int x = 0b12000;``
This will not compile: 2 is not one of the symbols used in the binary encoding.

In addition to octal and binary encodings, literals can also be written in hexadecimal. This is a very popular encoding.

That is because although binary notation is as close as possible to how numbers are actually stored, it is too difficult for humans to effectively work with such numbers: in binary, the number one million contains 20 digits, not not 7.

That's why programmers came up with the hexadecimal system. After all, 16 is 2 raised to the 4th power, so exactly 4 bits correspond to one hexadecimal digit. Roughly speaking, every 4 bits can now be written as a single hexadecimal digit.

The hexadecimal encoding also has its own unique prefix: 0x. Examples:

Decimal number Binary notation Hexadecimal notation
17 0b00010001 0x11
41 0b00101001 0x29
85 0b01010101 0x55
256 0b100000000 0x100

Ok, you say, it's clear enough how we got the octal system: we just threw out the numbers 8 and 9, but where do we get the 6 extra symbols for the hexadecimal system? I would like to see them!

It's all straightforward. The first 6 letters of the English alphabet were taken as the 6 missing symbols: A (10), B (11), C (12), D (13), E (14), F (15).

Examples:

Hexadecimal notation Binary notation Decimal number
0x1 0b00000001 1
0x9 0b00001001 9
0xA 0b00001010 10
0xB 0b00001011 11
0xC 0b00001100 12
0xD 0b00001101 13
0xE 0b00001110 14
0xF 0b00001111 15
0x1F 0b00011111 31
0xAF 0b10101111 175
0xFF 0b11111111 255
0xFFF 0b111111111111 4095

## 4. How to convert a number from hexadecimal

Converting a number from hexadecimal to decimal is very easy. Let's say you have the number 0xAFCF. How much is that in decimal?

First, we have a positional number system, which means the contribution of each digit to the overall number increases by a factor of 16 as we move from right to left:

A*163 + F*162 + C*161 + F

The symbol A corresponds to the number 10, the letter C corresponds to the number 12, and the letter F represents fifteen. We get:

10*163 + 15*162 + 12*161 + 15

Raising 16 to the various powers that correspond to the digits, we get:

10*4096 + 15*256 + 12*16 + 15

We sum everything up and get:

45007

You know how this number is stored in memory:

0xAFCF

But now let's convert it to binary. In binary it would be:

0b1010111111001111

Every set of four bits corresponds to exactly one hexadecimal character. That's super convenient. Without any multiplication or exponentiation."