Table Of Contents

Basics

Decimal (base-10)

We as humans count in the base-10 format viz. 1, 2, 3, ... 10, 11 etc, we start counting from 0, and go up to 9. After nine, we run out of numbers, so we add one to the left, and start over from 0 again
The base is also known as radix

8653   = (8 x 10^3) + (6 x 10^2) + (5 x 10^1) + (3 x 10^0)
212.11 = (2 x 10^2) + (1 x 10^1) + (2 x 10^0) + (1 x 10^-1) + (1 x 10^-2)

Binary (base-2)

Starting at 0, we count to 1, after that we add a 1 to the left and then roll back to 0. i.e. 0, 1, 10, 11, 100, 101, 111, 1000, 1001, 1011, 1111, 10000 etc.
Every single operation that takes place inside a computer involves binary, everything is expressed in zeros and ones.
We can use electrical circuits to represent binary numbers when they're powered on. There are certain range-in volts, for example: If the voltage is b/w 0-1, it will mean 0 in binary, and if 3-4, it will mean 1 in binary.

Representing an ASCII string in binary
Each letter is represented by eight bits (because it's ASCII)
   L        o        o        k       space      o        u        t      !
--------------------------------------------------------------------------------   
01001100 01101111 01101111 01101011 00100000 01101111 01110101 01110100 00100001
 8-bit     8-bit    8-bit   8-bit     8-bit   8-bit    8-bit     8-bit   8-bit

Hexadecimal (base-16)

Everything is done by a computer in binary, but reading and understanding them is very inconvenient for us. Hexadecimal is the solution for better readability and conserving monitor space whilst reading the same.
Hexadecimals are base-16 numbers, but we have only digits upto 9, after that we use letters A-F

0 1 2 3 4 5 6 7 8 9 A B C D E F

Uppercase and lowercase Hex digits have the same meaning, but we should be consistent with the one we choose.
base-16 relies on 4-bit system because 16 unique combinations of bit patterns are possible, think about this, don't panic. Hex is possible between 0 - 15, for representing 15 in binary, we need four placeholders _ _ _ _, so for Hex, 4-bit placeholder system for each character is followed.

Binary    Hex Value   Decimal
-----------------------------------
0000      0           0
0001      1           1
0010      2           2
0011      3           3
0100      4           4
0101      5           5
0110      6           6
0111      7           7
1000      8           8
1001      9           9
1010      A           10
1011      B           11
1100      C           12
1101      D           13
1110      E           14
1111      F           15

Octal (base-8)

0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 16, 17, 20   ...

Notation

For representing different numbering systems, a standard is to be followed in order to avoid confusion.

**Hexadecimal** 
1.  Place 0x before the hex numeral:  **0x33**
2.  Append a lowercase letter h after the numeral: **33h**
3.  Use the 16 subscript: **(33)_16**
4.  Specify base 16: **33 (base 16)**

---------------------------------------------------------

**Binary**
1. Place 0b before the binary: **0b1101**
2. Append a lowercase b at the end: 1000b
3. Use the 2 subscript: **(11)_2**
4. Specify the base beforehand: **10 (base 2)**

---------------------------------------------------------

Decimal
1. Place a 0d before the decimal: **0d112**
2. Mention base-10 in the subscript: **(112)_10**
3. Append a lowercase d after the decimal: **112d**
4. Specify the base beforehand: 10 (base 10)

Conversion b/w anything to decimal

The thinking behind it

We follow the same method as seen in this one, just the weight changes, nothing else does.
This is what we have normally been doing since years $(number)_{10} = digit * (weight) ^ {place} + .......$
For converting a number of some other number system, we follow the same technique, just the base-number changes. If it has some sort of decimal, we use the negative powers as we did in at the start, in the base-10 section.

The methods

<aside> 👉 Method 1

</aside>

We use the same method in Binary, only difference is the base.

Untitled

<aside> 👉 Method 2

</aside>