Disclaimer

I’m not an expert. I’m learning alongside you, and everything written here reflects how I currently understand the topic.
I’m learning by trying to teach others (often called the Feynman method).

Why are you even learning this?

I’m learning x86-64 assembly, so I need to refresh my understanding of numerical systems first. This is just part of a series, and I’ll be documenting everything I learn along the way. If you can’t explain what you’ve learned, then you don’t fully understand it.

The Number Systems

Decimal

The decimal number system is a base-10 system.
This means it uses 10 distinct digits:


0 1 2 3 4 5 6 7 8 9

This is the system we use for everyday counting, like 33, 45, etc.

Binary

“Bi” means two, so the binary system is a base-2 system.

0 1

0 usually represents off
1 usually represents on

Binary is heavily used in computers because hardware naturally maps to two states.

Hexadecimal

Hexadecimal is a base-16 number system.

hexa = 6
decimal = 10
6 + 10 = 16

First, we reuse the decimal digits:


0 1 2 3 4 5 6 7 8 9

Then we add six letters to represent values beyond 9:


A B C D E F

Which correspond to:

A = 10
B = 11
C = 12
D = 13
E = 14
F = 15

So hexadecimal digits represent:


0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

That’s 16 values total, including zero.

Converting a Decimal Number to Binary

To convert decimal to binary, we use successive division by 2.

Rules:

Divide the number by 2
Write down the remainder
Use integer division only
Stop when the quotient becomes 0
Read the remainders from bottom to top

Example: Convert 15 (decimal) to binary


15 ÷ 2 = 7 remainder 1
7  ÷ 2 = 3 remainder 1
3  ÷ 2 = 1 remainder 1
1  ÷ 2 = 0 remainder 1

Now read the remainders bottom to top:


1111b

So:


15 = 1111b

Another Example: Convert 12 (decimal) to binary


12 ÷ 2 = 6 remainder 0
6  ÷ 2 = 3 remainder 0
3  ÷ 2 = 1 remainder 1
1  ÷ 2 = 0 remainder 1

Read bottom to top:


1100b

So:


12 = 1100b

Converting a Decimal Number to Hexadecimal

The method is the same as binary conversion, but instead of dividing by 2, we divide by 16.

Example: Convert 15 (decimal) to hexadecimal


15 ÷ 16 = 0 remainder 15

Since the quotient is 0, we stop.

Now convert the remainder:

15 corresponds to F in hexadecimal

So the result is:


0x0F

Key Idea

If the number is smaller than the base, the quotient will be 0, and the remainder will be the number itself.

Another example: Convert 13 (decimal) to hexadecimal

Since 13 is smaller than the base (16), the quotient will be 0 and the remainder will be the number itself.


13 ÷ 16 = 0 remainder 13

In hexadecimal:

13 corresponds to D

So the result is: 0x0D

Converting a Hexadecimal Number to Decimal

Example 1: Convert 0x100 to decimal

Write out the hexadecimal digits:


1 0 0

List out the powers of 16 from left to right:


1     0     0
16²   16¹   16⁰

So it will be:


1 × 16² + 0 × 16¹ + 0 × 16⁰

List out the corresponding powers first:


1 × 256 + 0 × 16 + 0 × 1

(Any non-zero number raised to the power of 0 is equal to 1.)

Simplify it and add:


256 + 0 + 0

Meaning:


0x100 is 256 in decimal.

Example 2: Convert 0x1337 to decimal

Write out the hexadecimal digits:

1 3 3 7

List out the powers of 16 from left to right:

1    3    3    7
16³  16²  16¹  16⁰

So it will be:

1 × 16³ + 3 × 16² + 3 × 16¹ + 7 × 16⁰

Solve it until everything becomes single numbers, step by step so you won’t get confused:

4096 + 3 × 256 + 3 × 16 + 7

Last step:

4096 + 768 + 48 + 7

Which is:

Converting Hexadecimal to Binary

Let’s use 0x0D as an example.

Convert letters to numbers first if there are any

D corresponds to 13.

Now use this as a base (4-bit values):

8 4 2 1

What numbers add up to 13? 8, 4, and 1

8 4 2 1 
1 1 0 1

Even if a digit is 0, we still apply the same principle and always write 4 bits.

Let’s try 0xDF

D = 13, F = 15

For D(13):

8 4 2 1 
1 1 0 1

because 8 + 4 + 1 is 13 .

For F(15):

8 4 2 1
1 1 1 1

Because 8 + 4 + 2 + 1 is 15.

So 0xDF = 11011111

Last practice: 0x1337

No letter, so we use the digits directly.

8 4 2 1 
0 0 0 1

For 3:

8 4 2 1 
0 0 1 1

For 3 again:

8 4 2 1 
0 0 1 1

For 7

8 4 2 1 
0 1 1 1

Putting everything together: 0001 0011 0011 0111 so the binary reperesentation of hexaddecimal 1337 is: 0001001100110111