Bismillahir Rahmanir Rahim

Understanding Number Systems: Decimal, Binary, and Hexadecimal

Nice to meet you again! While you are likely accustomed to working with everyday decimal numbers, digital systems rely heavily on Binary (0s and 1s) and Hexadecimal systems. This is primarily due to their convenience and efficiency in hardware design and programming.

Let's break down how these systems work and how to seamlessly convert between them.

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1. Decimal Digits (Base 10)

The decimal system uses ten digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.

Each column represents a value that is 10 times more than the previous column (reading from right to left). These are also called Base 10 numbers because we represent each column as a multiple of a power of 10.

• Capacity: An N-digit decimal number can represent up to 10ᴺ possibilities.

• Example: A 3-digit number has 10³ = 1000 possibilities (ranging from 000 to 999).

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2. Binary Numbers (Base 2)

In the binary system, "bits" represent only one of two values: 0 or 1. Each column carries twice the weight of the previous column, making this a Base 2 system.

The Powers of 2 Table

If you work in programming or IT, you will save significant time by memorizing the powers of 2 up to 2¹⁶:

Power	Value		Power	Value
2⁰	1		2⁹	512
2¹	2		2¹⁰	1024 (1k)
2²	4		2¹¹	2048
2³	8		2¹²	4096
2⁴	16		2¹³	8192
2⁵	32		2¹⁴	16384
2⁶	64		2¹⁵	32768
2⁷	128		2¹⁶	65536
2⁸	256

Converting Binary to Decimal: Let's convert 1101 (binary) to decimal: = (1 * 2³) + (1 * 2²) + (0 * 2¹) + (1 * 2⁰) = 8 + 4 + 0 + 1 = 13 (decimal)

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3. Hexadecimal Numbers (Base 16)

Binary numbers can become very long and prone to errors when written out. To simplify this, we use Hexadecimal (Base 16).

A group of 4 bits gives us 16 possibilities (2⁴ = 16). With just one hexadecimal digit, we can represent an entire 4-bit sequence!

Hexadecimal Conversion Table

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

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4. Practical Conversions

Here is how you can practically convert numbers between these three systems.

A. Hexadecimal to Binary and Decimal

Goal: Convert 2ED (hex) to binary and decimal.

• Hex to Binary: Map each character using the table above.

  • 2 = 0010

  • E = 1110

  • D = 1101

  • Result: 001011101101 (binary)

• Hex to Decimal: Multiply by powers of 16.

  • = (2 * 16²) + (14 * 16¹) + (13 * 16⁰)

  • = 512 + 224 + 13

  • Result: 749 (decimal)

B. Binary to Hexadecimal

Goal: Convert 1111010 (binary) to hexadecimal.

1. Group into 4 bits: Start from the right. Pad with leading zeros if necessary: 0111 and 1010.

2. Map to Hex: * 1010 = A

   • 0111 = 7

3. Result: 7A (hex)

C. Decimal to Hexadecimal and Binary

Goal: Convert 333 (decimal) to hexadecimal and binary.

There are two primary methods for doing this: the Subtraction Method and the Repeated Division Method.

Method 1: The Subtraction Method (Descending Powers)

You subtract the largest possible power of the target base until you reach zero.

• Decimal to Hexadecimal:

  1. The largest power of 16 less than 333 is 256 (which is 16²).

  2. 256 goes into 333 once (1 * 256). Remainder: 333 - 256 = 77.

  3. 16 goes into 77 four times (4 * 16 = 64). Remainder: 77 - 64 = 13.

  4. 13 in decimal is D in hex.

  5. Hex Result: 14D (hex)

• Decimal to Binary:

  1. 333 - 256 (which is 2⁸) = 77

  2. 77 - 64 (which is 2⁶) = 13

  3. 13 - 8 (which is 2³) = 5

  4. 5 - 4 (which is 2²) = 1

  5. 1 - 1 (which is 2⁰) = 0

  • Since binary digits exist at the positions for 2⁸, 2⁶, 2³, 2², and 2⁰, we place a 1 in those columns and a 0 everywhere else.

  • Binary Result: 101001101

Method 2: Repeated Division (The Remainder Method)

This is often the easiest method for programming or working with large numbers. You repeatedly divide the decimal number by the target base (2 or 16) and record the remainders. Read the remainders from bottom to top to get your answer.

• Decimal to Hexadecimal (Divide by 16):

  1. 333 ÷ 16 = 20, remainder 13 (which is Hex D)

  2. 20 ÷ 16 = 1, remainder 4 (which is Hex 4)

  3. 1 ÷ 16 = 0, remainder 1 (which is Hex 1)

  • Read bottom to top Result: 14D (hex)

• Decimal to Binary (Divide by 2):

  1. 333 ÷ 2 = 166, remainder 1

  2. 166 ÷ 2 = 83, remainder 0

  3. 83 ÷ 2 = 41, remainder 1

  4. 41 ÷ 2 = 20, remainder 1

  5. 20 ÷ 2 = 10, remainder 0

  6. 10 ÷ 2 = 5, remainder 0

  7. 5 ÷ 2 = 2, remainder 1

  8. 2 ÷ 2 = 1, remainder 0

  9. 1 ÷ 2 = 0, remainder 1

  • Read bottom to top Result: 101001101 (binary)