NUMBER BASE

We use different number systems to count and write numbers. These systems help us work with numbers in different ways. The main number systems are:

1. Decimal System
2. Binary System
3. Octal System
4. Hexadecimal System

1. Decimal Numeral System (Base-10)

This is the system we use every day. It has 10 digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9.

We call it “Base-10” because it uses 10 different digits.

How it works:

Each position in a decimal number has a value that is a power of 10.

Position values (from right to left):

• First position: 10⁰ = 1
• Second position: 10¹ = 10
• Third position: 10² = 100
• Fourth position: 10³ = 1000
• And so on…

Example:

Let’s look at the number 2538:

2538₁₀ = 2×10³ + 5×10² + 3×10¹ + 8×10⁰ = 2×1000 + 5×100 + 3×10 + 8×1 = 2000 + 500 + 30 + 8 = 2538

2. Binary Numeral System (Base-2)

Binary uses only 2 digits: 0 and 1.

We call it “Base-2” because it uses 2 different digits.

Computers use binary because electronic circuits can easily represent two states: ON (1) and OFF (0).

How it works:

Each position in a binary number has a value that is a power of 2.

Position values (from right to left):

• First position: 2⁰ = 1
• Second position: 2¹ = 2
• Third position: 2² = 4
• Fourth position: 2³ = 8
• Fifth position: 2⁴ = 16
• And so on…

Examples:

Let’s convert some binary numbers to decimal:

1. 10101₂ = 1×2⁴ + 0×2³ + 1×2² + 0×2¹ + 1×2⁰ = 16 + 0 + 4 + 0 + 1 = 21
2. 10111₂ = 1×2⁴ + 0×2³ + 1×2² + 1×2¹ + 1×2⁰ = 16 + 0 + 4 + 2 + 1 = 23
3. 100011₂ = 1×2⁵ + 0×2⁴ + 0×2³ + 0×2² + 1×2¹ + 1×2⁰ = 32 + 0 + 0 + 0 + 2 + 1 = 35

3. Octal Numeral System (Base-8)

Octal uses 8 digits: 0, 1, 2, 3, 4, 5, 6, and 7.

We call it “Base-8” because it uses 8 different digits.

How it works:

Each position in an octal number has a value that is a power of 8.

Position values (from right to left):

• First position: 8⁰ = 1
• Second position: 8¹ = 8
• Third position: 8² = 64
• Fourth position: 8³ = 512
• And so on…

Examples:

Let’s convert some octal numbers to decimal:

1. 27₈ = 2×8¹ + 7×8⁰ = 16 + 7 = 23
2. 30₈ = 3×8¹ + 0×8⁰ = 24 + 0 = 24
3. 4307₈ = 4×8³ + 3×8² + 0×8¹ + 7×8⁰ = 4×512 + 3×64 + 0×8 + 7×1 = 2048 + 192 + 0 + 7 = 2247

4. Hexadecimal Numeral System (Base-16)

Hexadecimal (often called “hex”) uses 16 digits: 0-9 and A-F.

• 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 are the same as in decimal
• A = 10, B = 11, C = 12, D = 13, E = 14, F = 15

We call it “Base-16” because it uses 16 different digits.

How it works:

Each position in a hexadecimal number has a value that is a power of 16.

Position values (from right to left):

• First position: 16⁰ = 1
• Second position: 16¹ = 16
• Third position: 16² = 256
• Fourth position: 16³ = 4096
• And so on…

Examples:

Let’s convert some hexadecimal numbers to decimal:

1. 28₁₆ = 2×16¹ + 8×16⁰ = 32 + 8 = 40
2. 2F₁₆ = 2×16¹ + F×16⁰ = 2×16 + 15×1 = 32 + 15 = 47
3. BC12₁₆ = B×16³ + C×16² + 1×16¹ + 2×16⁰ = 11×4096 + 12×256 + 1×16 + 2×1 = 45056 + 3072 + 16 + 2 = 48146

Why Do We Use Different Number Systems?

1. Decimal (Base-10): Easy for humans to use because we have 10 fingers.
2. Binary (Base-2): Used by computers because electronic circuits can easily represent two states: ON and OFF.
3. Octal (Base-8): Used to represent binary data in a shorter form. Every 3 binary digits can be converted to 1 octal digit.
4. Hexadecimal (Base-16): Used to represent binary data in an even shorter form. Every 4 binary digits can be converted to 1 hexadecimal digit.

Practice Questions

1. Convert 42₁₀ to binary.
2. Convert 1010₂ to decimal.
3. Convert 75₈ to decimal.
4. Convert 3F₁₆ to decimal.
5. Convert 100₁₀ to hexadecimal.