Preparing for My Final Exam… Number Systems: Decimal, Binary, Octal, and Hexadecimal

From Decimal to …

Binary (example: 45₁₀ → ?₂)

Rule: Repeatedly divide by 2 and record the remainder each time. Read remainders bottom → top.

Read up: 1 0 1 1 0 1 → 45₁₀ = 101101₂

Octal (example: 45₁₀ → ?₈)

Divide by 8 instead:

Read up → 55₈
45₁₀ = 55₈

Hexadecimal (example: 45₁₀ → ?₁₆)

Divide by 16:

Read up → 2 D → 45₁₀ = 2D₁₆

From Binary to …

Decimal (example: 110101₂ → ?₁₀)

Multiply each place value by a power of 2 and sum:

• 1×32 + 1×16 + 0×8 + 1×4 + 0×2 + 1×1
  = 32 + 16 + 0 + 4 + 0 + 1 = 53
  110101₂ = 53₁₀

Octal (example: 110101₂ → ?₈)

Group 3 bits from the right, convert each group to a single octal digit.

• 110101 → (110)(101) → 6, 5 → 65₈
• 1⋅4+1⋅2+0⋅1=4+2+0=6
• 1⋅4+0⋅2+1⋅1=4+0+1=5
  110101₂ = 65₈

Hexadecimal (example: 110101₂ → ?₁₆)

Group 4 bits from the right (pad left with zeros if needed).

• 110101 → 0011 0101 → 3, 5 → 35₁₆
• 0011₂ = 0×8+0×4+1×2+1×1=30×23+0×22+1×21+1×20=3
• 0101₂ = 0×8+1×4+0×2+1×1= 50×23+1×22+0×21+1×20=5 → 35₁₆
• 110101₂ = 35₁₆

From Octal to …

Decimal (example: 745₈ → ?₁₀)

Use powers of 8:

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• 7×64 + 4×8 + 5×1
  = 448 + 32 + 5 = 485
  745₈ = 485₁₀

Binary (example: 756₈ → ?₂)

Convert each octal digit to 3-bit binary:

• 7→111, 5→101, 6→110 → 111101110₂

756₈ = 111101110₂

Hexadecimal (example: 756₈ → ?₁₆)

Go via binary, then group in 4 bits:

• 756₈ → 111101110₂ → pad left: 0001 1110 1110
• 0001→1, 1110→E, 1110→E → 1EE₁₆
• 0001₂ = 0×8+0×4+0×2+1×1= 1
• 1110₂ = 1×8+1×4+1×2+0×2=14 → E
• 1110₂ = 14 → E
• 756₈ = 1EE₁₆

From Hexadecimal to …

Decimal (example: 2F₁₆ → ?₁₀)

Use powers of 16 (F = 15):

• 2×2=16+ 15×1 = 32 + 15 = 47
  2F₁₆ = 47₁₀

Binary (example: 2F₁₆ → ?₂)

Each hex digit → 4-bit binary:

• 2→0010, F→1111 → 0010 1111₂ (leading zeros optional → 101111₂)

2F₁₆ = 00101111₂

Octal (example: 2F₁₆ → ?₈)

Hex → Binary → group 3 bits:

• 2F₁₆ → 0010 1111₂ → group: 000 101 111 → 0, 5, 7
• 000=>0×4+0×2+0×1=0
• 101=>1×4+0×2+1×1=4+0+1=5
• 111​=1×4+1×2+1×1=4+2+1=7
• drop leading 0 → 57₈
  2F₁₆ = 57₈