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Worked Examples: Number Conversions

Step-by-step solutions for all conversion types

Example 1 Foundation

Binary to Denary

Question

Convert the binary number 11010110 to denary.

Solution

1 Write out the place values

1286432168421

2 Write the binary digits underneath

1286432168421

11010110

3 Add place values where there's a 1

128 + 64 + 16 + 4 + 2 = 214

Final Answer

11010110₂ = 214₁₀

Example 2 Foundation

Denary to Binary (Division Method)

Question

Convert denary 179 to an 8-bit binary number.

Solution

1 Repeatedly divide by 2, recording remainders

179 ÷ 2 = 89remainder 1

89 ÷ 2 = 44remainder 1

44 ÷ 2 = 22remainder 0

22 ÷ 2 = 11remainder 0

11 ÷ 2 = 5remainder 1

5 ÷ 2 = 2remainder 1

2 ÷ 2 = 1remainder 0

1 ÷ 2 = 0remainder 1

2 Read remainders from bottom to top

Reading up: 1 → 0 → 1 → 1 → 0 → 0 → 1 → 1 = 10110011

Final Answer

179₁₀ = 10110011₂

Verify: 128 + 32 + 16 + 2 + 1 = 179 ✓

Example 3 Core

Binary Addition

Question

Add the binary numbers 01011011 and 00101110.

State whether overflow occurs.

Solution

1 Set up the addition (rightmost column first)

  01011011
+ 00101110
----------

2 Add column by column, carrying when needed

Column 1 (rightmost): 1 + 0 = 1

Column 2: 1 + 1 = 10, write 0, carry 1

Column 3: 0 + 1 + 1(carry) = 10, write 0, carry 1

Column 4: 1 + 0 + 1(carry) = 10, write 0, carry 1

Column 5: 1 + 1 + 1(carry) = 11, write 1, carry 1

Column 6: 0 + 0 + 1(carry) = 1

Column 7: 1 + 1 = 10, write 0, carry 1

Column 8: 0 + 0 + 1(carry) = 1

3 Write the result

    1 1 1 1  ← carries
  01011011  (91)
+ 00101110  (46)
----------
  10001001  (137)

Final Answer

01011011 + 00101110 = 10001001

Verify: 91 + 46 = 137 ✓

No overflow - result (137) fits within 8 bits (max 255)

Example 4 Core

Hexadecimal to Denary

Question

Convert hexadecimal B7 to denary.

Solution

1 Identify the hex digits and their denary values

B = 11, 7 = 7

2 Multiply by place values (powers of 16)

B (11) is in the 16s column: 11 × 16 = 176

7 is in the 1s column: 7 × 1 = 7

3 Add the results

176 + 7 = 183

Final Answer

B7₁₆ = 183₁₀

Example 5 Core

Binary to Hexadecimal

Question

Convert binary 11011001 to hexadecimal.

Solution

1 Split into groups of 4 bits (from right)

1101 | 1001

2 Convert each group to hex

1101 = 8 + 4 + 1 = 13 = D

1001 = 8 + 1 = 9 = 9

Final Answer

11011001₂ = D9₁₆

Verify: D9 = (13×16) + 9 = 208 + 9 = 217

Binary: 128+64+16+8+1 = 217 ✓

Example 6 Core

Binary Shifts

Question

a) Perform a left shift by 2 on 00010100 (20)

b) Perform a right shift by 2 on 01100100 (100)

Solution (a) - Left Shift

1 Move all bits left by 2 positions

Original: 0 0 0 1 0 1 0 0

After: 0 1 0 1 0 0 0 0

Fill empty positions on right with 0s

2 Calculate expected result

Left shift by 2 = multiply by 2² = multiply by 4

20 × 4 = 80

Answer (a)

00010100 << 2 = 01010000 (80)

Solution (b) - Right Shift

1 Move all bits right by 2 positions

Original: 0 1 1 0 0 1 0 0

After: 0 0 0 1 1 0 0 1

Fill empty positions on left with 0s, rightmost bits are lost

2 Calculate expected result

Right shift by 2 = divide by 2² = divide by 4

100 ÷ 4 = 25

Answer (b)

01100100 >> 2 = 00011001 (25)

Worked Examples: Number Conversions

Binary to Denary

Question

Solution

Final Answer

Denary to Binary (Division Method)

Question

Solution

Final Answer

Binary Addition

Question

Solution

Final Answer

Hexadecimal to Denary

Question

Solution

Final Answer

Binary to Hexadecimal

Question

Solution

Final Answer

Binary Shifts

Question

Solution (a) - Left Shift

Answer (a)

Solution (b) - Right Shift

Answer (b)

Practice Problems

1. Convert 10101100 to denary

2. Convert 145 to binary

3. Add 01110011 + 00011010

4. Convert C4 to denary

5. Convert 10111010 to hex

6. Shift 00011000 left by 1