floating-point-conversions

Denary to floating-point binary

How do you convert denary to floating-point binary?

• For the following conversions we are assuming:

  • 12-bit mantissa (bit 0 is the sign: 0 = +, 1 = -)

  • 4-bit exponent (in two’s complement, unbiased)

1. Convert the number into binary (split into integer part + fractional part)

   • Example: 5 = 101₂, 0.25 = 0.01₂, so 5.25 = 101.01₂

2. Normalise the binary into the form 0.xxxxx × 2ⁿ

   • Move the binary point until there is one non-zero digit to the right of it

   • Count how many places the binary point moved = the exponent

3. Build the mantissa (12 bits):

   • First bit = sign (0 for +, 1 for −)

   • Next 11 bits = binary digits after the binary point

   • Pad with zeros on the right if needed

4. Convert the exponent to 4-bit two’s complement (unbiased)

   • Positive exponents are normal binary

   • Negative exponents use two’s complement

5. Write the final floating-point number as (Mantissa | Exponent)

Example: +5.25

• Step 1: Convert to binary

  • 5.25 = 101.01₂

• Step 2: Normalise

  • 101.01₂ = 0.10101 × 2³

• Step 3: Build mantissa (12 bits)

  • Sign = 0 (positive)

  • Fraction = 10101 → pad → 10101000000

  • Mantissa = 010101000000

• Step 4: Exponent

  • +3 = 0011

• Final representation:
  Mantissa = 010101000000
  Exponent = 0011

Example: −5.25

• Step 1: Convert the magnitude (ignore the sign for now)

  • First, convert +5.25 to binary → 101.01₂

  • Normalise: 101.01₂ = 0.10101 × 2³

  • You always start by working with the positive version, then apply the negative sign later

• Step 2: Build the mantissa (12 bits)

  • Sign = 1 (negative)

  • Fraction = 10101 → pad → 10101000000

  • Mantissa = 110101000000

• Step 3: Exponent

  • Exponent is the same as for +5.25 → +3 = 0011

• Final representation:
  Mantissa = 110101000000
  Exponent = 0011

Summary table

Floating-point binary to denary

How do you convert floating-point binary to denary?

• For the following conversions we are assuming:

  • 12-bit mantissa (sign + 11 fraction bits)

  • 4-bit exponent (two’s complement, unbiased)

1. Read the exponent (4 bits)

   • Convert from two’s complement into denary

   • Positive values = normal binary

   • Negative values = two’s complement

2. Check the mantissa sign

   • First bit = sign (0 = +, 1 = −)

   • Remember: the rest of the mantissa is just the fraction, not in two’s complement

3. Restore the fraction

   • Write it as 0.<fraction bits> in binary

   • Example: mantissa 010101000000 → fraction bits 10101000000 → 0.10101₂

4. Apply the exponent

   • Shift the binary point right if exponent is positive, left if exponent is negative

   • Multiply the fraction by 2^(exponent)

5. Apply the sign

   • If the sign bit was 1, make the final result negative

Convert the following to denary (+)

Mantissa: 010101000000
Exponent: 0011

• Step 1: Exponent

  • 0011₂ = +3

• Step 2: Mantissa sign

  • First bit = 0 → positive

• Step 3: Restore fraction

  • Fraction bits = 10101000000

  • Fraction = 0.10101₂

• Step 4: Apply exponent

  • 0.10101 × 2³ = 101.01₂ = 5.25₁₀

• Step 5: Apply sign

  • Positive → stays +5.25

• Final result: +5.25

Convert the following to denary (-)

Mantissa: 110101000000
Exponent: 0011

• Step 1: Exponent

  • 0011₂ = +3

• Step 2: Mantissa sign

  • First bit = 1 → negative

• Step 3: Restore fraction

  • Fraction bits = 10101000000

  • Fraction = 0.10101₂

• Step 4: Apply exponent

  • 0.10101 × 2³ = 101.01₂ = 5.25₁₀

• Step 5: Apply sign

  • Negative → −5.25

• Final result: −5.25

Summary table