Floating-point numbers, representation and manipulation — practice questions

1. Convert to Binary:
   • Integer part: $13_{10} = 8 + 4 + 1 = 1101_2$
   • Fractional part: $0.5_{10} = 1/2 = .1_2$
   • Combined: $13.5_{10} = 1101.1_2$
2. Normalise the Binary Number:
   • To make the mantissa start with '0.1', we must shift the binary point 4 places to the left: $1101.1 \rightarrow .11011$
   • The number is positive, so the mantissa starts '0.1...'. This format is correct: $0.11011$.
3. Determine the Exponent:
   • We shifted the point 4 places to the left, so the exponent is $+4$.
   • Convert 4 to 6-bit two's complement: $4_{10} = 000100_2$.
4. Format the Mantissa:
   • The normalised value is $0.11011$.
   • In a 10-bit two's complement format, we pad with zeros: $0110110000_2$.

Final Answer:

• Mantissa: 0110110000
• Exponent: 000100

1. Convert the positive equivalent to Binary:
   • First, consider +5.25.
   • Integer part: $5_{10} = 4 + 1 = 101_2$
   • Fractional part: $0.25_{10} = 1/4 = .01_2$
   • Combined: $5.25_{10} = 101.01_2$
2. Normalise the positive binary number:
   • Shift the binary point 3 places to the left: $101.01 \rightarrow .10101$
   • The mantissa for +5.25 is $0.10101$.
3. Determine the Exponent:
   • We shifted the point 3 places, so the exponent is $+3$.
   • Convert 3 to 4-bit two's complement: $3_{10} = 0011_2$.
4. Format the Mantissa for the negative number:
   • First, write the positive mantissa in 8-bit form: $01010100_2$.
   • Now, convert to a negative number using two's complement:
     • Invert the bits: 10101011
     • Add 1: 10101100
   • This mantissa starts with '10', which is the correct normalised form for a negative number.

Final Answer:

• Mantissa: 10101100
• Exponent: 0011