Binary floating-point representation is used to store real numbers that include fractional parts. It works in a similar way to standard form (scientific notation) in denary.
In denary, a number may be written as:
\[
\text{Number} = \text{Mantissa} \times 10^{\text{Exponent}}
\]
In binary floating-point, the base is 2 instead of 10:
\[
\text{Number} = \text{Mantissa} \times 2^{\text{Exponent}}
\]
The mantissa contains the significant digits, and the exponent shows how far the binary point moves.
A floating-point number is stored in two parts:
Both are usually stored using two’s complement.
A floating-point number is normalised when:
This ensures maximum precision.
Examples:
Valid normalised mantissa:
\[
0.101101
\]
Invalid (not normalised):
\[
0.001011
\]
\[
\text{Number} = \text{Mantissa} \times 2^{\text{Exponent}}
\]
Mantissa:
\[
0.101_2
\]
Exponent:
\[
010_2
\]
Step 1: Convert mantissa
\[
0.101_2 =
(1 \times 2^{-1}) +
(0 \times 2^{-2}) +
(1 \times 2^{-3})
\]
\[
= 0.5 + 0 + 0.125
\]
\[
= 0.625
\]
Step 2: Convert exponent
\[
010_2 = 2
\]
Step 3: Apply formula
\[
0.625 \times 2^2
\]
\[
= 0.625 \times 4
\]
\[
= 2.5
\]
Final Answer: 2.5 denary
Convert:
\[
6.5_{10}
\]
Step 1: Convert to binary
\[
6 = 110
\]
\[
0.5 = 0.1
\]
\[
6.5 = 110.1
\]
Step 2: Normalise
\[
0.1101 \times 2^3
\]
Step 3: Result
Mantissa:
\[
0.1101
\]
Exponent:
\[
011
\]
If the number is negative:
Exponent remains normal signed integer.
Floating point allows representation of:
Example:
\[
0.0000000000000000001
\]
Floating-point numbers have limited precision.
This causes:
Example:
\[
0.1_{10}
\]
Cannot be stored exactly in binary.
\[
\text{Number} =
\text{Mantissa}
\times
2^{\text{Exponent}}
\]
1. Convert the following floating-point number to denary:
Mantissa:
\[
0.101
\]
Exponent:
\[
011
\]
2. Convert the denary number 9.25 to floating-point form.
3. Explain why normalisation is used.
4. Convert:
\[
0.01101 \times 2^4
\]
to denary.