In simple terms
A friendly intro before the formal notes — no formulas yet.
Scientific Notation for Computers
Floating-point is how computers handle numbers that are very large, very small, or have a fractional part. It's like scientific notation, splitting a number into a significant value (the mantissa) and a power-of-two scale factor (the exponent).
Imagine writing down the distance to the sun. You wouldn't write 150,000,000,000 metres; you'd write it in scientific notation as metres. This separates the core digits (1.5) from the scale (the power of 11). Floating-point representation does the exact same thing for computers, but using binary (base 2) instead of denary (base 10).
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Convert the denary number into a fixed-point binary number (e.g., 9.75 becomes 1001.11).
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Move the binary point so it's at the beginning, creating a fractional number. This is called normalisation.
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Count how many places the binary point moved. This value becomes the exponent.
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Store the normalised mantissa and the exponent as two's complement binary numbers using the specified number of bits.
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Key formulas
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$Value = Mantissa \times 2^{Exponent}$
Full topic notes
Formal explanation with the rigour you need for the exam.
The Structure of a Floating-Point Number
A floating-point number is not stored as a single binary value. Instead, the bits allocated to it are split into two distinct parts: the mantissa and the exponent. Both parts are typically stored as signed binary numbers using the two's complement system.
$Value = Mantissa \times 2^{Exponent}$
Mantissa: A fixed-point binary fraction that holds the significant digits of the number. Its sign bit (Most Significant Bit) indicates whether the number is positive (0) or negative (1). It determines the number's precision.
Exponent: A binary integer that represents the power of 2 by which the mantissa should be multiplied. It determines the position of the binary point and thus the number's range (magnitude).
Conversion and Normalisation
Converting a denary number to floating-point involves several steps. The most critical is normalisation, which standardises the format to maximise precision. For the Cambridge syllabus, a normalised positive number has a mantissa starting '01' (representing 0.1...), and a normalised negative number has a mantissa starting '10' (representing -0.1...). This is achieved by shifting the binary point and adjusting the exponent accordingly.
Impact of Bit Allocation: Range vs. Precision
The total number of bits for a floating-point number is fixed. How these bits are divided between the mantissa and the exponent creates a fundamental trade-off. Allocating more bits to one part means fewer bits are available for the other.
More bits for Mantissa: Increases precision. The number can store more significant figures, reducing rounding errors. However, this leaves fewer bits for the exponent, decreasing the range.
More bits for Exponent: Increases range. The system can represent much larger and smaller numbers (further from zero). However, this leaves fewer bits for the mantissa, decreasing precision.
This trade-off means a floating-point system is designed for either high precision or a wide range, but cannot maximise both simultaneously within a fixed bit length.
Exam questions frequently ask about the consequences of changing the bit allocation. For example, 'What would be the effect of increasing the number of bits used for the exponent?'. The answer should always mention the impact on both range (increases) and precision (decreases).
Worked examples
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Represent the denary number 13.5 in normalised floating-point format using a 10-bit mantissa and a 6-bit exponent. Both should use two's complement.
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Convert to Binary:
Represent the denary number -5.25 in normalised floating-point format using an 8-bit mantissa and a 4-bit exponent. Both should use two's complement.
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Convert the positive equivalent to Binary:
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Glossary
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What are the two components of a floating-point number?
The mantissa and the exponent.
Key takeaways
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Mantissa: A fixed-point binary fraction that holds the significant digits of the number. Its sign bit (Most Significant Bit) indicates whether the number is positive (0) or negative (1). It determines the number's precision.
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Exponent: A binary integer that represents the power of 2 by which the mantissa should be multiplied. It determines the position of the binary point and thus the number's range (magnitude).
Practice — then mark it
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Test Your Knowledge on Floating-Point Numbers
Test Your Knowledge on Floating-Point Numbers
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