9618 · 13.3
Floating-point numbers, representation and manipulation flashcards
Revision flashcards for Cambridge 9618 Floating-point numbers, representation and manipulation (syllabus 13.3). Flip, recall, then mark a real past-paper question.
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What are the two components of a floating-point number?
The mantissa and the exponent.
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What does the mantissa represent?
The significant digits of the number. It holds the value, or precision, of the number.
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What does the exponent represent?
The power to which the base (usually 2) is raised. It indicates how many places to shift the binary point, determining the number's magnitude or range.
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What is normalisation in floating-point representation?
The process of adjusting the mantissa and exponent so that the number is represented in a standardised format, maximising precision. For Cambridge A-Level, this means the mantissa must start with '0.1' for positive numbers and '1.0' for negative numbers.
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Why is normalisation important?
It ensures a unique representation for each number and maximises the number of significant bits stored in the mantissa, thereby increasing precision.
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How is a negative floating-point number typically represented?
The mantissa is stored using two's complement representation. The exponent can also be negative and is also stored using two's complement.
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What is the effect of allocating more bits to the mantissa?
It increases the precision of the number, meaning more significant digits can be stored accurately.
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What is the effect of allocating more bits to the exponent?
It increases the range of the number, meaning much larger and much smaller magnitudes can be represented.
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What is floating-point overflow?
An error that occurs when a number is too large in magnitude to be represented by the available exponent bits.
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What is floating-point underflow?
An error that occurs when a number is too small in magnitude (too close to zero) to be represented. The number is rounded to zero.
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How do you convert a normalised positive mantissa (e.g., 01101000) to its negative equivalent?
Use the two's complement method: flip all the bits (10010111) and then add 1 (10011000).
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What is the relationship between the mantissa, exponent, and the real number?
Real Number = Mantissa × 2^Exponent