Learning objective
Rounding errors: Know and be able to explain why both fixed point and floating point representation of decimal numbers may be inaccurate. Use binary fractions. For a real number to be represented exactly by the binary number system, it must be capable of being represented by a binary fraction in the given number of bits. Some values cannot ever be represented exactly, for example 0.110.
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Binary number system
Subtopic
Rounding errors
Aqa A Level Computer ScienceFundamentals of data representation
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Quick explanation
Rounding errors: Know and be able to explain why both fixed point and floating point representation of decimal numbers may be inaccurate. Use binary fractions. For a real number to be represented exactly by the binary number system, it must be capable of being represented by a binary fraction in the given number of bits. Some values cannot ever be represented exactly, for example 0.110
- This point belongs to Binary number system, especially Rounding errors.
- You need to be able to rounding errors: Know and be able to explain why both fixed point and floating point representation of decimal numbers may be inaccurate. Use binary fractions. For a real number to be represented exactly by the binary number system, it must be capable of being represented by a binary fraction in the given number of bits. Some values cannot ever be represented exactly, for example 0.110.
- Use the linked flashcards and practice questions to check recall, then practise applying the idea in an exam-style answer.
Why it matters
This objective helps connect Rounding errors to exam-style questions, flashcards, and revision notes for Binary number system.
Quick student answer
Which binary fraction has the decimal value 0.625?
Direct answer
0.101
Key terms
- Binary fraction: A fractional value represented using binary digits, with each fractional position contributing a binary fraction value.
- Rounding error: The inaccuracy introduced when a value that cannot be represented exactly is stored as an approximation.
Common trap
Assuming floating point is always exact: Floating point can still be inaccurate because a value may not be representable as a binary fraction in the available number of bits.
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Related learning objectives
- Unsigned binary: Know the difference between unsigned binary and signed binary. Students are expected to be able to convert between unsigned binary and decimal and vice versa. Know that in unsigned binary the minimum and maximum values for a given number of bits, n, are 0 and 2n -1 respectively.
Unsigned binary
- Unsigned binary arithmetic: Be able to: • add two unsigned binary integers • multiply two unsigned binary integers. 66
Unsigned binary arithmetic
- Signed binary using two’s complement: Know that signed binary can be used to represent negative integers and that one possible coding scheme is two’s complement. This is the only representation of negative integers that will be examined. Students are expected to be able to convert between signed binary and decimal and vice versa. Know how to: • represent negative and positive integers in two’s complement • perform subtraction using two’s complement • calculate the range of a given number of bits, n.
Signed binary using two’s complement
- Numbers with a fractional part: Know how numbers with a fractional part can be represented in: • fixed point form in binary in a given number of bits • floating point form in binary in a given number of bits. Students are not required to know the Institute of Electrical and Electronic Engineers (IEEE) standard, only to know, understand and be able to use a simplified floating representation consisting of mantissa + exponent. Be able to convert for each representation from: • decimal to binary of a given number of bits • binary to decimal of a given number of bits. Exam questions on floating point numbers will use a format in which both the mantissa and exponent are represented using two's complement.
Numbers with a fractional part
- Absolute and relative errors: Be able to calculate the absolute error of numerical data stored and processed in computer systems. Be able to calculate the relative error of numerical data stored and processed in computer systems. Compare absolute and relative errors for large and small magnitude numbers, and numbers close to one.
Absolute and relative errors