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Binary number system
Official AQA 7517 section 4.5.4.
0
Objectives
10
Flashcards
10
Questions
90 min
Study time
AqaA LevelComputer ScienceFundamentals of data representation
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Unsigned binary1 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 arithmetic1 objectives
- Unsigned binary arithmetic: Be able to: • add two unsigned binary integers • multiply two unsigned binary integers. 66
Signed binary using two’s complement1 objectives
- 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.
Numbers with a fractional part1 objectives
- 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.
Rounding errors1 objectives
- 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.
Absolute and relative errors1 objectives
- 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.
Range and precision1 objectives
- Range and precision: Compare the advantages and disadvantages of fixed point and floating point forms in terms of range, precision and speed of calculation.
Normalisation of floating point form1 objectives
- Normalisation of floating point form: Know why floating point numbers are normalised and be able to normalise un- normalised floating point numbers with positive or negative mantissas.
Underflow and overflow1 objectives
- Underflow and overflow: Explain underflow and overflow and describe the circumstances in which they occur.
Key terms
Unsigned binaryBitCarryPartial productTwo’s complementSigned binaryFixed pointMantissaBinary fractionRounding errorAbsolute errorRelative error
Exam tips
- Check the number of bits first: Before converting or finding a range, write down the place values for the exact number of bits stated.
- Align before calculating: Write binary values in columns with matching place values aligned, and add leading zeroes where needed.
Common mistakes
- Using the wrong maximum-value formula: Use 2^n - 1. There are 2^n possible bit patterns, but the values start at 0, so the largest value is one less than 2^n.
- Forgetting positional shifts in multiplication: Shift each partial product left according to the position of the corresponding 1 in the multiplier, then add the aligned rows.
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