Learning objective
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.
Read the explanation, check the common trap, then practise with flashcards and questions.
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Topic
Binary number system
Subtopic
Normalisation of floating point form
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Quick explanation
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
- This point belongs to Binary number system, especially Normalisation of floating point form.
- You need to be able to 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.
- 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 Normalisation of floating point form to exam-style questions, flashcards, and revision notes for Binary number system.
Quick student answer
Why are floating point numbers normalised?
Direct answer
To give each value a consistent representation and maximise the available precision
Key terms
- Normalisation: The process of moving the binary point in a floating point mantissa to its required standard position and adjusting the exponent so that the value remains unchanged.
- Mantissa: The part of a floating point representation that contains the significant digits of the number.
Common trap
Changing the exponent in the wrong direction: Moving the point right increases the mantissa by a power of two, so decrease the exponent by the same number of places. Moving the point left requires the exponent to increase.
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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
- 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.
Rounding errors