4.5 Fundamentals of data representation
4.5.1 Number systems
4.5.1.1 Natural numbers
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Be familiar with the concept of a natural number and the set ℕ of natural numbers (including zero). 
ℕ = {0, 1, 2, 3, … } 
4.5.1.2 Integer numbers
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Be familiar with the concept of an integer and the set ℤ of integers. 
ℤ = { …, 3, 2, 1, 0, 1, 2, 3, … } 
4.5.1.3 Rational numbers
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Be familiar with the concept of a rational number and the set ℚ of rational numbers, and that this set includes the integers. 
ℚ is the set of numbers that can be written as fractions (ratios of integers). Since a number such as 7 can be written as 7/1, all integers are rational numbers. 
4.5.1.4 Irrational numbers
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Be familiar with the concept of an irrational number. 
An irrational number is one that cannot be written as a fraction, for example √2. 
4.5.1.5 Real numbers
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Be familiar with the concept of a real number and the set ℝ of real numbers, which includes the natural numbers, the rational numbers and the irrational numbers. 
ℝ is the set of all 'possible real world quantities'. 
4.5.1.6 Ordinal numbers
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Be familiar with the concept of ordinal numbers and their use to describe the numerical positions of objects. 
When objects are placed in order, ordinal numbers are used to tell their position. For example, if we have a wellordered set S = {‘a’, ‘b’, ‘c’, ‘d’}, then ‘a’ is the 1st object, ‘b’ the 2nd, and so on. 
4.5.1.7 Counting and measurement
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Be familiar with the use of:

4.5.2 Number bases
4.5.2.1 Number base
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Be familiar with the concept of a number base, in particular:

Students should be familiar with expressing a number’s base using a subscript as follows: Base 10: Number_{10}, eg 67_{10} Base 2: Number_{2}, eg 10011011_{2} Base 16: Number_{16}, eg AE_{16} 
Convert between decimal, binary and hexadecimal number bases. 

Be familiar with, and able to use, hexadecimal as a shorthand for binary and to understand why it is used in this way. 
4.5.3 Units of information
4.5.3.1 Bits and bytes
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Know that:

A bit is either 0 or 1. 
Know that the 2^{n} different values can be represented with n bits. 
For example, 3 bits can be configured in 2^{3} = 8 different ways. 000, 001, 010, 011, 100, 101, 110, 111 
4.5.3.2 Units
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Know that quantities of bytes can be described using binary prefixes representing powers of 2 or using decimal prefixes representing powers of 10, eg one kibibyte is written as 1KiB = 2^{10} B and one kilobyte is written as 1 kB = 10^{3} B. Know the names, symbols and corresponding powers of 2 for the binary prefixes:

Historically the terms kilobyte, megabyte, etc have often been used when kibibyte, mebibyte, etc are meant. 
4.5.4 Binary number system
4.5.4.1 Unsigned binary
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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 2^{n }1 respectively. 
4.5.4.2 Unsigned binary arithmetic
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Be able to:

4.5.4.3 Signed binary using two’s complement
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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:

4.5.4.4 Numbers with a fractional part
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Know how numbers with a fractional part can be represented in:

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:

Exam questions on floating point numbers will use a format in which both the mantissa and exponent are represented using two's complement. 
4.5.4.5 Rounding errors
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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 0.1_{10}. 
4.5.4.6 Absolute and relative errors
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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. 
4.5.4.7 Range and precision
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Compare the advantages and disadvantages of fixed point and floating point forms in terms of range, precision and speed of calculation. 
4.5.4.8 Normalisation of floating point form
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Know why floating point numbers are normalised and be able to normalise unnormalised floating point numbers with positive or negative mantissas. 
4.5.4.9 Underflow and overflow
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Explain underflow and overflow and describe the circumstances in which they occur. 
4.5.5 Information coding systems
4.5.5.1 Character form of a decimal digit
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Differentiate between the character code representation of a decimal digit and its pure binary representation. 
4.5.5.2 ASCII and Unicode
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Describe ASCII and Unicode coding systems for coding character data and explain why Unicode was introduced. 
4.5.5.3 Error checking and correction
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Describe and explain the use of:

4.5.6 Representing images, sound and other data
4.5.6.1 Bit patterns, images, sound and other data
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Describe how bit patterns may represent other forms of data, including graphics and sound. 
4.5.6.2 Analogue and digital
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Understand the difference between analogue and digital:

4.5.6.3 Analogue/digital conversion
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Describe the principles of operation of:


Know that ADCs are used with analogue sensors. 

Know that the most common use for a DAC is to convert a digital audio signal to an analogue signal. 
4.5.6.4 Bitmapped graphics
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Explain how bitmaps are represented. 

Explain the following for bitmaps:

The size of an image is also alternatively sometimes described as the resolution of an image. Size of an image in pixels is width of image in pixels x height of image in pixels. Resolution is expressed as number of dots per inch where a dot is a pixel. Colour depth = number of bits stored for each pixel. 
Calculate storage requirements for bitmapped images and be aware that bitmap image files may also contain metadata. 
Ignoring metadata, storage requirements = size in pixels x colour depth where size in pixels is width in pixels x height in pixels. 
Be familiar with typical metadata. 
eg width, height, colour depth. 
4.5.6.5 Vector graphics
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Explain how vector graphics represents images using lists of objects. 
The properties of each geometric object/shape in the vector graphic image are stored as a list. 
Give examples of typical properties of objects. 

Use vector graphic primitives to create a simple vector graphic. 
4.5.6.6 Vector graphics versus bitmapped graphics
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Compare the vector graphics approach with the bitmapped graphics approach and understand the advantages and disadvantages of each. 

Be aware of appropriate uses of each approach. 
4.5.6.7 Digital representation of sound
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Describe the digital representation of sound in terms of:


Calculate sound sample sizes in bytes. 
4.5.6.8 Musical Instrument Digital Interface (MIDI)
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Describe the purpose of MIDI and the use of event messages in MIDI. 

Describe the advantages of using MIDI files for representing music. 
4.5.6.9 Data compression
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Know why images and sound files are often compressed and that other files, such as text files, can also be compressed. 

Understand the difference between lossless and lossy compression and explain the advantages and disadvantages of each. 

Explain the principles behind the following techniques for lossless compression:

4.5.6.10 Encryption
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Understand what is meant by encryption and be able to define it. 
Students should be familiar with the terms cipher, plaintext and ciphertext. Caesar and Vernam ciphers are at opposite extremes. One offers perfect security, the other doesn’t. Between these two types are ciphers that are computationally secure – see below. Students will be assessed on the two types. Ciphers other than Caesar may be used to assess students' understanding of the principles involved. These will be explained and be similar in terms of computational complexity. 
Be familiar with Caesar cipher and be able to apply it to encrypt a plaintext message and decrypt a ciphertext. Be able to explain why it is easily cracked. 

Be familiar with Vernam cipher or onetime pad and be able to apply it to encrypt a plaintext message and decrypt a ciphertext. Explain why Vernam cipher is considered as a cypher with perfect security. 
Since the key k is chosen uniformly at random, the ciphertext c is also distributed uniformly. The key k must be used once only. The key k is known as a onetime pad. 
Compare Vernam cipher with ciphers that depend on computational security. 
Vernam cipher is the only one to have been mathematically proved to be completely secure. The worth of all other ciphers ever devised is based on computational security. In theory, every cryptographic algorithm except for Vernam cipher can be broken, given enough ciphertext and time. 