Fundamentals of Data Representation | AQA A-Level Computer Science (7517)

Fundamentals of Data Representation

240 questions
16 subtopics
Paper 2: the written exam
Paper 2

Number bases and binary arithmetic, two's complement, fixed and floating point, rounding error, character codes, error checking, images, sound and encryption.

Examined on Paper 2.

Sample questions from Fundamentals of Data Representation

Answer each one closed book first, then open the answer.

Number systems and the sets of numbers

Explain why every integer is also a rational number.

Show the answer
Any integer can be written as a fraction with a denominator of 1. For example 7 = 7/1 and −3 = −3/1. Since both parts are integers and the denominator is not zero, the number satisfies the definition of a rational number, so ℤ is a subset of ℚ.
Number bases: decimal, binary and hexadecimal

Describe the repeated division method for converting a decimal number to binary.

Show the answer
Divide the number by 2 and write down the remainder, which is 0 or 1. Divide the quotient by 2 and write down the next remainder. Keep going until the quotient is 0, then read the remainders from the last one written back to the first — that string of remainders is the binary number.
Unsigned binary and binary arithmetic

Convert 181₁₀ to an 8-bit unsigned binary number.

Show the answer
10110101₂. Work down the place values: 181 − 128 = 53, 53 − 32 = 21, 21 − 16 = 5, 5 − 4 = 1, 1 − 1 = 0, so there is a 1 in the 128, 32, 16, 4 and 1 columns and a 0 in the 64, 8 and 2 columns.
Signed binary using two's complement

Write −1 and −128 as 8-bit two's complement numbers.

Show the answer
−1 is 11111111: inverting 00000001 gives 11111110 and adding 1 gives 11111111, and the place values sum to −128 + 127 = −1. −128 is 10000000, the most negative value, whose only set bit carries the weight −128.
Normalisation of floating point numbers

State the test for whether a mantissa is normalised, for both signs.

Show the answer
The first two bits of the mantissa must differ. A positive normalised mantissa begins 0.1, and a negative normalised mantissa begins 1.0. If the first two bits are the same — 0.0 or 1.1 — the number is not normalised and the mantissa can be shifted further left.
Rounding errors, accuracy, range and precision

Compare absolute and relative error for a large magnitude number, using an absolute error of 100 in a value of 5 000 000.

Show the answer
The absolute error of 100 sounds substantial, but the relative error is 100 ÷ 5 000 000 = 0.00002, that is 0.002%. For large magnitude numbers a large absolute error can still be insignificant, so absolute error alone is misleading and relative error is the better guide.
Error checking and correction

Why are the digits in a check digit calculation usually given different weights?

Show the answer
Because an unweighted sum cannot detect a transposition. If two adjacent digits are swapped — a very common typing error — the plain total is unchanged and the error slips through. Giving each position a different weight means a swap changes the weighted total, so transposed digits as well as single mistyped digits are caught.
Bitmapped graphics

Calculate the storage needed for an image 100 pixels by 200 pixels with a colour depth of 8 bits, ignoring metadata.

Show the answer
The size in pixels is 100 × 200 = 20 000. At 8 bits per pixel that is 20 000 × 8 = 160 000 bits, which is 160 000 ÷ 8 = 20 000 bytes, or about 19.5 KiB. With a colour depth of exactly one byte, the byte count equals the pixel count.

The 16 subtopics

One subtopic is one session. Work down the list.

Subtopic	What it covers	Questions
Number systems and the sets of numbers	Recall questions on the sets ℕ, ℤ, ℚ and ℝ, why every integer is rational, whether a recurring decimal is rational, and ordinal against cardinal numbers.	16
Number bases: decimal, binary and hexadecimal	Recall questions on what a base is, the digits each system uses, and converting between decimal, binary and hexadecimal in both directions.	20
Bits, bytes and units of information	Recall questions on how many values n bits can hold, choosing the smallest workable number of bits, and the binary prefixes against the decimal ones.	15
Unsigned binary and binary arithmetic	Recall questions on the range of an n-bit unsigned number, converting in both directions, adding and multiplying in binary, and recognising overflow.	15
Signed binary using two's complement	Recall questions on what the most significant bit contributes, the invert-and-add-one and copy-and-invert methods, the range formula, and subtraction by addition.	16
Fixed point and floating point representation	Recall questions on place values after the binary point, representing a given decimal in a fixed point format, and how the mantissa and exponent divide the work.	18
Normalisation of floating point numbers	Recall questions on why numbers are normalised, the test for both signs, and normalising positive and negative mantissas without losing the sign.	11
Rounding errors, accuracy, range and precision	Recall questions on why 0.1 cannot be held exactly, absolute against relative error, the trade-off between range and precision, and why floats are not compared for equality.	19
Character coding systems	Recall questions on ASCII and Unicode, why letters are coded in alphabetical order, converting between case, and why a character code is not a numeric value.	10
Error checking and correction	Recall questions on parity bits, majority voting, checksums and check digits, what each detects, and which of them can correct an error rather than only spot one.	16
Bit patterns, analogue and digital	Recall questions on how one bit pattern can mean many things, analogue against digital data and signals, sampling, and the error quantisation introduces.	11
Bitmapped graphics	Recall questions on pixels, resolution and colour depth, how many colours a given depth allows, and calculating the storage an image needs.	13
Vector graphics	Recall questions on drawing primitives and their properties, describing a picture in vector terms, and why a vector image enlarges without loss of quality.	12
Digital representation of sound	Recall questions on sampling rate and sample resolution, the Nyquist theorem and the CD sampling rate, and calculating the size of a sound file.	12
MIDI and Data Compression	Recall questions on the purpose of MIDI, its event messages and files, and on lossy and lossless compression and run length encoding.	18
Encryption	Recall questions on the Caesar and Vernam ciphers, frequency analysis, why the one-time pad alone is unbreakable, and symmetric against asymmetric keys.	18

Fundamentals of Data Representation is 240 of the 1,516 questions in the guide.Get the guide, £9

How the guide is worked

Answering a question from memory stores it far better than reading the answer again. The guide runs that as a fixed procedure on one subtopic at a time, about twenty minutes a session.

Step 1 · Closed book
Cover the answers. Work through one subtopic and write down what you can. Leave blanks where you have nothing.
Step 2 · Open book
Go back to the top. Read each printed answer and write it out in full, including the ones you had right.
Step 3 · Closed book again
Same questions, same order, from memory. The gap between pass one and pass three is the session result.

Read the full method, the return schedule and the research behind it.