4.5 - Fundamentals of data representation

Question 1

What are natural numbers? (2 points)

Answer 1

Symbols used to count objects

ℕ of natural numbers = {0, 1, 2, 3, …}

Question 2

What are whole numbers? (2 points)

Answer 2

Any positive or negative whole numbers (integers)

ℤ = { …, -3, -2, -1, 0, 1, 2, 3, … }

Question 3

What are rational numbers? (2 points)

Answer 3

Values that can be expressed as fractions or ratios

ℚ - the set of numbers that can be written as fractions (ratios of integers)

Question 4

What are irrational numbers?

Answer 4

Numbers that cannot be expressed exactly as a fraction

Question 5

What are real numbers? (2 points)

Answer 5

Numbers used for measurement/any positive or negative number with/without a fractional part

ℝ - the set of all ‘possible real world quantities’

Question 6

What are ordinal numbers?

Answer 6

Numbers that describe the numerical position of objects

Question 7

What are the 3 number systems in Computer Science?

Answer 7

Denary - base-10 (0-9)

Binary - base-2 (0-1)

Hexadecimal - base-16 (0-9, A-F)

Question 8

How do you convert from denary to binary? (3 steps)

Answer 8

1. Divide number by 2 and take remainder as last digit.
2. Keep dividing result by 2 until result of division = 0.
3. Write digits right to left to get the binary number.

Question 9

How do you convert from denary to hexadecimal and vice-versa? (4 steps)

Answer 9

Denary -> Hexadecimal
1. Divide the number by 16 to get the quotient and remainder.

2. Convert the quotient and remainder into hexadecimal digits and put them together.

Question 10

How do you convert from binary to hexadecimal and vice-versa?

Answer 10

Split the number into 2 halves, convert each and then put the halves back together

Question 11

What are the 4 advantages of hexadecimal over binary?

Answer 11

Much easier to remember than a binaryvalue

Quicker to write/type - digits only takeup 1 character, not 4

Less likely to make an error - fewer characters

Easy to convert to and from binary

Question 12

Define ‘bit’.

Answer 12

The fundamental unit of information

Question 13

What are the 4 SI-established binary prefixes?

Answer 13

Kilo = 10^3 B

Mega = 10^6 B

Giga = 10^9 B

Tera = 10^12 B

Question 14

What are the 4 IEC-established binary prefixes?

Answer 14

Kibi = 2^10 B

Mebi = 2^20 B

Gibi = 2^30 B

Tebi = 2^40 B

Question 15

How do you multiply two binary numbers together? (6 points)

Answer 15

First number = multiplicand

Second number = multiplier

If multiplier bit = 1: Write down multiplicand

If multiplier bit = 0: Write down a line of 0s with the same amount of bits as the multiplicand

Shift the partial product to the left by one bit with every bit multiplied

Add the partial products together to attain the product

Question 16

What is the advantage and disadvantage of fixed point binary?

Answer 16

Advantage:
Binary point is set in a fixed position - does not need to be stored in memory

Disadvantage:
Allocation of the bits between the whole part and fractional part cannot change - trade-off between the number of bits allocated to the
the whole and fractional part
- Large whole part = large values without fractional detail
- Large fractional part = precise decimal values represented with small whole numbers

Question 17

What are the 2 advantages of floating point binary?

Answer 17

More flexible technique - allows the way that the bits are allocated to vary so that both very large and very small numbers can be represented

2’s complement used for both the mantissa and the exponent - allows positive and negative values to be stored

Question 18

Why may both fixed point and floating point representations of decimal numbers be inaccurate? (3 points)

Answer 18

In fixed point form, truncation can occur when numbers with large fractional points (or recurring binary sequences) needing to be represented using a specific number of bits

The bits that come after the available places are filled are dropped or rounding is used

The mantissa in floating point follows the same form

Question 19

Define ‘absolute error’ and ‘relative error’.

Answer 19

Absolute error - the difference between the approximate value of the binary representation of a number and the original value of the number

Relative error - the absolute error divided by the original value of the number

Question 20

Define ‘under/overflow’.

Answer 20

When the result of a calculation is too small/large to be represented with the available number of bits

Question 21

What are the 4 advantages of vector graphics over bitmap images?

Answer 21

Can be resized without distortion

File sizes are usually smaller - especially when they do not include many intricate shapes

Much easier to create and edit images - each object in a vector graphic can be adapted independently by using its set of attributes

High-quality resolution - stays good quality, regardless of the resolution of the medium that they are displayed on

Question 22

What is the advantage of bitmapped graphics over vector graphics? (2 points)

Answer 22

Can depict almost any level of complexity and detail

Vector graphics produced by combining limited number of shapes - variety of images they can represent is restricted

Question 23

How are digital sound files created from analogue sound waves? (3 points)

Answer 23

1. Microphone picks up sound waves - converts the waves into analogue electrical signals
2. Signals are processed by an ADC which does sampling - takes measurements of level of analogue signal at regular time intervals
3. Measurements are assigned a binary pattern and stored in the computer’s memory

Question 24

What are MIDI files and what do they explain? (4 points)

Answer 24

Files consisting of a list of instructions orevent messages

They explain:
- What notes must be played
- When they must be played
- How long or loud each note should be

Question 25

What 3 types of signals do MIDI files hold to produce sound?

Answer 25

Digital signals that specify notation, pitch, and intensity

Control signals for parameters such as volume and vibrato

Clock signals to set the tempo