Topic 5 - Fundamentals of Data Representation

Question 1

What are natural numbers?

Answer 1

Natural numbers are a set of whole numbers consisting of positive numbers as well as zero.

Question 2

What is the symbol for a natural number set?

Answer 2

ℕ is the symbol for natural numbers.

Question 3

Define a set.

Answer 3

A set is an unordered collection of data which contains each item only once.

Question 4

What are integer numbers?

Answer 4

Integer numbers are a set of whole numbers consisting of positive and negative numbers as well as zero.

Question 5

Give the symbol of an integer number set.

Answer 5

ℤ is the symbol for integer numbers.

Question 6

What are rational numbers?

Answer 6

Rational numbers (a.k.a quotients) are a set of numbers consisting of positive and negative numbers as well as zero. Rational numbers can be in the form of a fraction or decimal.

Question 7

Give the symbol for a rational numbers set.

Answer 7

ℚ is the symbol for rational numbers.

Question 8

Define irrational numbers.

Answer 8

Irrational numbers are opposite to rational numbers as they can’t be expressed as fractions and are never-ending and non-repeating; e.g. π, √2. Irrational numbers DON’T have a symbol.

Question 9

Describe the real numbers set.

Answer 9

Real numbers are a number set consisting of all the number word quantities; integer, natural, irrational and natural number sets are all in the real number set.

Question 10

State the real numbers set symbol.

Answer 10

ℝ is the symbol for real numbers.

Question 11

What are ordinal numbers?

Answer 11

Ordinal numbers are a set of numbers which specify positions for certain quantities or objects; e.g. 1st, 2nd, 3rd, 4th…

Question 12

Give a use of natural numbers.

Answer 12

Natural numbers are used in counting quantities or things as you can’t count using negative numbers but you can using positive numbers.

Question 13

Give a use of real numbers.

Answer 13

Real numbers are used in measuring quantities because not every accurate measurement is a whole number; there are many measurements that are in the form of a decimal.

Question 14

What are number bases?

Answer 14

Number bases are simply differing forms of representation of numbers.

Question 15

Define the denary number system.

Answer 15

Denary, also known as decimal, number system is a system extending from 0 to 9 and is used by humans. It is a base 10 system.

Question 16

Describe the binary number system.

Answer 16

The binary number system is a system used by computers consisting of only two digits: 0 and 1. It is used to process data for the computer to understand. This system is base 2.

Question 17

Describe the hexadecimal number system.

Answer 17

The hexadecimal number system is a system which extends from 0 - 9 but uses uppercase characters to represent the decimal numbers from 10 to 15. It is a base 16 system and the most compact of all the systems as it can easily represent huge numbers in smaller ways.

Question 18

Describe how to convert a binary number to a denary number.

Answer 18

Write down the binary number with each integer spaced apart a little, and above each 1 or 0, write a place value header (a collection of values starting from the right, and increasing in powers of 2). Next, take note of any place value headers with a 1 below them and at the end, add these headers, giving you the decimal number.

Question 19

Describe how to convert a denary number to a binary number.

Answer 19

Write down the place value headers and one by one, under the headers decide whether the number is equal to or more than the header, and based on that, subtract the header from the number and place a 1 under the header or place a 0 if the header is larger than the number. Continue doing this for the rest of the headers.

Question 20

How can a binary number be converted to hexadecimal?

Answer 20

To convert a binary number to hex, firstly split the binary number into nibbles (a nibble is 4 bits), and using smaller place value headers, convert the nibbles to denary. Then, using the hex number system, convert the denary numbers to hex if needed.

Question 21

How can you convert a hex number to denary?

Answer 21

Write the hex number and above each hex integer, write the base-16 place values with the smallest at the far right, increasing in multiples of 16 (1, 16, 256…). Then convert any hex digits to denary and multiply the hex digits by the place values, and add these multiplication results to get the denary result.

Question 22

Describe the method of converting a denary number to hexadecimal.

Answer 22

To convert from decimal to hex, firstly convert the denary number to binary, then split the binary number into two smaller nibbles which will be converted to decimal before converting the decimal numbers (if needed) to hex.

Question 23

What is a bit?

Answer 23

A bit the smallest and most fundamental piece of information/memory in a computer. A bit can be in the form of either a 1 or 0, used and represented by a high or low current computer.

Question 24

What is equal to 4 bits?

Answer 24

4 bits = a nibble.

Question 25

What is equal to 8 bits?

Answer 25

8 bits = a whole byte.

Question 26

How can you represent a bit and byte?

Answer 26

A bit is represented by a lowercase ‘b’.

A byte is represented by an uppercase ‘B’.

Question 27

State the relationship between the number of bits added to a number with the number of different representations of the value.

Answer 27

The number of bits added to a number is directly proportional to the number of different available representations of the value.

Question 28

How many bytes are there in a kilobyte?

Answer 28

A kilobyte has 1000 bytes (10^3 bytes).

Question 29

How many kilobytes are in a megabyte?

Answer 29

There are 1000 kilobytes in a megabyte (10^6 bytes)

Question 30

How many megabytes are in a gigabyte?

Answer 30

A gigabyte has 1024 megabytes and 10^9 bytes.

Question 31

How many gigabytes are in a terabyte?

Answer 31

A terabyte = 1000 GB.

Question 32

Why are there binary prefixes?

Answer 32

Binary prefixes are used so computers can understand the different quantities of data. To eliminate confusion between the denary and binary prefixes, computer scientists produced the binary prefix system, representing multiples of base-2.

Question 33

Give the names of the binary prefixes.

Answer 33

Kibi, mebi, gibi and tebi.

Question 34

How many bits are in a kibibit?

Answer 34

A Ki (kibibit) has 1024 bits. (2^10)

Question 35

How many kibibits are in a mebibit?

Answer 35

There are 1024 Ki in a mebibit. (2^20)

Question 36

How many mebibits are in a gibibit?

Answer 36

There are 1024 Mi in a gibibit. (2^30)

Question 37

How many gibibits are in a tebibit?

Answer 37

There are 1024 Gi in a tebibit. (2^40)

Question 38

What is an unsigned binary number?

Answer 38

An unsigned binary number is a binary number which can ONLY represent POSITIVE numbers.

Question 39

Describe a signed binary number.

Answer 39

A signed binary number is a binary number that ONLY represents NEGATIVE numbers.

Question 40

If the unsigned binary number ‘1011’ were assigned, what would be the decimal result?

Answer 40

The decimal result of 1011 would be 8 + 2 + 1 = 11.

Question 41

How many decimal values may be represented with the 2 binary bits?

Answer 41

With 2 binary bits, 4 denary values can be represented; 0, 1, 2, 3.

Question 42

What is the formula for the range of denary numbers of n unsigned binary numbers that can be represented?

Answer 42

For n unsigned bits, decimal numbers from 0 to 2^n -1 can be represented.

Question 43

State the 4 primary rules of unsigned binary arithmetic.

Answer 43

0 + 0 = 0
0 + 1 = 1
1 + 1 = 10 (2 in binary)
1 + 1 + 1 = 11 (3 in binary)

Question 44

Add the binary integers 1011 and 1110.

Answer 44

1011 + 1110 = 11001

Question 45

What is ‘two’s complement’?

Answer 45

Two’s complement is a rule used to represent positive and negative numbers in binary.

Question 46

How is the ‘two’s complement’ rule used in signed binary?

Answer 46

Two’s complement is used in binary by giving the most significant place value header a negative sign; e.g. for 4 bits (8, 4, 2, 1), 8 will turn into -8 (-8, 4, 2, 1).

Question 47

If the signed binary number ‘1101’ were assigned, what would be the decimal result?

Answer 47

The decimal result would be -8 + 4 + 1 = -3.

Question 48

How does subtraction occur using ‘two’s complement’?

Answer 48

Computers always add integers, therefore, subtraction is performed by simply adding negative numbers.

Question 49

For n signed binary bits, what is the range of two’s complement denary number?

Answer 49

For n signed binary bits, the range is 2^n-1 -1, to -2^n-1.

Question 50

Give two methods of binary to represent decimal point denary numbers.

Answer 50

• Fixed point form

- Floating point form

Question 51

What is fixed point form binary?

Answer 51

Fixed point form binary is a method used to represent decimal point numbers. It works like this; the binary number is split into 2 sets of nibbles with one nibble under normal place value headers (8, 4, 2, 1) and the other nibble under fractional headers (1/2, 1/4, 1/8, 1/16).

Question 52

Define mantissa.

Answer 52

The mantissa is the main binary number you want to convert.

Question 53

Define exponent.

Answer 53

The exponent is the binary number which deduces the position of the binary point in the main number (mantissa).

Question 54

What is floating point binary?

Answer 54

Floating point binary is a method used to represent decimal point denary numbers where a ‘floating’ binary point (decimal point) moves around a binary number (mantissa), its position decided by a seperate, small binary exponent.

Question 55

Are fixed point and floating point binary always accurate?

Answer 55

Although fixed and floating point binary can represent a large variety of decimal numbers, there are some decimal numbers such as 1/3 (0.3 recurring) or 0.1 which both fixed and floating point binary fail to accurately represent but can approximately represent.

Question 56

What are absolute and relative errors?

Answer 56

Absolute and relative errors are types of errors used to calculate the difference between a fixed/floating point binary number representation and the actual denary value.

Question 57

Define the absolute error.

Answer 57

An absolute error is simply the difference of inaccuracy between the approximate binary representation and the accurate denary number.

Question 58

How do you calculate the absolute error?

Answer 58

You convert the fixed point binary number to a denary value and then subtract this denary value from the accurate, actual value, giving the absolute error.

Question 59

Define the relative error.

Answer 59

A relative error is simply the absolute error divided by the actual value.

Question 60

Give an advantage of floating point binary.

Answer 60

Floating point binary has a larger range than fixed point binary due to its compatibility with both positive and negative numbers.

Question 61

State the difference in range and precision in floating point binary.

Answer 61

The length of the mantissa and exponent can significantly affect range and precision;

• A larger mantissa and smaller exponent can cause a smaller range but higher precision.
• A smaller mantissa and larger exponent may cause a higher range but smaller precision.

Question 62

State the difference in range and precision in fixed point binary.

Answer 62

• Moving the binary point to the left will increase precision but decrease range.
• Moving the binary point to the right will increase range but decrease precision.

Question 63

What is normalization?

Answer 63

Normalisation is a process used in floating point binary, ensuring the floating point begins with 01 (for positive unsigned binary), or 10 (for negative signed binary). The process is used to ensure maximum precision for binary bits.

Question 64

How would you normalize a floating point binary number?

Answer 64

Depending on whether the number is signed or unsigned, you should move the number a certain number of places to the left to achieve either 01 or 10 at the front of the number, allowing a more precise conversion to denary.

Question 65

Define an underflow error.

Answer 65

An underflow error is a type of binary error where there are too little quantity of bits available for representation.

Question 66

Define an overflow error.

Answer 66

An overflow error is a type of binary error where there are too many bits available for representation.

Question 67

What do computers use to represent characters such as ^, $, K?

Answer 67

Computers use information coding systems to represent characters, and this is done by assigning character codes to each individual character.

Question 68

What is a character code?

Answer 68

A character code is a denary value assigned to each character.

Question 69

How do computers understand these character representations?

Answer 69

Generally, a computer will convert each character code to binary, so it can understand the character;
e.g. The letter D is assigned the character code 4, which in binary = 0100.

Question 70

Define ASCII.

Answer 70

ASCII (American Standard Code Information Interchange), is an information coding system which uses 7 bits to represent 128 characters (2^7). This includes A-Z, a-z, 0-9, many other symbols and foreign alphabets.

Question 71

Define Unicode.

Answer 71

Unicode is an information coding system which uses 8-48 bits (1-6 bytes) to represent an individual character, giving it a significantly wider range of characters than ASCII but causing it to take up more space.

Question 72

Define a parity bit.

Answer 72

A parity bit is a singular bit added to a sequence of bits, its main functions to check for transmission errors and ensure an odd or even quantity of 1s or 0s. The parity bit and the rest of the data will be sent to the receiving computer, and if the data is valid and no errors are detected, then it will be approved otherwise it will be resent back to the sender if one error is found.

Question 73

Give a problem with using parity bits.

Answer 73

During a parity check, if there is one error then it will be detected as an error, yet if 2 errors are present in the data, there will still be odd or even parity maintained thus it will be treated as valid and the computer will not find any errors.

Question 74

What is majority voting?

Answer 74

Majority voting is a method of error checking, involving a piece of data being sent, and each bit transmitted multiple times before being received. Upon being received, if some of the data is corrupted, a majority vote will be held where the most common bit will be selected from each piece of data and the 4 bits will be put together.

Question 75

Give a problem with majority voting.

Answer 75

In majority voting, the volume of data being transmitted will increase with the repetition of bits, and this can cause a substantial amount of time being used up.

Question 76

What is a checksum?

Answer 76

A checksum is a mathematical algorithm applied to a sequence/block of data.

Question 77

How does the checksum work?

Answer 77

The checksum is calculated using an algorithm, and is transmitted with the data to the receiver. The receiver applies the same initial algorithm to check whether the data is correct; if it isn’t, then the receiver will resend the data back to the sender.

Question 78

What is a check digit?

Answer 78

A check digit is a type of checksum which only consists of one digit added to a sequence of data, and check digit also has significantly less complex algorithms, making it more efficient but simultaneously reducing the variety of errors able to detect.

Question 79

What is analogue data?

Answer 79

Analogue data is data which is continuous, variable and accepts all kinds of values. In its graph, it appears as a smooth curve.

Question 80

What is digital data?

Answer 80

Digital data is data which tends to be discrete, accepts certain kinds of values. In its graph, digital data is characterized by its straight lines and changes at specified intervals.

Question 81

How can you convert digital data to analogue data?

Answer 81

You will need to use a DAC (Digital to Analogue Converter), to convert digital data to analogue data. This works by the converter reading a bit pattern of analogue data, then reverse quantising the bit pattern into a continuous loop of data, allowing it to create an analogue, alternating electrical current. DACs are used commonly in audio, to convert digital audio signals to analogue signals.

Question 82

How can you convert analogue data to digital data?

Answer 82

To convert analogue data to digital data, you will need an ADC (Analogue to Digital Converter), which reads analogue data at regular intervals and records this to produce samples. Upon recording, the recording is digitally stored in the form of a bit pattern, producing digital data. Common uses of ADCs are in microphones or temperature sensors.

Question 83

Describe the frequency the samples are taken at during conversion of analogue data to digital data.

Answer 83

When analogue data is converted to digital data, sampling occurs at a specific frequency in Hertz to get the sampling per second. Generally, the frequency should be high as this helps produce a stronger reproduction of analogue data.

Question 84

Give one way computers represent images.

Answer 84

Bitmap graphics is one way computers represent images.

Question 85

Define bitmap graphics.

Answer 85

Bitmap graphics is a form of image representation when an image is broken down into pixels, with a binary value assigned to each pixel.

Question 86

What is the resolution of an image?

Answer 86

The resolution of an image can either be defined as the number of dots per square inch or simply the number of pixels within an image.

Question 87

What do the binary values in each pixel represent?

Answer 87

In an image, the binary values assigned to each pixel generally represent the colour; e.g. in an image, 1 could represent a black pixel, while 0 could represent a white pixel.

Question 88

Define a pixel.

Answer 88

A pixel is the smallest element of an image.

Question 89

What is colour/bit depth?

Answer 89

Colour/bit depth, is the amount of bits assigned to each pixel: with 1 bit, there are 2^1 (2) different representations, but with 2 bits, there are 2^2 (4) different representations.

Question 90

Give the formula needed to calculate the storage in a bitmap graphics image.

Answer 90

Storage = Number of pixels (length x width) x bit depth.

Question 91

Why is the calculated storage for bitmap images from the formula, the minimum storage?

Answer 91

The storage calculated by the formula is minimum as the image may sometimes include extra important metadata (length value, width value, date created, colour depth…).

Question 92

Give another way computers represent images.

Answer 92

Another way images are represented by computers are by vector graphics.

Question 93

Define vector graphics.

Answer 93

Vector graphics are a method of image representation where images are represented by geometric shapes such as circles, rectangles, lines etc. The properties of each shape (fill colour, fill style, dimensions), are stored in a list.

Question 94

Give an advantage of vector graphics.

Answer 94

Vector graphics is more efficient as less storage is used up due to storage being concentrated in each shape, while bitmap graphics have storage concentrated in each pixel and of course, there are nowhere near as much geometric shapes as there are pixels in an image!

Question 95

Give an advantage of bitmap graphics.

Answer 95

Vector graphics only make use of shapes, allowing them to be used in very simple pictures like company logos, while bitmap graphics use a large variety of pixels thus allowing them to have lots more properties such as real-life photography.

Question 96

How can computers represent sound?

Answer 96

Computers represent sound via a sequence of samples, taking a digital value.

Question 97

What is the sampling rate?

Answer 97

The sampling rate is the number of samples recorded per second.

Question 98

Define the sample resolution.

Answer 98

The sample resolution is the number of bits assigned to each sample.

Question 99

How can you calculate the storage/size of sound?

Answer 99

Storage of sound = Duration of sound in seconds x sample rate (Hz) x sample resolution
Just like image graphics, this storage is minimum as metadata can be added and increase the sound size.

Question 100

Describe and explain the Nyquist theorem.

Answer 100

The Nyquist theorem is a theorem that states the sample rate of a digital audio file must be at least twice the highest frequency of the sound, to ensure an accurate representation.

Question 101

What is MIDI?

Answer 101

Musical Instrument Digital Interface, also known as MIDI is a method of sound representation used by electronic musical instruments connected to a computer. Unlike other sound representation methods, MIDI doesn’t use sampling yet it does store several event messages requested by instruments such as the duration of a note or volume of a note.

Question 102

Give an advantage of using MIDI.

Answer 102

MIDI is perfect in easy manipulation of music with no loss in quality.

Question 103

Give a disadvantage of using MIDI.

Answer 103

MIDI can’t represent speech and compared to other sampled sound recordings, it sounds very unrealistic.

Question 104

Can smaller or larger files be compressed faster?

Answer 104

Smaller files can be compressed faster and transferred over the internet.

Question 105

Give the two types of compression.

Answer 105

The two types of compression are lossy compression and lossless compression.

Question 106

Describe lossy compression.

Answer 106

Lossy compression is a type of compression where the file size is decreased but the quality of the file also decreases; the sample resolution of sound falls, the quality of an image decreases, the information of a text file can also fall. It involves removing non-essential data such as different shades of colour.

Question 107

Describe lossless compression.

Answer 107

Lossless compression is a type of compression where the file size is decreased yet the quality/information stays the same with no loss. Patterns are recorded in the data and removed.

Question 108

State the two types of lossless compression.

Answer 108

• RLE (Run-Length Encoding)

- Dictionary Based Methods

Question 109

Define RLE (Run-Length Encoding).

Answer 109

RLE is a type of lossless compression which reduces repetitions of certain elements in a file, instead replacing it with one occurrence followed by a number of times repeated.

Question 110

Define dictionary based methods.

Answer 110

Dictionary based methods is a type of lossless compression where a dictionary is created next to the file, containing repeated data. The algorithm reduces repetitions of data to

Question 111

Give a disadvantage of lossless compression over lossy compression.

Answer 111

Unfortunately, lossless compression has a limit of how much the file can be compressed.

Question 112

Define encryption.

Answer 112

Encryption is the process of scrambling data so it can be safely and securely transmitted and not intercepted.

Question 113

What is plaintext?

Answer 113

Plaintext is unencrypted text.

Question 114

What is ciphertext?

Answer 114

Ciphertext is encrypted text.

Question 115

What is a cipher?

Answer 115

A cipher is an encryption method to encrypt text.

Question 116

What must you know and have to decrypt text?

Answer 116

To decrypt text, you must have the key and must know the encryption method.

Question 117

Describe the ‘Caesar cipher’.

Answer 117

The Caesar cipher is an encryption method consisting of replacing characters, with each character being replaced by the same character.

Question 118

State the two types of Caesar cipher.

Answer 118

• Shift cipher

- Substitution cipher.

Question 119

Define one type of the Caesar cipher.

Answer 119

The shift cipher is a type of Caesar cipher, where each character will be shifted the same amount of times.

Question 120

Define another type of the Caesar cipher.

Answer 120

The substitution cipher is a type of Caesar cipher, where each character is substituted by a completely random character.

Question 121

How easy is it to crack the Caesar ciphers?

Answer 121

It is relatively easy to crack the Caesar ciphers with the shift cipher being the easier one as each character is shifted the same amount of times, so once one character is cracked, instantly you can crack the others. The substitution cipher is more complex as the letters are randomly substituted but still isn’t that hard either.

Question 122

What is the Vernam cipher?

Answer 122

The Vernam cipher is a type of cipher, referred to as a ‘one-time pad cipher’, referring to the key of the cipher.
The key of the cipher should be truly unique and random, should be as long or longer as the plaintext and destroyed after being used.

Question 123

Describe how the Vernam cipher is carried out.

Answer 123

1) Ensure you align the plaintext and the key.
2) Convert the plaintext and key to binary.
3) Apply an XOR logic gate when combining the two bit patterns.
4) Convert the newly formed binary from the XOR combination, to encrypted characters.
5) Destroy the key used.

Question 124

Define XOR.

Answer 124

XOR is a type of logic gate which only inputs true if one of its inputs is true. This means ‘exclusive OR’.

Question 125

How can you decrypt the Vernam cipher?

Answer 125

To decrypt the Vernam cipher, you use the same key used to encrypt the cipher.

Question 126

Why is the Vernam cipher so secure?

Answer 126

The Vernam cipher is very mathematically secure as the key used each time is completely random, so the ciphertext formed will also be random. Also, destroying the key after use is beneficial as it won’t be detected by interceptors.

Question 127

Define computational security.

Answer 127

Computational security is a mathematical measure of the time frame a cipher can be cracked. A cipher which is computationally secure is said to have a low probability of being cracked in a constant time by a computer. Currently, the Vernam cipher is the most secure cipher out there.

Question 128

State the difference between asymmetric and symmetric encryption.

Answer 128

Symmetric encryption involves only one key to encrypt and decrypt while asymmetric encryption uses 2 keys to encrypt and decrypt.