Topic 5 - Fundamentals of Data Representation Flashcards
What are natural numbers?
Natural numbers are a set of whole numbers consisting of positive numbers as well as zero.
What is the symbol for a natural number set?
ℕ is the symbol for natural numbers.
Define a set.
A set is an unordered collection of data which contains each item only once.
What are integer numbers?
Integer numbers are a set of whole numbers consisting of positive and negative numbers as well as zero.
Give the symbol of an integer number set.
ℤ is the symbol for integer numbers.
What are rational numbers?
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.
Give the symbol for a rational numbers set.
ℚ is the symbol for rational numbers.
Define irrational numbers.
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.
Describe the real numbers set.
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.
State the real numbers set symbol.
ℝ is the symbol for real numbers.
What are ordinal numbers?
Ordinal numbers are a set of numbers which specify positions for certain quantities or objects; e.g. 1st, 2nd, 3rd, 4th…
Give a use of natural numbers.
Natural numbers are used in counting quantities or things as you can’t count using negative numbers but you can using positive numbers.
Give a use of real numbers.
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.
What are number bases?
Number bases are simply differing forms of representation of numbers.
Define the denary number system.
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.
Describe the binary number system.
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.
Describe the hexadecimal number system.
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.
Describe how to convert a binary number to a denary number.
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.
Describe how to convert a denary number to a binary number.
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.
How can a binary number be converted to hexadecimal?
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.
How can you convert a hex number to denary?
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.
Describe the method of converting a denary number to hexadecimal.
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.
What is a bit?
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.
What is equal to 4 bits?
4 bits = a nibble.
What is equal to 8 bits?
8 bits = a whole byte.
How can you represent a bit and byte?
A bit is represented by a lowercase ‘b’.
A byte is represented by an uppercase ‘B’.
State the relationship between the number of bits added to a number with the number of different representations of the value.
The number of bits added to a number is directly proportional to the number of different available representations of the value.
How many bytes are there in a kilobyte?
A kilobyte has 1000 bytes (10^3 bytes).
How many kilobytes are in a megabyte?
There are 1000 kilobytes in a megabyte (10^6 bytes)
How many megabytes are in a gigabyte?
A gigabyte has 1024 megabytes and 10^9 bytes.
How many gigabytes are in a terabyte?
A terabyte = 1000 GB.
Why are there binary prefixes?
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.
Give the names of the binary prefixes.
Kibi, mebi, gibi and tebi.
How many bits are in a kibibit?
A Ki (kibibit) has 1024 bits. (2^10)
How many kibibits are in a mebibit?
There are 1024 Ki in a mebibit. (2^20)
How many mebibits are in a gibibit?
There are 1024 Mi in a gibibit. (2^30)
How many gibibits are in a tebibit?
There are 1024 Gi in a tebibit. (2^40)
What is an unsigned binary number?
An unsigned binary number is a binary number which can ONLY represent POSITIVE numbers.
Describe a signed binary number.
A signed binary number is a binary number that ONLY represents NEGATIVE numbers.
If the unsigned binary number ‘1011’ were assigned, what would be the decimal result?
The decimal result of 1011 would be 8 + 2 + 1 = 11.
How many decimal values may be represented with the 2 binary bits?
With 2 binary bits, 4 denary values can be represented; 0, 1, 2, 3.
What is the formula for the range of denary numbers of n unsigned binary numbers that can be represented?
For n unsigned bits, decimal numbers from 0 to 2^n -1 can be represented.
State the 4 primary rules of unsigned binary arithmetic.
0 + 0 = 0
0 + 1 = 1
1 + 1 = 10 (2 in binary)
1 + 1 + 1 = 11 (3 in binary)
Add the binary integers 1011 and 1110.
1011 + 1110 = 11001
What is ‘two’s complement’?
Two’s complement is a rule used to represent positive and negative numbers in binary.
How is the ‘two’s complement’ rule used in signed binary?
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).
If the signed binary number ‘1101’ were assigned, what would be the decimal result?
The decimal result would be -8 + 4 + 1 = -3.
How does subtraction occur using ‘two’s complement’?
Computers always add integers, therefore, subtraction is performed by simply adding negative numbers.
For n signed binary bits, what is the range of two’s complement denary number?
For n signed binary bits, the range is 2^n-1 -1, to -2^n-1.
Give two methods of binary to represent decimal point denary numbers.
- Fixed point form
- Floating point form
What is fixed point form binary?
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).
Define mantissa.
The mantissa is the main binary number you want to convert.
Define exponent.
The exponent is the binary number which deduces the position of the binary point in the main number (mantissa).
What is floating point binary?
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.
Are fixed point and floating point binary always accurate?
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.
What are absolute and relative errors?
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.
Define the absolute error.
An absolute error is simply the difference of inaccuracy between the approximate binary representation and the accurate denary number.
How do you calculate the absolute error?
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.
Define the relative error.
A relative error is simply the absolute error divided by the actual value.
Give an advantage of floating point binary.
Floating point binary has a larger range than fixed point binary due to its compatibility with both positive and negative numbers.
State the difference in range and precision in floating point binary.
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.
State the difference in range and precision in fixed point binary.
- 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.
What is normalization?
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.
How would you normalize a floating point binary number?
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.
Define an underflow error.
An underflow error is a type of binary error where there are too little quantity of bits available for representation.
Define an overflow error.
An overflow error is a type of binary error where there are too many bits available for representation.
What do computers use to represent characters such as ^, $, K?
Computers use information coding systems to represent characters, and this is done by assigning character codes to each individual character.
What is a character code?
A character code is a denary value assigned to each character.
How do computers understand these character representations?
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.
Define ASCII.
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.
Define Unicode.
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.
Define a parity bit.
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.
Give a problem with using parity bits.
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.
What is majority voting?
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.
Give a problem with majority voting.
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.
What is a checksum?
A checksum is a mathematical algorithm applied to a sequence/block of data.
How does the checksum work?
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.
What is a check digit?
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.
What is analogue data?
Analogue data is data which is continuous, variable and accepts all kinds of values. In its graph, it appears as a smooth curve.
What is digital data?
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.
How can you convert digital data to analogue data?
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.
How can you convert analogue data to digital data?
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.
Describe the frequency the samples are taken at during conversion of analogue data to digital data.
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.
Give one way computers represent images.
Bitmap graphics is one way computers represent images.
Define bitmap graphics.
Bitmap graphics is a form of image representation when an image is broken down into pixels, with a binary value assigned to each pixel.
What is the resolution of an image?
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.
What do the binary values in each pixel represent?
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.
Define a pixel.
A pixel is the smallest element of an image.
What is colour/bit depth?
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.
Give the formula needed to calculate the storage in a bitmap graphics image.
Storage = Number of pixels (length x width) x bit depth.
Why is the calculated storage for bitmap images from the formula, the minimum storage?
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…).
Give another way computers represent images.
Another way images are represented by computers are by vector graphics.
Define vector graphics.
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.
Give an advantage of vector graphics.
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!
Give an advantage of bitmap graphics.
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.
How can computers represent sound?
Computers represent sound via a sequence of samples, taking a digital value.
What is the sampling rate?
The sampling rate is the number of samples recorded per second.
Define the sample resolution.
The sample resolution is the number of bits assigned to each sample.
How can you calculate the storage/size of sound?
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.
Describe and explain the Nyquist theorem.
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.
What is MIDI?
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.
Give an advantage of using MIDI.
MIDI is perfect in easy manipulation of music with no loss in quality.
Give a disadvantage of using MIDI.
MIDI can’t represent speech and compared to other sampled sound recordings, it sounds very unrealistic.
Can smaller or larger files be compressed faster?
Smaller files can be compressed faster and transferred over the internet.
Give the two types of compression.
The two types of compression are lossy compression and lossless compression.
Describe lossy compression.
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.
Describe lossless compression.
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.
State the two types of lossless compression.
- RLE (Run-Length Encoding)
- Dictionary Based Methods
Define RLE (Run-Length Encoding).
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.
Define dictionary based methods.
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
Give a disadvantage of lossless compression over lossy compression.
Unfortunately, lossless compression has a limit of how much the file can be compressed.
Define encryption.
Encryption is the process of scrambling data so it can be safely and securely transmitted and not intercepted.
What is plaintext?
Plaintext is unencrypted text.
What is ciphertext?
Ciphertext is encrypted text.
What is a cipher?
A cipher is an encryption method to encrypt text.
What must you know and have to decrypt text?
To decrypt text, you must have the key and must know the encryption method.
Describe the ‘Caesar cipher’.
The Caesar cipher is an encryption method consisting of replacing characters, with each character being replaced by the same character.
State the two types of Caesar cipher.
- Shift cipher
- Substitution cipher.
Define one type of the Caesar cipher.
The shift cipher is a type of Caesar cipher, where each character will be shifted the same amount of times.
Define another type of the Caesar cipher.
The substitution cipher is a type of Caesar cipher, where each character is substituted by a completely random character.
How easy is it to crack the Caesar ciphers?
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.
What is the Vernam cipher?
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.
Describe how the Vernam cipher is carried out.
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.
Define XOR.
XOR is a type of logic gate which only inputs true if one of its inputs is true. This means ‘exclusive OR’.
How can you decrypt the Vernam cipher?
To decrypt the Vernam cipher, you use the same key used to encrypt the cipher.
Why is the Vernam cipher so secure?
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.
Define computational security.
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.
State the difference between asymmetric and symmetric encryption.
Symmetric encryption involves only one key to encrypt and decrypt while asymmetric encryption uses 2 keys to encrypt and decrypt.
