Chapter 1 - Data representation

Number systems

The binary system

• Base 2 number system

• Two values 0 and 1 to represent all values

• Uses power of 2

Converting from binary to denary

• Each time a 1-value appears in a binary number column, the column value (heading) is added to a total

The equivalent denary number is 128 + 64 + 32 + 8 + 4 + 2 = 238

Converting from denary to binary

• Successive division by 2 until the value “0” is reached.

The hexadecimal system

• Base 16 number system

• Uses 16 different digits to represent each value

• 1 = 1

• 2 = 2

• 3 = 3

• 4 = 4

• 5 = 5

• 6 = 6

• 7 = 7

• 8 = 8

• 9 = 9

• 10 = A

• 11 = B

• 12 = C

• 13 = D

• 14 = E

• 15 = F

• A typical example of hex is 2 1 F 2 A

• Since 16 = 2^4 this means FOUR binary digits are equivalent to each hexadecimal digit.

Converting from binary to hexadecimal and from hexadecimal to binary

• Starting from the right, split the denary number into groups of 4 

• If the last group has less than 4 bits, then simply fill in with 0s from the left

• Take each group of 4 bits and convert it into the equivalent hexadecimal digit

1 0 1 1 1 1 1 0 0 0 0 1

First split this up into groups of 4 bits: 

1 0 1 1 1 1 1 0 0 0 0 1

Then find the equivalent hexadecimal digits: 

    B     E      1

1 0 1 1 1 1 1 1 0 0 0 0 1 = B E 1

Converting from hexadecimal to denary and from denary to hexadecimal

Convert hexadecimal numbers into denary

• Take each of the hexadecimal digits and multiple it by the heading values

• Add all the resultant totals together to give the denary number

• The hex digits A —> F need to be converted to 10 —> 15 before carrying out the multiplication.

1024 + 80 + 10 = 1114

Convert from denary to hexadecimal

• Successive division by 16 until the value “0” is reached.

Use of the hexadecimal system

• Error codes

• MAC addresses

• IPv6 addresses

• HTML colour codes

Error codes

• Memory location of the error and are usually automatically generated by the computer.

Media Access Control (MAC) addresses

• A number which uniquely identifies a device on a network

• Refers to the network interface card (NIC)

• Usually made up of 48 bits which are shown as 6 groups of two hexadecimal digits.

NN - NN - NN - DD - DD - DD

Or

NN:NN:NN:DD:DD:DD

• (NN - NN - NN) is the identity number of the manufacturer of the device and the (DD - DD - DD) is the serial number of the device

Internet Protocol (IP) addresses

• Each device connected to a network is given an address known as the Internet Protocol (IP) address.

• IPv4 is a 32-bit number written in denary or hexadecimal form

• IPv6 is a 128-bit number broken down into 16-bit chunks represented by a hexadecimal number.

• IPv6 uses a colon (:) and IPv4 uses a decimal point (.)

HyperText Mark-up Language (HTML) colour codes

• Used to represent colours of text on the computer screen

• All colours are made up of different combinations of three primary colours (red, green, blue)

• Intensity of each colour is determined by its hexadecimal value

# FF 00 00 represents primary colour red

# 00 FF 00 represents primary colour green

# 00 00 FF represents primary colour blue

# FF 00 FF represents fuchsia

# FF 80 00 represents orange 

# B1 89 04 represents a tan colour

Addition of binary numbers

• Addition of two binary digits

• Addition of three binary digits

Ex.

Answer : 0 1 1 1 0 0 0 1

Overflow

• An overflow error is when the data type used to store data was not large enough to hold the data.

Logic binary shifts

• Moving the binary number to the left or to the right

• Each shift left is equivalent to multiplying the binary number by 2 

• Each shift right is equivalent to dividing the binary number by 2

• As bits are shifted, any empty positions are replaced with a zero

• The denary number 21 is 00010101 in binary. If we put this into an 8-bit register:

• If we now shift the bits in this register one place to the left, we obtain:

• The value of the binary bits is now 21 x 2^1 = 42. We can see this is correct if we calculate the denary value of the new binary number 101010 (32 + 8 + 2 = 42)

• The denary number 200 is 11001000 in binary. If we put this into an 8-bit register:

• If we shift the bits in this register two places to the right, we obtain:

• The value of the binary bits is now 200 / 2^2 = 50. We can see this is correct if we calculated the denary value of the new binary number 00110010 (32 + 16 + 2 = 50).

Two’s complement (binary numbers)

• Allow the possibility of representing negative integers we make use of two’s complement.

• In two’s complement, the leftmost bit is changed to a negative value. 

• 128 is changed to -128 but all the other headings remain the same

• Range of new possible numbers from -128 (10000000) to +127 (01111111)

• Leftmost bit always the sign of the binary number

• 1-value in the leftmost bit indicates a negative number and a 0-value in the leftmost bit indicates a positive number

• Written in this format:

Writing positive binary numbers in two’s complement format

• Ex. 19 and 4 written in two’s complement:

Converting positive denary numbers to binary numbers in the two’s complement format

• Ex. 38

• Since the number is already a positive number, we put a zero in the -128 value and put the rest of the values as you would normally.

Converting positive binary numbers in the two’s complement format to positive denary numbers

• Ex. convert 01101110 in two’s complement binary into denary:

• Each time a 1 appears in a column, the column value is added to the total

• Ex. 64 + 32 + 8 + 4 + 2 = 110

Writing negative binary numbers in two’s complement format and converting to denary

• Ex. -109

• By following our normal rules, each time a 1 appears in a column, the column value is added to the total. So, we can see that in denary this is: -128 + 16 + 2 + 1 = -109

Converting negative denary numbers into binary numbers in two’s complement format

• Consider the number +67 in 8-bit (two’s complement) binary format:

• Method 1

Now let’s consider the number -67. One method of finding the binary equivalent to -67 is to simply put 1s in their correct places:

• Method 2

However, looking at the two binary numbers above, there is another possible way to find the binary representation of a negative denary number:

• First write the number as a positive binary value - in this case 67:     01000011

• We then invert each binary value, which means swap the 1s and 0s around:   10111100

• Then add 1 to that number:         1

• This gives us the binary for -67:

Text, sound and images

Character sets - ASCII code and Unicode

• The ASCII code system (American Standard Code for Information Interchange) is used for communication systems and computer systems.

• The standard ASCII code character set consists of 7-bit codes (0 to 127 in denary or 00 to 7F in hexadecimal) that represents the letters, numbers and characters found on a standard keyboard, together with 32 control codes (that use codes 0 to 31 (denary) or 00 to 19 (hexadecimal)).

• Unicode can represent all languages of the world, thus supporting many operating systems, search engines and internet browsers used globally. 

Representation of sound

• Sampling means measuring the amplitude of the sound wave

• The number of bits per sample is known as the sampling resolution (also known as the bit depth). 

• Sampling rate is the number of sound samples taken per second. This is measured in hertz (Hz), where 1 Hz means ‘one sample per second’.

Representation of (bitmap) images

• Bitmap images are made up of pixels (picture elements); an image is made up of a two-dimensional matrix of pixels

• The number of bits used to represent each colour is called the colour depth

• Image resolution refers to the number of pixels that make up an image

Data storage and file compression

Calculation of file size

• The file size of an image is calculated as:

Image resolution (in pixelsO x colour depth (in bits)

• The size of a mono sound file is calculated as:

Sample rate (in Hz) x sample resolution (in bits) x length of sample (in seconds)

• For a stereo sound file, you would then multiply the result by two.

Lossy and lossless file compression

Lossy file compression

• The file compression algorithm eliminates unnecessary data from the file. This means the original file cannot be reconstructed once it has been compressed.

Lossless file compression

• All the data from the original uncompressed file can be reconstructed. This is particularly important for files where any loss of data would be disastrous.

• Run-length encoding (RLE) can be used for lossless compression of a number of different file formats