2. Data Representation: Questions and Answers

b) binary digits (0 and 1)

So all data (numbers, characters, images, audio, video) must be converted to binary form.

b) 8

c) 15

A=10, B=11, C=12, D=13, E=14, F=15.

b) positional

c) 75

64 + 8 + 2 + 1 = 75

a) 101

b) Sum 0, Carry 1

b) (10000)2

13 + 3 = 16

b) negative number

0 indicates a positive number.

a) 10101000

c) 2's complement

b) floating point number

b) 7 bits

This gives 2^7 = 128 characters.

c) Unicode Consortium

It is a non-profit organization.

a) pixel

b) PNG

c) WAV

b) digitization

c) 24 to 60

d) GIF

Base or radix.

Decem.

Least Significant Digit (LSD); Most Significant Digit (MSD).

2.

7.

10; 11.

2; bottom to top.

3 bits; 4 bits.

1.

Word. The number of bits in it is the word length.

Mantissa and exponent.

American Standard Code for Information Interchange.

24 bits (8 bits each for Red, Green and Blue).

JPEG uses lossy compression; PNG uses lossless compression.

Frames.

A number system is a way of writing and representing numbers using a set of symbols or digits, based on value and base. Four number systems: decimal (base 10), binary (base 2), octal (base 8) and hexadecimal (base 16).

The number of symbols (digits) used in a number system is called its base or radix. For example, the decimal system uses 10 symbols (0 to 9), so its base is 10.

• LSB (Least Significant Bit): the rightmost bit of a binary number.
• MSB (Most Significant Bit): the leftmost bit of a binary number.

Computers are built from millions or billions of tiny electronic switches called transistors, which have only two states: ON (high voltage, 1) and OFF (low voltage, 0). This makes binary the simplest, most reliable and error-resistant way to represent data. Using more states (like the 10 digits of decimal) would need precise voltage levels that are hard to maintain because of electrical noise, heat or signal degradation, causing frequent errors.

Hexadecimal uses 16 symbols (0-9 and A-F, where A=10 to F=15). It is useful in computing because it gives a compact, human-friendly way to read and write binary values. Each hex digit stands for exactly 4 bits, so conversion to and from binary is easy.

• Computer language: binary is the native language of computers; all data and instructions are binary codes.
• Digital electronics: represents digital signals in circuits, the backbone of digital technology.
• Efficient data storage: used to store data in computers.
• Encryption: securing data by converting information into binary form.

Octal is more compact than binary, so it is easier to read, and it converts easily to and from binary. It is used to represent file permissions in Unix-like operating systems, memory addresses in some computer systems, and in embedded systems like microcontrollers for programming and debugging.

A number system in which the value of a digit depends on its position is a positional number system. Each position represents a power of the base, starting from base^0 at the rightmost. Example: 534 = 5x10^2 + 3x10^1 + 4x10^0 = 500 + 30 + 4. Here 5 has the largest place value (100) and 4 has the smallest (1).

1. Sign and magnitude representation
2. 1's complement representation
3. 2's complement representation

The MSB is the sign bit (0 = positive, 1 = negative) and the remaining 7 bits (in an 8-bit word) hold the magnitude. To represent -25: magnitude 25 = 0011001; sign is negative so MSB = 1; result = 10011001.

A negative number is formed by changing all 0s to 1s and all 1s to 0s in the 8-bit binary form of the positive number. Example: 87 = 01010111, so -87 = 10101000.

It is the most commonly used method for negative integers. Convert the absolute value to 8-bit binary, find the 1's complement, then add 1. Example: 25 = 00011001; 1's complement = 11100110; add 1 gives 11100111, so -25 = 11100111. Complement methods are used only for representing negative integers.

A word is a group of bits of fixed size handled as a unit by a processor. The number of bits in a word is the word length. Popular word lengths are 8, 16, 32 and 64.

A number with both an integer part and a fractional part is a floating point number. It has a mantissa (the actual number, normalised, e.g. 0.134) and an exponent (how many places the point moves, like scientific notation; computers use powers of 2). Example: 13.4 = 0.134 x 10^2, with mantissa 134 and exponent 2.

ASCII (American Standard Code for Information Interchange) is a character encoding standard that represents letters, numbers and symbols using 7-bit codes (128 characters). It was first published in 1963 by the American Standards Association. IBM popularised 8-bit encodings called Extended ASCII. Merits: simple and efficient, widely supported and compatible. Demerit: limited character set; does not adequately support multiple languages, special characters or symbols.

Unicode is a standard character encoding scheme that is universal, efficient, uniform and unambiguous, introduced because ASCII (7-bit) and Extended ASCII (8-bit, 256 characters) cannot represent characters of all scripts. It is maintained by the Unicode Consortium, defines up to 1,114,112 code points and is commonly encoded as UTF-8, UTF-16 or UTF-32. Merits: universal language support, platform independence, simplified application development and efficient data sharing.

An image is stored as a collection of tiny dots called pixels. Each pixel's colour and brightness is converted into binary data. Colours use the RGB model (Red, Green, Blue), each component from 0 to 255, so each pixel's colour takes 24 bits (8 bits each for R, G and B). A digital image has a header section (file format, width, height, colour depth, compression type), pixel data (the binary colour values) and metadata (date, camera settings, author, GPS location).

1920 x 1080 = 2,073,600 pixels.

Natural sound waves are analog signals. Since computers understand only binary, analog signals are converted into digital signals. This process is called digitization.

A video is a series of still images called frames, shown quickly (usually 24 to 60 frames per second) along with sound, creating the illusion of motion. Each frame is stored digitally, often in RGB colour format. Videos are compressed to reduce file size, and audio tracks are synchronised with the frames. The compressed video and audio streams are stored together in a single file container such as MP4, AVI or MOV.

OGG is a container format, and Vorbis is the audio compression format commonly used inside it. Ogg Vorbis is open-source and gives good compression efficiency; it is used in open-source applications and interactive games.

35 / 2 = 17, remainder 1
17 / 2 = 8, remainder 1
8 / 2 = 4, remainder 0
4 / 2 = 2, remainder 0
2 / 2 = 1, remainder 0
1 / 2 = 0, remainder 1
Read the remainders from bottom to top: 100011
(35)10 = (100011)2

45 / 2 = 22, remainder 1
22 / 2 = 11, remainder 0
11 / 2 = 5, remainder 1
5 / 2 = 2, remainder 1
2 / 2 = 1, remainder 0
1 / 2 = 0, remainder 1
Read the remainders from bottom to top: 101101
(45)10 = (101101)2

59 / 2 = 29, remainder 1
29 / 2 = 14, remainder 1
14 / 2 = 7, remainder 0
7 / 2 = 3, remainder 1
3 / 2 = 1, remainder 1
1 / 2 = 0, remainder 1
Read the remainders from bottom to top: 111011
(59)10 = (111011)2

Check: 32 + 16 + 8 + 2 + 1 = 59. (Some prints of the textbook show (11101)2, which equals 29. The correct answer is 111011.)

65 / 2 = 32, remainder 1
32 / 2 = 16, remainder 0
16 / 2 = 8, remainder 0
8 / 2 = 4, remainder 0
4 / 2 = 2, remainder 0
2 / 2 = 1, remainder 0
1 / 2 = 0, remainder 1
Read the remainders from bottom to top: 1000001
(65)10 = (1000001)2

78 / 2 = 39, remainder 0
39 / 2 = 19, remainder 1
19 / 2 = 9, remainder 1
9 / 2 = 4, remainder 1
4 / 2 = 2, remainder 0
2 / 2 = 1, remainder 0
1 / 2 = 0, remainder 1
Read the remainders from bottom to top: 1001110
(78)10 = (1001110)2

123 / 2 = 61, remainder 1
61 / 2 = 30, remainder 1
30 / 2 = 15, remainder 0
15 / 2 = 7, remainder 1
7 / 2 = 3, remainder 1
3 / 2 = 1, remainder 1
1 / 2 = 0, remainder 1
Read the remainders from bottom to top: 1111011
(123)10 = (1111011)2

(101101)2 = 1x32 + 0x16 + 1x8 + 1x4 + 0x2 + 1x1
        = 45
(101101)2 = (45)10

(110101)2 = 1x32 + 1x16 + 0x8 + 1x4 + 0x2 + 1x1
        = 53
(110101)2 = (53)10

Group in 4 bits from the right: 0011 0101
Hex digits:                    3    5
(110101)2 = (35)16

(10110101)2 = 1x128 + 0x64 + 1x32 + 1x16 + 0x8 + 1x4 + 0x2 + 1x1
        = 181
(10110101)2 = (181)10

Group in 4 bits from the right: 1011 0101
Hex digits:                    B    5
(10110101)2 = (B5)16

(10101101)2 = 1x128 + 0x64 + 1x32 + 0x16 + 1x8 + 1x4 + 0x2 + 1x1
        = 173
(10101101)2 = (173)10

Group in 4 bits from the right: 1010 1101
Hex digits:                    A    D
(10101101)2 = (AD)16

(10001)2 = 1x16 + 0x8 + 0x4 + 0x2 + 1x1
        = 17
(10001)2 = (17)10

Group in 4 bits from the right: 0001 0001
Hex digits:                    1    1
(10001)2 = (11)16

The decimal values are 10, 12, 14, so the series increases by 2.
Next: 16 and 18
16 = (10000)2
18 = (10010)2
Missing terms: (10000)2, (10010)2

Group in 3 bits from the right: 101 111 010
Octal digits:                 5   7   2
(101111010)2 = (572)8

Group in 4 bits from the right: 0001 0111 1010
Hex digits:                    1    7    A
(101111010)2 = (17A)16

Group in 3 bits from the right: 001 111 001
Octal digits:                 1   7   1
(1111001)2 = (171)8

Group in 4 bits from the right: 0111 1001
Hex digits:                    7    9
(1111001)2 = (79)16

Group in 3 bits from the right: 101 010 101
Octal digits:                 5   2   5
(101010101)2 = (525)8

Group in 4 bits from the right: 0001 0101 0101
Hex digits:                    1    5    5
(101010101)2 = (155)16

Group in 3 bits from the right: 001 000 101
Octal digits:                 1   0   5
(1000101)2 = (105)8

Group in 4 bits from the right: 0100 0101
Hex digits:                    4    5
(1000101)2 = (45)16

Group in 3 bits from the right: 101 101 010
Octal digits:                 5   5   2
(101101010)2 = (552)8

Group in 3 bits from the right: 001 110 010 101
Octal digits:                 1   6   2   5
(1110010101)2 = (1625)8

Group in 4 bits from the right: 1001 0111 1011
Hex digits:                    9    7    B
(100101111011)2 = (97B)16

Group in 4 bits from the right: 0011 0101 1001
Hex digits:                    3    5    9
(1101011001)2 = (359)16

Group in 4 bits from the right: 1111 0000
Hex digits:                    F    0
(11110000)2 = (F0)16

Octal digits:   5   2   3
3-bit binary:   101 010 011
(523)8 = (101010011)2

Octal digits:   7   0   4   6
3-bit binary:   111 000 100 110
(7046)8 = (111000100110)2

Octal digits:   3   4   5
3-bit binary:   011 100 101
(345)8 = (011100101)2

Hex digits:     9    A    F
4-bit binary:   1001 1010 1111
(9AF)16 = (100110101111)2

Hex digits:     1    2    D
4-bit binary:   0001 0010 1101
(12D)16 = (000100101101)2

Hex digits:     2    F
4-bit binary:   0010 1111
(2F)16 = (00101111)2

(123)8 = 1x8^2 + 2x8^1 + 3x8^0
       = 1x64 + 2x8 + 3x1
       = 64 + 16 + 3
       = 83

(258)16 = 2x16^2 + 5x16^1 + 8x16^0
        = 2x256 + 5x16 + 8x1
        = 512 + 80 + 8
        = 600

Magnitude 19 in 7 bits: 0010011
Sign: negative, so MSB = 1
-19 in sign and magnitude = 10010011

Magnitude 49 in 7 bits: 0110001
Sign: positive, so MSB = 0
+49 in sign and magnitude = 00110001

Magnitude 97 in 7 bits: 1100001
Sign: negative, so MSB = 1
-97 in sign and magnitude = 11100001

Magnitude 127 in 7 bits: 1111111
Sign: positive, so MSB = 0
+127 in sign and magnitude = 01111111

8-bit binary of 24: 00011000
Replace 0s with 1s and 1s with 0s: 11100111
-24 in 1's complement = 11100111

8-bit binary of 69: 01000101
Replace 0s with 1s and 1s with 0s: 10111010
-69 in 1's complement = 10111010

8-bit binary of 100: 01100100
Replace 0s with 1s and 1s with 0s: 10011011
-100 in 1's complement = 10011011

8-bit binary of 127: 01111111
Replace 0s with 1s and 1s with 0s: 10000000
-127 in 1's complement = 10000000

8-bit binary of 33: 00100001
1's complement:         11011110
Add 1:                  00000001
2's complement:         11011111
-33 in 2's complement = 11011111

8-bit binary of 71: 01000111
1's complement:         10111000
Add 1:                  00000001
2's complement:         10111001
-71 in 2's complement = 10111001

8-bit binary of 111: 01101111
1's complement:         10010000
Add 1:                  00000001
2's complement:         10010001
-111 in 2's complement = 10010001

8-bit binary of 127: 01111111
1's complement:         10000000
Add 1:                  00000001
2's complement:         10000001
-127 in 2's complement = 10000001

   1101
+  0011
-------
  10000
Check: 13 + 3 = 16 = (10000)2

   1111
+  1101
-------
  11100
Check: 15 + 13 = 28 = (11100)2

   1011
+  0110
-------
  10001
Check: 11 + 6 = 17 = (10001)2

   1010
+  1010
-------
  10100
Check: 10 + 10 = 20 = (10100)2

Each is a positional system: the value of a digit is the digit times a power of the base, based on its position.

• Decimal to binary: divide repeatedly by 2 until 0, then read the remainders from bottom to top.
• Binary to decimal: multiply each bit by its power of 2 and add.
• Octal to binary: replace each octal digit by its 3-bit binary equivalent and combine.
• Binary to octal: group bits in 3s from the right (add leading 0s if needed) and convert each group.
• Hexadecimal to binary: replace each hex digit by its 4-bit binary equivalent and combine.
• Binary to hexadecimal: group bits in 4s from the right (add leading 0s if needed) and convert each group.

Computers are built using millions or billions of tiny electronic switches called transistors, which can be in only two states: ON (high voltage, 1) or OFF (low voltage, 0). This two-state nature matches binary exactly, making it the simplest, most reliable and most error-resistant way to represent data. A system with more states, like decimal's 10 digits, would need precise voltage levels that are hard to maintain because of electrical noise, heat or signal degradation, causing frequent errors. In binary a signal is clearly 'on' or 'off'.

• Sign and magnitude: MSB is the sign (0 positive, 1 negative); remaining 7 bits are the magnitude. -25 = 10011001.
• 1's complement: flip every bit of the 8-bit positive form. -87: 01010111 becomes 10101000.
• 2's complement: 1's complement plus 1. -25: 00011001, then 11100110, plus 1 gives 11100111.

2's complement is the most commonly used by computers because it is simple, efficient and avoids many problems of other methods. Complement methods are used only for negative integers.

A floating point number has an integer part and a fractional part. It is stored in two parts: the mantissa, the actual digits normalised to the form 0.xxx (for 13.4 it is 0.134), and the exponent, which says how many places the point moves, like scientific notation. Decimal uses powers of 10 but computers use powers of 2. Example: 13.4 = 0.134 x 10^2, where 134 is the mantissa and 2 is the exponent. In a computer the same parts are stored in binary format.

Computers store characters as binary codes. ASCII uses 7 bits (128 characters) and was published in 1963; Extended ASCII uses 8 bits (256 characters) and mostly adds European characters. Because these cannot represent all scripts, Unicode was introduced. It is universal and unambiguous, maintained by the Unicode Consortium, defines up to 1,114,112 code points and is encoded as UTF-8, UTF-16 or UTF-32.

An image is a grid of pixels. Each pixel's colour is stored in binary using the RGB model: 8 bits each for Red, Green and Blue, i.e. 24 bits per pixel. A digital image file has a header, pixel data and metadata, and may be stored uncompressed or compressed (lossy or lossless).

• BMP: Microsoft; uncompressed; large size but high quality; printing and editing.
• JPEG: lossy; good balance of quality and size; photos and web images.
• PNG: lossless; supports transparency; logos and web graphics.
• GIF: 256 colours; supports animation and transparency; web animations and memes.
• TIFF: professional use; lossy and lossless; high quality, large size.

Natural sound is an analog signal. It is converted into a digital signal by digitization so computers can store, process and transmit it.

• WAV: Microsoft; uncompressed; very high quality, large files; professional editing and recording.
• MP3: lossy; removes inaudible sounds; widely supported; music players and online sharing.
• AAC: improved MP3 with better quality at the same bit rate; streaming platforms.
• FLAC: lossless; archiving and high-fidelity playback.
• OGG (Vorbis): open-source container with Vorbis compression; open-source applications and games.
• WMA: Microsoft; lossy and lossless options; works best in Windows applications.

A video is a sequence of frames (usually 24 to 60 per second) with sound. Each frame is stored digitally, often in RGB format; the video is compressed and the audio is synchronised with the frames. The video and audio streams are stored together in one container file (MP4, AVI, MOV).

Binary addition follows its own rules because the base is 2: 0+0=0, 0+1=1, 1+0=1, and 1+1=0 with a carry of 1. Carries move to the next column on the left.

   1101
+  0011
-------
  10000
Check: 13 + 3 = 16 = (10000)2

   1111
+  1101
-------
  11100
Check: 15 + 13 = 28 = (11100)2