Data Representation

Number systems

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Understand binary

Any form of data needs to be converted to binary to be processed by a computer.
The basic building block in all computers is the binary number system.

Switch

Switch ON is 1 and OFF is 0. Alt text

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Digit weight

Every one of us is used to the decimal or denary (base 10) number system. This uses the digits 0 to 9 which are placed in ‘weighted’ columns.

10000	1000	100	10	1
10⁴	10³	10²	10¹	10⁰
3	1	4	2	1

Example: 3x10000 + 1x1000 + 4x100 + 2x10 + 1x1 = 31421

Binary to denary

Except for denary, we use binary(base 2) and hexadecimal(base 16) number system in the computer.

Denary value	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
Binary value	0000	0001	0010	0011	0100	0101	0110	0111	1000	1001	1010	1011	1100	1101	1110	1111
Hexadecimal value	0	1	2	3	4	5	6	7	8	9	A	B	C	D	E	F

The binary system uses 1s and 0s only which gives these corresponding weightings.
We can convert binary number to denary according to digit weightings.

128	64	32	16	8	4	2	1
2⁷	2⁶	2⁵	2⁴	2³	2²	2¹	2⁰
1	1	1	0	1	1	1	0

Example: 1110 1110₂ = 128 + 64 + 32 + 8 + 4 + 2 = 238₁₀

Convert binary to denary: 0011 0011 ?

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binary

number system

Which number is closest to 0011 1111 ?

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Which numbers are less than 1011 1101 ?

170

180

190

200

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Hexadecimal to denary

The hexadecimal system is very closely related to the binary system.
Hexadecimal is a base 16 system.
Because it is a system based on 16 different digits, the numbers 0 to 9 and the letters A to F are used to represent hexadecimal digits.
A = 10, B = 11, C = 12, D = 13, E = 14 and F = 15.
We can convert hexadecimal number to denary according to digit weightings.

65536	4096	256	16	1
16⁴	16³	16²	16¹	16⁰
0	1	1	2	3

Example: 01123₁₆ = 1x4096 + 1x256 + 2x16 + 3x1 = 4387₁₀

Convert hexadecimal to denary: ABC ?

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Convert hexadecimal to denary: DE0 ?

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Use of the hexadecimal system

When the memory contents are output to a printer or monitor, this is known as a memory dump.

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Denary to binary

Converting from denary to binary is slightly more complex.
This method involves successive division by 2 until the result is 0; the remainders are then written from bottom to top to give the binary value.

Divider	Result	Remainder	Process
2	107
2	53	1	107 ➗ 2 = 53 ……1
2	26	1	53 ➗ 2 = 26 ……1
2	13	0	26 ➗ 2 = 13 ……0
2	6	1	13 ➗ 2 = 6 ……1
2	3	0	6 ➗ 2 = 3 ……0
2	1	1	3 ➗ 2 = 1 ……1
	0	1	1 ➗ 2 = 0 ……1

Example: 107₁₀ = 110 1011₂

Convert denary to binary: 41 ?

101010

101001

101000

101011

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Convert denary to binary: 252 ?

11111101

11111110

11111111

11111100

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Denary to hexadecimal

Converting from denary to hexadecimal is slightly similar with denary to binary.
This method involves successive division by 16; the remainders are then written from bottom to top to give the hexadecimal value.

Divider	Result	Remainder	Process
16	2004
16	125	4	2004 ➗ 16 = 125 ……4
16	7	13(D)	125 ➗ 16 = 7 ……13(D)
16	0	7	7 ➗ 16 = 0 ……7

Example: 2004₁₀ = 7D4₁₆

Convert denary to hexadecimal: 2022 ?

7E6

7E5

7146

7145

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Convert denary to hexadecimal: 252 ?

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Binary to hexadecimal

Since 16 = 2⁴, four binary digits are equivalent to each hexadecimal digit.

Binary value	0000	0001	0010	0011	0100	0101	0110	0111	1000	1001	1010	1011	1100	1101	1110	1111
Hexadecimal value	0	1	2	3	4	5	6	7	8	9	A	B	C	D	E	F
Denary value	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15

Example: 1111100001² = 0011 1110 0001² = 3E1¹⁶

Convert binary to hexadecimal: 1001011 ?

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Hexadecimal to binary

Since 16 = 2⁴, one hexadecimal digit are equivalent to four binary digits.
Example: 45A₁₆ = 0100 0101 1010₂

Convert hexadecimal to binary: 6C ?

1101110

1101101

1111100

1101100

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Positive numbers in binary

There are two methods to represent both positive and negative numbers.
In one’s complement, each digit in the binary number is inverted.
In two’s complement, each digit in the binary number is inverted and a ‘1’ is added to the right-most bit.

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Decimal value	Two's-complement representation
0	0000 0000
1	0000 0001
2	0000 0010
126	0111 1110
127	0111 1111
−128	1000 0000
−127	1000 0001
−126	1000 0010
−2	1111 1110
−1	1111 1111

Two’s complement

The two’s complement uses these weightings for an 8-bit number representation.

	-128	64	32	16	8	4	2	1
104	0	1	1	0	1	0	0	0
-104	1	0	0	1	1	0	0	0

For example, 104 in binary is 0 1 1 0 1 0 0 0.
To find the binary value for −104 using two’s complement:
Invert the digits: 1 0 0 1 0 1 1 1 (+104 in denary)
Add 1: 1
Which gives: 1 0 0 1 1 0 0 0 (−104 in denary)

Use two's complement to represent -114 with 8 bits.

10001110

01110010

11110010

00001110

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Binary addition

bianry addtion

0 + 0 = 0
1 + 0 = 1 (sum 1 and carry 0)
1 + 1 = 10 (sum 0 and carry 1 )
1 + 1 + (carry 1) = 11= sum 1 and carry 1

Add 0 0 1 0 0 1 0 1 (37 in denary) and 0 0 1 1 1 0 1 0 (58 in denary).

	-128	64	32	16	8	4	2	1
37	0	0	1	0	0	1	0	1
	+
58	0	0	1	1	1	0	1	0
	=
Carry	0	1	0	0	0	0	0	0
Sum	0	1	0	1	1	1	1	1

The sum is 0101 1111, which is 95 in denary.

Calculate 0011 1001 + 0010 1001.

0110 0001

0110 0110

0110 0010

0110 0011

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Binary subtraction

Carry out the subtraction 95 – 68 in binary.

		-128	64	32	16	8	4	2	1
95		0	1	0	1	1	1	1	1
		-
68		0	1	0	0	0	1	0	0
-68		1	0	1	1	1	1	0	0
Carry	1 (Drop it)	1	1	1	1	1	0	0	0
Sum	1 (Drop it)	0	0	0	1	1	0	1	1

The result is 0001 1011, which is 27 in denary.

Calculate 0110 0011 - 0011 0000.

0011 0010

0011 0011

0011 0001

0011 0111

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Overflow error

Add 0 1 0 1 0 0 1 0 (82 in denary) and 0 1 0 0 0 1 0 1 (69 in denary).

	-128	64	32	16	8	4	2	1
82	0	1	0	1	0	0	1	0
	+
69	0	1	0	0	0	1	0	1
	=
Carry	1	0	0	0	0	0	0	0
Sum	1	0	0	1	0	1	1	1

The sum is 1001 0111, which is -105 (Incorrect)

Overflow error

The expected answer for 82 + 69 is 151, which is out of range for the 8 bits register (-128~127), this is known as an overflow error.

Logical shifts

The positive binary integer is multiplied or divided according to the shift performed.
Bits shifted from the end of the register are lost and zeros are shifted in at the opposite end of the register.
The most significant bit(s) or least significant bit(s) are lost.

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Right shift 1011 0101 with 2 digits?

0110 1101

0010 1101

1010 1101

0010 1100

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Measurement of the size of computer memories

TIP

A binary digit is referred to as a BIT. 4 bits are 1 NIBBLE. 8 bits are 1 BYTE.

Storage Device

Measurement	Number of bytes
1 kilobyte(1 KB)	10³
1 megabyte(1 MB)	10⁶
1 gigabyte(1 GB)	10⁹
1 terabyte(1 TB)	10¹²
1 petabyte(1 PB)	10¹⁵

Computer System

Measurement	Number of bytes
1 kibibyte(1 KiB)	2¹⁰
1 mebibyte(1 MiB)	2²⁰
1 gibibyte(1 GiB)	2³⁰
1 tebibyte(1 TiB)	2⁴⁰
1 pebibyte(1 PiB)	2⁵⁰

KB vs KiB

KB is used in factory and Kib is used in computer system.

8GB = ( ) bits?

$6.4 \times 10^{10}$

$8.0 \times 10^{10}$

$6.4 \times 10^{9}$

$8.0 \times 10^{9}$

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8GiB = ( ) bytes?

$2^{31}$

$2^{32}$

$2^{33}$

$2^{34}$

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Binary-coded decimal (BCD) system

The binary-coded decimal (BCD) system uses a 4-bit code to represent each denary digit.

0 0 0 0	0	0 1 0 1	5
0 0 0 1	1	0 1 1 0	6
0 0 1 0	2	0 1 1 1	7
0 0 1 1	3	1 0 0 0	8
0 1 0 0	4	1 0 0 1	9

Therefore, the denary number 3 1 6 5 would be 0011 0001 0110 0101 in BCD format.

Use of BCD

The most obvious use of BCD is in the representation of digits on a calculator or clock display.

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Consider adding $0.37 and $0.94 together using fixed-point decimals.

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Convert denary number 271 to BCD format?

0010 0111 0001

0001 0000 1111

0001 0000 1110

0001 0000 1101

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ASCII codes and Unicodes

ASCII (American Standard Code for Information Interchange)

Text is converted to binary to be processed by a computer according to ASCII code.(7 bits per character).

Extended ASCII code

Extended ASCII code use 8 bits per character.
A is stored as 0100 0001

ASCII code

DEC	OCT	HEX	BIN	Symbol	HTML Number	Description
65	101	41	01000001	A	A	Uppercase A
66	102	42	01000010	B	B	Uppercase B
67	103	43	01000011	C	C	Uppercase C
68	104	44	01000100	D	D	Uppercase D
69	105	45	01000101	E	E	Uppercase E
70	106	46	01000110	F	F	Uppercase F

Unicode

Unicode allows for a greater range of characters and symbols than ASCII, including different languages and emojis. (16 bits per character)

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If A is stored as 65 in ASCII code, what is the binary value to store G?

01000111

01001000

01001001

01001010

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Which items are using hexadecimal?

Unicode

Memory Dump

HTML Color

Programming Language

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Tools

Data Representation ​

Number systems ​

Understand binary ​

Digit weight ​

Binary to denary ​

Hexadecimal to denary ​

Use of the hexadecimal system ​

Denary to binary ​

Denary to hexadecimal ​

Binary to hexadecimal ​

Hexadecimal to binary ​

Positive numbers in binary ​

Two’s complement ​

Binary addition ​

Binary subtraction ​

Overflow error ​

Logical shifts ​

Measurement of the size of computer memories ​

Storage Device ​

Computer System ​

Binary-coded decimal (BCD) system ​

Use of BCD ​

ASCII codes and Unicodes ​

ASCII code ​

Unicode ​

Data Representation

Number systems

Understand binary

Digit weight

Binary to denary

Hexadecimal to denary

Use of the hexadecimal system

Denary to binary

Denary to hexadecimal

Binary to hexadecimal

Hexadecimal to binary

Positive numbers in binary

Two’s complement

Binary addition

Binary subtraction

Overflow error

Logical shifts

Measurement of the size of computer memories

Storage Device

Computer System

Binary-coded decimal (BCD) system

Use of BCD

ASCII codes and Unicodes

ASCII code

Unicode