Number Systems

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# Number Systems - PowerPoint PPT Presentation

Number Systems. Character Representation. ASCII American Standard Code for Information Interchange Standard encoding scheme used to represent characters in binary format on computers 7-bit encoding, so 128 characters can be represented

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### Number Systems

Character Representation
• ASCII
• American Standard Code for Information Interchange
• Standard encoding scheme used to represent characters in binary format on computers
• 7-bit encoding, so 128 characters can be represented
• 0 to 31 (& 127) are "control characters" (cannot print)
Binary Data
• Decimal, hex & character representations are easier for humans to understand; however…
• All the data in the computer is binary
• An int is typically 32 binary digits
• int y = 5; (y = 0x00000005;)
• In computer y = 00000000 00000000 00000000 00000101
• A char is typically 8 binary digits
• char x = 5; (or char x = 0x05;)
• In computer, x = 00000101
How Computers Store Numbers
• Computer systems are constructed of digital electronics. That means that their electronic circuits can exist in only one of two states: on or off.
• Most computer electronics use voltage levels to indicate their present state. For example, a transistor with five volts would be considered "on", while a transistor with no voltage would be considered "off.”
• These patterns of "on" and "off" stored inside the computer are used to encode numbers using the binary number system.
• Because of their digital nature, a computer\'s electronics can easily manipulate numbers stored in binary by treating 1 as "on" and 0 as "off.”
Number System
• A number system defines how a number can be represented using distinct symbols. A number can be represented differently in different systems. For example, the two numbers (2A)16 and (52)8 both refer to the same quantity, (42)10, but their representations are different.
Common Number Systems

S = {0, 1, 2, 3, 4, 5, 6, 7}

Decimal
• The word decimal is derived from the Latin root decem (ten). In this system the base b = 10 and we use ten symbols:
• The symbols in this system are often referred to as decimal digits or just digits.

S = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}

Binary
• The word binary is derived from the Latin root bini (or two by two). In this system the base b = 2 and we use only two symbols:
• The symbols in this system are often referred to as binary digits or bits (binary digit).

S = {0, 1}

• The word hexadecimal is derived from the Greek root hex (six) and the Latin root decem (ten). In this system the base b = 16 and we use sixteen symbols to represent a number. The set of symbols is:
• Note that the symbols A, B, C, D, E, F are equivalent to 10, 11, 12, 13, 14, and 15 respectively. The symbols in this system are often referred to as hexadecimal digits.
• Each hexadecimal digit represents four binary digits (bits), and the primary use of hexadecimal notation is a human-friendly representation of binary-coded values in computing and digital electronics.

S = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F}

Octal
• The word octal is derived from the Latin root octo (eight). In this system the base b = 8 and we use eight symbols to represent a number. The set of symbols is:
• Each octal digit represents three binary digits (bits)

S = {0, 1, 2, 3, 4, 5, 6, 7}

Types of Number System

S = {0, 1, 2, 3, 4, 5, 6, 7}

Conversion Among Bases

The possibilities:

Decimal

Octal

Binary

Quick Example

2510=110012=318=1916

Base

Binary to Decimal

Decimal

Octal

Binary

Binary to Decimal

Technique

• Multiply each bit by 2n, where n is the “power” of the bit
• The power is the position of the bit, starting from 0 on the right

Example

1010112=> 1 x 20 = 1 1 x 21 = 2 0 x 22 = 0 1 x 23 = 8 0 x 24 = 0 1 x 25 = 32

4310

Bit 0

FRACTIONSBinary to Decimal

Technique

• Multiply each bit by 2n, where n is the “power” of the bit
• The power is the position of the bit, starting from 0 on the right

Example

21

20

2-1

2-2

2-3

2-4

10.01102 => 1 0 . 0 1 1 0

• 0 x (1/16) = 0
• 1 x (1/8) = 1/8
• 1 x (1/4) = 1/4
• 0 x (1/2) = 0
• 0 x (1) = 0
• 1 x (2) = 2

2-4

Bit 0

2-3

2-2

2-1

20

21

Ans: 2.375

Octal to Decimal

Decimal

Octal

Binary

Octal to Decimal

Technique

• Multiply each bit by 8n, where n is the “power” of the bit
• The power is the position of the bit, starting from 0 on the right

Example

7248 => 4 x 80 = 4 2 x 81 = 16 7 x 82 = 448 46810

Bit 0

Decimal

Octal

Binary

Technique

• Multiply each bit by 16n, where n is the “power” of the bit
• The power is the position of the bit, starting from 0 on the right

Example

Bit 0

ABC16=>

C x 160 = 12 x 1 = 12B x 161 = 11 x 16 = 176A x 162 = 10 x 256 = 2560

274810

Decimal to Binary

Decimal

Octal

Binary

Decimal to Binary

Technique

• Divide by the base 2, keep track of the remainder
• Keep dividing until the quotient is 0.
• Take the remainder from the bottom and move upwards as the answer

E.g.: Convert 12510 to binary

Take the remainder from bottom upwards as answer

Ans:

11111012

FRACTIONSDecimal to Binary

Technique

• For the numbers after the point, multiply it by 2
• From the answer, take again the fraction part and multiply it by 2 again
• Keep on multiplying the fraction by 2 until the fraction part is 0

E.g.: Convert 12.2510to binary

12 . 25

12 / 2 = 6

6 / 2 = 3

3 / 2 = 1

1 / 2 = 0

R0

R0

R1

R1

0.25 X 2 = 0.50

0.50 X 2 = 1.00

01

1100

Ans:

1100.012

FRACTIONSDecimal to Binary

.14579x 20.29158x 20.58316x 21.16632x 20.33264x 20.66528x 21.33056

etc.

3.14579

11.001001...

Octal to Binary

Decimal

Octal

Binary

Octal to Binary

Technique

• Convert each octal digit to a 3-bit equivalent binary representation
• 1 octal digit = 3 binary digits

E.g.: Convert 7058 to binary

Start from the right ‘0’ bit

4 2 1

4 2 1

4 2 1

7 0 5

111 000 101

Ans: 111 000 1012

Binary to Octal

Decimal

Octal

Binary

Binary to Octal

Technique

• Divide the binary bits in group of 3’s, starting from the RIGHT
• Add 0’s to the last group to make it 3 bits
• Convert each grouped binary to their octal digits

E.g.: 10110101112

Start from the right ‘0’ bit

Divide the binary numbers into groups of 3’s. Add ‘0’ to the last group to make it 3 bits

001 011 010 111

4 2 1

4 2 1

4 2 1

4 2 1

001 011 010 111

1 3 2 7

Ans: 13278

Decimal

Octal

Binary

Technique

• Convert each hexadecimal digit to a 4-bit equivalent binary representation

4 binary digits

E.g.: Convert 3A816 to binary

Start from the right ‘0’ bit

8 4 2 1

8 4 2 1

8 4 2 1

3 A (10) 5

0011 1010 0101

Ans: 0011 1010 01012

Decimal

Octal

Binary

Technique

• Divide the binary bits in group of 4’s, starting from the RIGHT
• Add 0’s to the last group to make it 4 bits
• Convert each grouped binary to their hexadecimal digits

E.g.: 10101110112

Start from the right ‘0’ bit

Divide the binary numbers into groups of 4’s. Add ‘0’ to the last group to make it 4 bits

00101011 1011

8 4 2 1

8 4 2 1

8 4 2 1

0010 1011 1011

2 11 (B) 11 (B)

Ans: 2BB16

Decimal to Octal

Decimal

Octal

Binary

Decimal to Octal

Technique

• Divide by the base 8, keep track of the remainder
• Keep dividing until the quotient is 0.
• Take the remainder from the bottom and move upwards as the answer

E.g.: Convert 123410 to octal

Take the remainder from bottom upwards as answer

Ans:

23228

Decimal

Octal

Binary

Technique

• Divide by the base 16, keep track of the remainder
• Keep dividing until the quotient is 0.
• Take the remainder from the bottom and move upwards as the answer

E.g.: Convert 207910 to binary

Take the remainder from bottom upwards as answer

Ans:

81F16

Decimal

Octal

Binary

Technique

• First step:

Convert the octal digits to their 3-bits binary

• Second step:

Combined the binary obtained

• Third step:

Divide the binary into groups of 4 (hexadecimal) starting from the RIGHT

• Fourth step:

Find the hexadecimal digit from the grouped binary

E.g.: 10768

1 0 7 6

001

000

111

110

Divide the binary numbers into groups of 4’s. Add ‘0’ to the last group to make it 4 bits

Start from the right ‘0’ bit

8 4 2 1

8 4 2 1

8 4 2 1

0010 / 0011 / 1110

2

3

14 (E)

Ans: 23E16

Decimal

Octal

Binary

Technique

• First step:

Convert the hexadecimal digits to their 4-bits binary

• Second step:

Combined the binary obtained

• Third step:

Divide the binary into groups of 3 (octal ) starting from the RIGHT

• Fourth step:

Find the octal digit from the grouped binary

E.g.: C4516

C (12) 4 5

1100

0100

0101

Start from the right ‘0’ bit

Divide the binary numbers into groups of 4’s. Add ‘0’ to the last group to make it 4 bits

4 2 1

4 2 1

4 2 1

4 2 1

110 / 0 01 / 00 0 / 101

6

1

0

5

Ans: 61058

• Binary addition between 2 1-bit values:

“TWO”

Two n-bit values

• Propagate carries
• E.g.:

1

1

10101 + 11001

101110

21 + 25

46

Binary Subtraction

• Binary subtraction between 2 1-bit values:
Binary Subtraction

In situations like A – B, whereby the binary number ‘B’ is larger than A, swap the number so that B is in front: B – A. Then negate the answer: B – A = -Y

10111 is larger than 1001. So swap so that the larger number is in front and the smaller number is after.

Borrow ‘1’ from the next binary digit. Hence the number will become 10 which is equals to ‘2’.

E.g.: 1001 – 10111 = ?

1

1011 - 101

110

10111 - 1001

- 1110