Number systems
Download
1 / 45

Number Systems - PowerPoint PPT Presentation


  • 111 Views
  • Uploaded on

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

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Number Systems' - ursala


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
Number systems

Number Systems


Character representation
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
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
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
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
Common Number Systems

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


Decimal
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
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}


Hexadecimal
Hexadecimal

  • 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
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
Types of Number System

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


Conversion among bases
Conversion Among Bases

The possibilities:

Decimal

Octal

Binary

Hexadecimal


Quick example
Quick Example

2510=110012=318=1916

Base


Binary to decimal
Binary to Decimal

Decimal

Octal

Binary

Hexadecimal


Binary to decimal1
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

  • ADD the results

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


Fractions binary to decimal
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

  • ADD the results

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
Octal to Decimal

Decimal

Octal

Binary

Hexadecimal


Octal to decimal1
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

  • Add the results

Example

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

Bit 0


Hexadecimal to decimal
Hexadecimal to Decimal

Decimal

Octal

Binary

Hexadecimal


Hexadecimal to decimal1
Hexadecimal to Decimal

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

  • Add the results

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 to Binary

Decimal

Octal

Binary

Hexadecimal


Decimal to binary1
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


Fractions decimal to binary
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


Number systems

FRACTIONSDecimal to Binary

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

etc.

3.14579

11.001001...


Octal to binary
Octal to Binary

Decimal

Octal

Binary

Hexadecimal


Octal to binary1
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 octa l
Binary to Octal

Decimal

Octal

Binary

Hexadecimal


Binary to octal
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


Hexadecimal to binary
Hexadecimal to Binary

Decimal

Octal

Binary

Hexadecimal


Hexadecimal to binary1
Hexadecimal to Binary

Technique

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

  • 1 hexadecimal digit =

    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


Binary to hexadecimal
Binary to Hexadecimal

Decimal

Octal

Binary

Hexadecimal


Binary to hexadecimal1
Binary to Hexadecimal

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 octa l
Decimal to Octal

Decimal

Octal

Binary

Hexadecimal


Decimal to octal
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 to hexadecimal
Decimal to Hexadecimal

Decimal

Octal

Binary

Hexadecimal


Decimal to hexadecimal1
Decimal to Hexadecimal

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


Octal to hexadecimal
Octal to Hexadecimal

Decimal

Octal

Binary

Hexadecimal


Octal to hexadecimal1
Octal to Hexadecimal

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


Hexadecimal to octal
Hexadecimal to Octal

Decimal

Octal

Binary

Hexadecimal


Hexadecimal to octal1
Hexadecimal to Octal

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


Number systems

Binary Addition

  • Binary addition between 2 1-bit values:

“TWO”


Binary addition
Binary Addition

Two n-bit values

  • Add individual bits

  • Propagate carries

  • E.g.:

1

1

10101 + 11001

101110

21 + 25

46


Number systems

Binary Subtraction

  • Binary subtraction between 2 1-bit values:


Binary subtraction
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.

Subtract the numbers and negate the answer by putting the ‘-’ sign in front.

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