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# Numbering System - PowerPoint PPT Presentation

Numbering System. Base Conversion. Number systems. Decimal – 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 Binary – 0, 1 Octal – 0, 1, 2, 3, 4, 5, 6, 7 Hexadecimal system – 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F. Why different number systems?.

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### Numbering System

Base Conversion

• Decimal – 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

• Binary – 0, 1

• Octal – 0, 1, 2, 3, 4, 5, 6, 7

• Hexadecimal system – 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F

• Binary number result in quite a long string of 0s and 1s

• Easier for the computer to interpret input from the user

• In daily life, we use decimal (base 10) number system

• Computer can only read in 0 and 1

• Number system being used inside a computer is binary (base 2)

• Octal (base 8) and hexadecimal (base 16) are used in programming for convenience

• Conversion

• Binary number,

• Octal number,

• Hexadecimal number, and

• Decimal number.

For example:

62 = 111110 = 76 = 3E

decimal binary octal hexadecimal

1 For Decimal:

62 = 6x101 + 2x100

2 For Binary:

111110 = 1x25 + 1x24 + 1x23 + 1x22 + 1x21 + 0x20

3 For Octal:

76 = 7x81+ 6x80

3E = 3x161 + 14x160

• Since for hexadecimal system, each digit contains number from 1 to 15, thus we use A, B, C, D, E and F to represent 10, 11, 12, 13, 14 and 15.

• Binary to decimal

• X . 27 + X . 26+ X . 25+ X . 24 + X . 23+ X . 22+ X . 21 + X . 20

• Decimal to binary

• Keep dividing the number by two and keep track of the remainders.

• Arrange the remainders (0 or 1) from the least significant (right) to most significant (left) digits

• Binary to Octal (8 = 23)

• Every 3 binary digit equivalent to one octal digit

• Binary to Hexadecimal (16 = 24)

• Every 4 binary digit equivalent to one hexadecimal digit

• Octal to binary

• Every one octal digit equivalent to 3 binary digit

• Hexadecimal to binary

• Every one hexadecimal digit equivalent to 4 binary digits

• How to convert the decimal number to other number system

• e.g. convert 1810 in binary form

2 |18 ----0

2 |09 ----1

2 |04 ----0

2 |02 ----0

1

• 1810 = 100102

• e.g. convert 1810 in octal form

• Since for octal form, one digit is equal to 3 digits in binary number, we can change binary number to octal number easily.

e.g. 10010 = 010010

2 2

Thus, 100102 = 228

• e.g. convert 1810 in hexadecimal form

• Similarly, for hexadecimal form, one digit is equal to 4 digits in binary number.

e.g. 10010 = 00010010

1 2

Thus, 100102 = 1216

### Numbering System

• What is going on?

• 1 1 1 (carry)

• 3 7 5 8

• + 4 6 5 7

• 14 11 15

• 10 10 10 (subtract the base)

• 8 4 1 5

111

3758

+ 4657

8415

Rules.

• 0 + 0 = 0

• 0 + 1 = 1

• 1 + 0 = 1

• 1 + 1 = 2 = 102 = 0 with 1 to carry

• 1 + 1 + 1 = 3 = 112 = 1 with 1 to carry

• Verification

• 5510

• + 2810

• 8310

• 64 32 16 8 4 2 1

• 1 0 1 0 0 1 1

• = 64 + 16 + 2 +1

• = 8310

1 1 1 1

1 1 0 1 1 1

+ 0 1 1 1 0 0

2 3 2 2

- 2 2 2 2

1 0 1 0 0 1 1

ex Verification

1 0 0 1 1 1

+ 0 1 0 1 1 0 + ___

___________

128 64 32 16 8 4 2 1

=

=

1 1

6 4 3 78

+ 2 5 1 08

9 9

- 8 8 (subtract Base (8))

1 1 1 4 78

ex

3 5 3 68

+ 2 4 5 78

- (subtract Base (8))

1 1

7 C 3 916

+ 3 7 F 216

20 18 11

- 16 16 (subtract Base (16))

B 4 2 B16

8 A D 416

+ 5 D 616

- (subtract Base (16))

16

7 13 10

8 4 1 15

- 4 6 5 7

3 7 5 8

• How it was done?

• ( add the base 10 when borrowing)

• 1010

• 73010

• 8 41 5131015

• - 4 6 5 7

• 3 7 5 8

• Verification

• 8310

• - 2810

• 5510

• 64 32 16 8 4 2 1

• 1 1 0 1 1 1

• = 32 + 16 + + 4 + 2 +1

• = 5510

1 2 1

02 0 2 2

1 0 1 0 0 1 1

- 0 1 1 1 0 0

1 1 0 1 1 1

ex Verification

1 0 0 1 1 1

- 0 1 0 1 1 0 - ___

___________

128 64 32 16 8 4 2 1

=

=

8

0 0 8

1 11 4 78

89

- 6 4 3 78

2 5 1 08

ex

3 5 3 68

- 2 4 5 78

B 16

7 C 3 916

19

- 3 7 F 216

4 4 4 716

8 A D 416

- 5 D 616

16