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## PowerPoint Slideshow about ' Numbering System' - alan-brooks

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### Numbering System

### 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?

- Binary number result in quite a long string of 0s and 1s
- Easier for the computer to interpret input from the user

Base Conversion

- 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

Base Conversion

- Conversion
- Binary number,
- Octal number,
- Hexadecimal number, and
- Decimal number.

Base Conversion

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

4 For Hexadecimal:

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 and decimal system

- 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

Octal and Hexadecimal system

- 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

Base Conversion

- 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 binary form

Base Conversion

- 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

Base Conversion

- 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

Addition & Subtraction

Decimal Addition

- 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

Binary Addition

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

Binary Addition

- 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

Binary Addition

ex Verification

1 0 0 1 1 1

+ 0 1 0 1 1 0 + ___

___________

128 64 32 16 8 4 2 1

=

=

Decimal Subtraction

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

Binary Subtraction

- 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

Binary Subtraction

ex Verification

1 0 0 1 1 1

- 0 1 0 1 1 0 - ___

___________

128 64 32 16 8 4 2 1

=

=

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