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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

Numbering System

Base Conversion

number systems
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
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
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 conversion1
Base Conversion
  • Conversion
    • Binary number,
    • Octal number,
    • Hexadecimal number, and
    • Decimal number.
base conversion3
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 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
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 conversion4
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
base conversion5
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 conversion6
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

numbering system1

Numbering System

Addition & Subtraction

decimal addition
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
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 addition1
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 addition2
Binary Addition

ex Verification

1 0 0 1 1 1

+ 0 1 0 1 1 0 + ___

___________

128 64 32 16 8 4 2 1

=

=

octal addition
Octal Addition

1 1

6 4 3 78

+ 2 5 1 08

9 9

- 8 8 (subtract Base (8))

1 1 1 4 78

octal addition1
Octal Addition

ex

3 5 3 68

+ 2 4 5 78

- (subtract Base (8))

hexadecimal addition
Hexadecimal Addition

1 1

7 C 3 916

+ 3 7 F 216

20 18 11

- 16 16 (subtract Base (16))

B 4 2 B16

hexadecimal addition1
Hexadecimal Addition

8 A D 416

+ 5 D 616

- (subtract Base (16))

16

decimal subtraction
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
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 subtraction1
Binary Subtraction

ex Verification

1 0 0 1 1 1

- 0 1 0 1 1 0 - ___

___________

128 64 32 16 8 4 2 1

=

=

octal subtraction
Octal Subtraction

8

0 0 8

1 11 4 78

89

- 6 4 3 78

2 5 1 08

octal subtraction1
Octal Subtraction

ex

3 5 3 68

- 2 4 5 78

hexadecimal subtraction
Hexadecimal Subtraction

B 16

7 C 3 916

19

- 3 7 F 216

4 4 4 716

hexadecimal subtraction1
Hexadecimal Subtraction

8 A D 416

- 5 D 616

16

let s do some exercises

Let’s do some exercises!

Octal, Hexadecimal, Binary

Addition & Subtraction