Numbering System

Base Conversion

Number systems

Decimal – 0, 1, 2, 3, 4, 5, 6, 7, 8, 9Binary – 0, 1Octal – 0, 1, 2, 3, 4, 5, 6, 7Hexadecimal 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 1Number 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

ConversionBinary number, Octal number, Hexadecimal number, and Decimal number.

Base ConversionBinary Octal Hexadecimal Decimal

0000 0 0 0

0001 1 1 1

0010 2 2 2

0011 3 3 3

0100 4 4 4

0101 5 5 5

0110 6 6 6

0111 7 7 7

1000 10 8 8

1001 11 9 9

1010 12 A 10

1011 13 B 11

1100 14 C 12

1101 15 D 13

1110 16 E 14

1111 17 F 15

Base Conversion

For example:62 = 111110 = 76 = 3Edecimal 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 binaryKeep 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 systeme.g. convert 1810 in binary form

2 |18 ----0

2 |09 ----1

2 |04 ----0

2 |02 ----0

1

1810 = 100102

Base Conversion

e.g. convert 1810 in octal formSince 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 = 010 010

2 2

Thus, 100102 = 228

Base Conversion

e.g. convert 1810 in hexadecimal formSimilarly, for hexadecimal form, one digit is

equal to 4 digits in binary number.e.g. 10010 = 0001 0010

1 2

Thus, 100102 = 1216

Numbering System

Addition & Subtraction

Decimal Addition

111

3758

+ 4657

8415

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

Binary Addition

Rules. 0 + 0 = 00 + 1 = 11 + 0 = 11 + 1 = 2 = 102 = 0 with 1 to carry

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

Binary Addition

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

Verification

5510

+ 2810

8310

64 32 16 8 4 2 1 1 0 1 0 0 1 1

= 64 + 16 + 2 +1

= 8310

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

1 1

6 4 3 78 + 2 5 1 08 9 9 - 8 8 (subtract Base (8)) 1 1 1 4 78

Octal Addition

ex

3 5 3 68 + 2 4 5 78

- (subtract Base (8))

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 Addition

8 A D 416

+ 5 D 616 - (subtract Base (16))

16

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)

10 10 7 3 0 10

8 4 1 5 13 10 15

- 4 6 5 7

3 7 5 8

Binary Subtraction

1 2 1

0 2 0 2 2 1 0 1 0 0 1 1 - 0 1 1 1 0 0 1 1 0 1 1 1

Verification

8310

- 2810

5510

64 32 16 8 4 2 1

1 1 0 1 1 1

= 32 + 16 + + 4 + 2 +1

= 5510

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

8 0 0 8

1 1 1 4 78

8 9

- 6 4 3 78

2 5 1 08

Octal Subtraction

ex

3 5 3 68

- 2 4 5 78

Hexadecimal Subtraction

B 16

7 C 3 916 19

- 3 7 F 216

4 4 4 716

Hexadecimal Subtraction

8 A D 416

- 5 D 616

16

Let’s do some exercises!

Octal, Hexadecimal, Binary

Addition & Subtraction