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