1. Numeral systems (radix)

2. Commonly used systems Base / radix Name Symbols 2 Binary 0, 1 8 Octal
0, 1, 2, 3, 4, 5, 6, 7 10
Decimal
0, 1, 2, 3, 4, 5, 6, 7, 8, 9 16
Hexadecimal
0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F
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3. Binary (base 2) Positional number system
Used mainly in digital circuitry
Each digit is usually referred to as a bit
4 bits is a nibble, 8 bits is a byte
Native format for most of the computers these days
Typically written with either b suffix (00101b) or with base indication in subscript (01012)
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4. Conversion (1)
77 =
Divide the decimal number by the radix of the desired system
Repeat until the division result is less than 1 (integer division results to 0)
Division remainders form the new number
Start from the last remainder
division
result
remainder
77 / 2 38 1
38 / 2 19
19 / 2 9
9 / 2 4
4 / 2 2
2 / 2
1 / 2
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5. Conversion (2) 77 = 4D16 21980 = 55DC16 division result remainder
in hex
77 / 16 4 13 D
4 / 16
division
result
remainder
in hex
21980 / 16
1373 12 C
1373 / 16 85 13 D
85 / 16 5
5 / 5
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6. Conversion (3)
Positional number system – each position has a different value. In the case of binary, it’s the powers of two.
= 0*25 + 1*24 + 0*23 + 1*22 +1*21 +0*20 = 22
02178 = 0*83 + 2*82+ 1*81 + 7*80 = 143
24A16 = 2* * *160 = 586
Easy to convert between binary and hexadecimal
C B F
= 26 CB 41 F616
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7. Base 10
Base 2
Base 8
Base 16 00 1 01 2 02 3 03 4 04 5 05 6 06 7 07 8 10 08 9 11 09 12 0A 13 0B 15 17 0F 16 20 21 18 22