(Actually, “Numeral Systems”)
Unary◦ Each item is represented by an instance of a symbol
Example: 7 might be |||||||◦ Also called “tally”
Sign-value notation◦ An abbreviated form of Unary◦ Extra symbols replace groups of Unary symbols
Example: + might represent 5 unary | symbols, and * might represent 10 unary | symbols, so 68 could be represented at ******+|||
◦ In both Unary and Sign-value notation, 0 isn’t used
Roman numerals are a type of sign-value notation◦ I is 1, V is 5, X is 10, etc.◦ Added the concept of subtracting a smaller
number from a larger one, if the smaller symbol was placed in front of the larger one: IX is 9, a shorter way of writing VIIII
◦ Very difficult to calculate anything other than small values and simple calculations
◦ Fractions are difficult to represent and calculate
Two developments by Indian mathematicians led to our current number system
In the 5th century: place-value notation◦ Placement of a symbol gave it added meaning
In the 6th century: the concept of zero
Relatively small set of symbols used The placement of each symbol adds
additional meaning◦ Examples:
342 means three hundred forty two423 means four hundred twenty three
◦ In a sign-value notation, each of these would add up to 9, the sum of the value of each symbol
◦ The value of placement makes a big difference
The value of each position depends on the base used
The system needs an ordered set of symbols ◦ There must be as many symbols as the base◦ One of the symbols must be zero◦ Example:
A base three system might use the symbols 0, 1, 2 Counting: 0, 1, 2, 10, 11, 12, 20, 21, 22, 100, 101,
102, 110, 111, 112, 120, 121, 122, 200, …
The method of determining a value for a particular base and set of symbols is:1. Number the positions from right to left, starting with
zero2. Each position then has a value of the base to the
power of the number of that position
Example using base 3:
The value of the symbol in each position is multiplied by the position value
Position 4 Position 3 Position 2 Position 1 Position 0
Position value:base4
Position value:base3
Position value:base2
Position value:base1
Position value:base0
Position 4 Position 3 Position 2 Position 1 Position 0
Position value:34
Position value:33
Position value:32
Position value:31
Position value:30
Determining a value for a particular base and set of symbols (cont):◦ Base 10
To convert from another base to base 10, calculate position value, multiply position value times symbol value, and add them all together
Example: converting 12021 in base 3 to decimal
Position 4 Position 3 Position 2 Position 1 Position 0
Position value:104 = 10000
Position value:103 = 1000
Position value:102 = 100
Position value:101 = 10
Position value:100 = 1
Position 4 Position 3 Position 2 Position 1 Position 0
Position value:34 = 81
Position value:33 = 27
Position value:32 = 9
Position value:31 = 3
Position value:30 = 1
81 * 1 = 81 27 * 2 = 54 9 * 0 = 0 3 * 2 = 6 1 * 1 = 1
12021 in base 3 = 81 + 54 + 0 + 6 + 1 = 142 in base 10
Base 2 used in computers because of the easy conversion of electrical switch state on/off to 1 and 0
Early attempts to use base 10 not successful◦ Difficult to judge graduations in power from 0 to 9
(none to all)◦ Easier to judge on/off state, even with noise in the
measurement◦ Base 10 might be more successful now with
advanced tools, but binary is solidly established
Translation from binary (base 2) to decimal (base 10)
Example: 10011101
Position 7 Position 6 Position 5 Position 4 Position 3 Position 2 Position 1 Position 0
Position value:
27 = 128
Position value:
26 = 64
Position value:
25 = 32
Position value:
24 = 16
Position value:23 = 8
Position value:22 = 4
Position value:21 = 2
Position value:20 = 1
128 * 1 = 128
64 * 0 = 0 32 * 0 = 0 16 * 1 = 16
8 * 1 = 8 4 * 1 = 4 2 * 0 = 0 1 * 1 = 1
10011101 binary = 128 + 0 + 0 + 16 + 8 + 4 + 0 + 1 = 157 decimal
In computers, a binary number can represent◦ Data
Number Character Sound Color
◦ Program instruction◦ Memory address◦ Screen location (pixel)◦ A computer (IP address)◦ etc
Hexadecimal means 16; hexadecimal number system (hex) is base 16
First four positions in binary can represent 16 digits (0 – 15)
Hex often used in place of binary for humans◦ A single hex digit can replace 4 binary digits◦ Easier to see/read/remember hex than binary
Because base 16 system needs 16 symbols, the letters A-F are used in addition to 0-9: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F
Binary Hex Decimal
0000 0 0
0001 1 1
0010 2 2
0011 3 3
0100 4 4
0101 5 5
0110 6 6
0111 7 7
1000 8 8
1001 9 9
1010 A 10
1011 B 11
1100 C 12
1101 D 13
1110 E 14
1111 F 15
Some sequences of binary digits are represented as hex digits for convenience◦ MAC: 00-24-2B-08-C7-4A; 00-1E-EC-DA-93-51◦ Memory addresses
Often hex numbers have special characters added to make sure they are understood as hex◦ Followed by a lowercase h◦ Preceded by 0x (the number zero and lowercase x)
Other sequences of binary digits are represented as decimal digits◦ IP addresses: 127.0.0.1
In the past, base 8 (octal) numbering system was sometimes used
It could easily represent three binary digits (23 = 8)
Rarely used now
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