(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 two
423 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
zero
2. Each position then has a value of the base to the power
of the number of that position
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
Example using base 3:
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
The value of the symbol in each position is multiplied by
the position value

Determining a value for a particular base and set of
symbols (cont):
◦
Base 10
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

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:
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:
7
2 = 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