(Actually, “Numeral Systems”). Unary ◦ Each item is represented by an instance of a symbol...

(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: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 value:104 = 10000











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





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

(Actually, “Numeral Systems”). Unary ◦ Each item is represented by an instance of a symbol...

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