Number Systems

In mathematical number systems, the radix or base is the total number of unique symbols used to represent numbers. In this context, the words radix and base are synonymous.

Unary – Base 1

Probably the simplest number system is the Unary system, consisting of only one single symbol (I), meaning 1, used to represent different numbers or quantities. Numbers are written like so:

I, II, III, IIII, IIIII, IIIIII, IIIIIII, IIIIIIII, IIIIIIIII, IIIIIIIIII, ad infinitum.

Roman Numerals

The Romans changed the Unary system a bit.  They still used “I” to represent the number 1, but added other unique symbols such as:

V for 5
X for 10
L for 50
C for 100 and
M for 1,000.

I, II, III, IV, V, VI, VII, VIII, IX, X, XI, XII, XIII, XIV, XVI, XVII, XVIII, XIX, XX, etc.

The idea was that each number can be its own distinct concept. But you can’t assign a unique symbol for every number, there are just too many numbers! So what the Romans did was use the position of symbols to create unique numbers:

IV means subtract 1 from 5.
VI means add 1 to 5.
IX means subtract 1 from 10.
XI means add 1 to 10.
XIV means subtract 1 from 5 and add it to 10.
XVI means add 1 to 5 and add it to 10.
MCM means subtract 100 from 1,000 and add it to 1,000.
LXXXXIV means 50 + 10 + 10 + 10 + (5 – 1) or 84.
MCMLXXXXIV means 1984.

Decimal – Base 10

The Base 10 system uses 10 unique symbols.

0, 1, 2, 3, 4, 5, 6, 7, 8, 9

The Hindu–Arabic number system is a positional number system invented between the 1st and 4th centuries by Indian mathematicians and adopted in Arabic mathematics by the 9th century. This is the system we use today. We use the position of symbols in a similar way the Romans did but we always add, never subtract. The Base 10 system was the first system to incorporate the symbol for zero (0). We use zero as an empty place holder.

A good way to demonstrate this concept is to use an odometer, that gauge in a car that records the number of miles you have driven. As you drive, the numbers in the right column will “fill up” to 9, then change to 0 and the second column begins to fill up. The symbols “carry over” into the next column.

The first column (red) represents tenths of a mile, the second column represents single miles, the third column 10s, the fourth column 100s, the fifth 1,000s, etc. This is the positioning of symbols by always adding and “carrying over” to the next column.

Binary – Base 2

This is the simplest number system that has the concept of “carrying over”. In the Unary system there was no carrying over, the one symbol went on forever:

IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII (50).

Roman numerals made things only slightly more efficient:

MCMLXXXIV (1984).

Writing that number in Unary might take a while!

The Binary system uses only two symbols: 0 and 1.

Decimal
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16

Binary
0
1 (1 column full)
10 (carry over)
11 (2 columns full)
100 (carry over)
101
110
111 (3 columns full)
1000
1001
1010
1011
1100
1101
1110
1111 (4 columns full)
10000

This is the system that is used by computers. Computers are electronic devices. Electricity either travels through wires and circuits (1) or it doesn’t flow (0). A circuit is either opened (1) or closed (0). ON = 1, OFF = 0.

As can be seen in the chart above, “one zero” (10) in Binary is the same as 2 in Decimal.

Thus, there are 10 types of people in this world, those that understand binary and those that don’t!

Other Number Systems

There are many other Number Systems. Here’s one you use every day.

Base 60

You may not have realized it but common time clocks use the Base 60 Number System, a total of 60 unique numbers. There are 3 columns separated by colons in this system.

5:49:30 = (5 x 60 x 60) + (49 x 60) + 30 = 20,970 seconds past midnight.

There are 86,400 seconds in a day. (24 x 60 x 60)