Notes for Neil Zwillinger's Fifth Grade Class: Jean-Claude Chetrit 2/26/98
We have looked at the binary system (base 2).
0 1 10 11 100 101 110 111 1000 1001 1010
Today we are going to look at some other counting systems.
1) The Roman system:
I II III IV V VI VII VIII IX X XI XII
One disadvantage is that we need two new letters for every factor of 10:
1 and 5 I and V
10 and 50 X and L
100 and 500 C and D
1000 and 5000 M and ?.
Hence it is not possible to have very large numbers. Another problem is that there is no way to give a rule for addition or for multiplication. Try III times III. Doing arithmetic must have been real hard then. Yet one of the greatest mathematicians of all times, Archimedes, lived at that time and discovered many beautiful theorems. He was killed by the ruthless Romans who are also famous for burning down several times Alexandria, the city Alexander built with the greatest library of antiquity.
2) The unary system:
This starts the same way but just keeps going all the way.
l ll lll llll lllll llllll lllllll llllllll
It is the simplest system in the world. It has many advantages: nothing much to explain, addition is very simple (just concatenate) and so is multiplication (build a rectangle, then rearrange the l's. Prime numbers are best explained with this system. The disadvantage is that writing 1000 takes a long time. It is more an abstract concept used in certain branches of mathematics than a practical system.
3) The modern positional systems:
They include the binary, the trinary and the decimal system. They are a result of the invention of zero by the Arabs in the Middle Ages. The Arabs also gave us the words algebra and algorithm.
4) The trinary (or ternary) system (base 3):
0 1 2 10 11 12 20 21 22 100 101 102
Notice that
2 + 2 = 11
5) The decimal system (base 10):
1/3 = 0.3333...
where the "..." means go on forever (there are an infinite number of 3 after the period). Now if we multiply each side by 3 we get
1 = 0.9999...
which means that the number 1 has 2 different representations in the same counting system!!! This is why we need the concept of limit in math:
What is the difference between 1 and 0.99 ? 0.01
What is the difference between 1 and 0.9999 ? 0.0001
What is the difference between 1 and 0.9999… ? 0
Last updated on January 8, 1999
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