Our number system

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Website owner:  James Miller

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Our number system

What is this number system that we use everyday which was invented in ancient India? What are its basic assumptions and characteristics? Ancient peoples such as the ancient Egyptians and Sumerians developed number systems in which they were able to perform arithmetic operations but they were complicated, difficult to use, hard to learn.

Our number system represents a major intellectual breakthrough. Fundamental to it was the invention of the idea of a zero, appending this zero to the set of integers, and treating it like another integer.

What the inventor of this number system gave us was a ingenious naming system for the integers which employed a new invented number that we call zero which allows one to rather easily add, subtract, multiply, and divide numbers.

The first and main task in creating a number system is to create some scheme for providing a name for all integers 1, 2, 3, 4, ... We then devise procedures for adding, subtracting, multiplying, and dividing integers in this system we have created.

Since there are an infinite number of integers it is obvious that to provide a unique name for each integer we must create a finite set of symbols and then create a name for every integer by some scheme utilizing that set. In the case of our number system we use the symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 (where 0 is an invented number representing “nothing” — an idea representing a major breakthrough in thought) and 1, 2, 3, ... , 9 are symbols for our usual concepts of the integers 1 through 9. From these ten symbols we are able to provide a unique name for any possible integer. How do we do this? Well, the first nine integers correspond to our symbols 1, 2, 3, ... , 9. Now we have to provide a name for the number that comes after 9. What might we name it? We give it the name 10 which is made up of the two symbols 1 and 0. Thus 9 + 1 = 10. Now we want to provide a name to the number that comes after 10. We give it the name 11. Thus 10 + 1 = 11. To arrive at this name we use the fact that 0 + 1 = 1 where we are utilizing the property of our invented 0 of meaning nothing. Then the name for the number that comes after 11 is given by 11 + 1 = 12. We proceed in this way until we get to 19. What number comes after 19? We give that number the name 20. To arrive at that name we add a 1 to 19 to get 19 +1 = 20 where we first add 1 to 9 to get 10 and carry the 1 to get 20. Thus what we are doing is to use our standard rule for adding one to an integer to provide names to all integers. After doing this we can then derive the usual rules of arithmetic for subtracting, multiplying, and dividing integers.

Note all of the cycling, all of the cycles, in the names we have created for naming the integers. We have repeated cycles and cycles within cycles — the names cycle on the 10's, then on the 100's, then on the 1000's, etc. It is a very ingenious way for naming the integers and it is all made possible by the 0 that we introduced into the system.

In our number system any n digit integer abc ...q can be written as

abc...q = a∙10^n-1 + b∙10^n-2 + c∙10^n-3 ... + q∙10⁰

Example.

5673 = 5∙10³ + 6∙10² + 7∙10¹ + 3∙10⁰

= 5∙1000 + 6∙100 + 7∙10 + 3∙1

= 5000 + 600 + 70 + 3

July 2020