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Re: [HM] Counting from Zero
Posted:
Jul 26, 2005 9:12 AM


Dear Sam:
Your query brings to mind the conundrum about naming sets of numbers. In the preparation of elementary school teachers, we (!) instruct them first about the natural numbers {1, 2, 3, ...}, then move on to whole numbers {0, 1, 2, 3, ....}, and next the integers { 2, 1, 0, 1, 2, . . .}. Now the word INTEGER is from the Latin language and means WHOLE. Ergo, the whole numbers are integers?
Then of course, we have the counting numbers {1, 2, 3, . . .} (pace JHC). They are also called the NATURAL numbers, because that is the way we count. Now when we count, there are objects in front of us. (If there are no objects, then we don't count.) Suppose there are a bell, book, and candle on a able and we are asked to count them. So, pointing to the bell, we say "One," to the book "Two", and to the candle "Three". Why is the book TWO? or the candle THREE?
This example I have used many times with future elementary school teachers to get across to them the Greek concept of number, a collection of units. Each of the above three is ONE. When I add the second one to the first one, I have created a collection and its number is two; etc. This is the natural way of counting. (Hence, the identity: natural numbers = counting numbers) For the first time, many of the students understood what it meant to count.
Years ago I read a statement by by a mathematician (Edward Kastner??), "In teaching mathematics, it is not necessary to tell the whole truth. But what you teach must be truthful." Bertrand Russel contributed greatly to the development of number theory. It is best left to theoreticians and not exposed to children.
Best wishes!
Barney Hughes



