Background information:
In year 2013, I was a relief teacher at RI. There was a year 1 topic on ‘Real number system I’. During the teachers’ sharing session, Victor, the level coordinator, mentioned that
“Zero is NOT a natural number.”, as a reminder to teachers not to be confused on this point.
My response then was that “It depends on which mathematicians you ask.” This article is a discussion on issues relating to this question.
“Is Zero a natural number?” Why not?
Due to the late creation of the numeral, 0, Zero has always been excluded from the set of natural numbers. Often, the argument in supporting this view is that “natural numbers are counting numbers, numbers that use in the process of counting.” Since there is nothing to count when there is nothing, zero is not to be counted. Therefore, zero is not a counting number, thus, it does not belong to natural number.
However, more and more mathematicians define natural number inclusive of zero.
Dr Ng Wee Seng, NUS Math Dept, in his book “Introductory Mathematics” (second edition), on page 1, he wrote “
N the set of natural numbers = {0, 1, 2, 3 ….}. But, Dr Toh Tin Lam,
NTU, NIE, in his book “Calculus for Secondary School Teachers” (second edition), on page
8, he stated natural numbers are all positive integers. On a footnote below, Dr Toh mentioned “Some books include 0 in the set of Natural Numbers. However, in line with the
Cambridge O level syllabus, we shall not use this notation.” It appears that Dr Toh, has finally set the record straight that, with reference to Cambridge examination, 0 is NOT a natural number.
This appears to have given a definitive verdict that fully resolves this question, in relation to the teaching of mathematics in secondary schools. On the other hand, it has also been suggested that whether 0 is a natural number, depends on the branch of mathematics studied.
So, Secondary school mathematics will treat natural number as positive integers, everywhere else, it is NOT certain. This is problematic in relation to students who take examinations, such as AP, SAT or many mathematics competitions. In other words, teachers still need to know that taking examinations, other than Cambridge, one must check if 0 is defined as a natural number. Regarding mathematics competition, it becomes very difficult to decide which way to go.
This story seems to end here, but not really.
Natural Numbers has been removed from the Singapore version of O and A level examinations. In other words, Natural Numbers should not be taught in secondary school in
Singapore. If you need to use the natural numbers, excluding zero, then called it positive integers. If you need the zero and the positive integers, then you may use non-negative
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integers. By staying away from the “Natural Numbers” set, MOE Singapore, has avoided this issue. So, secondary school mathematics should NOT use the word – natural numbers when referring to the positive integers.
RI, which still using the “old” content, can either update itself to be more aligned with
MOE’s latest syllabus, or choose to continue with the usage of the word “natural numbers”, but need to make it clear to students that this set of numbers can both include zero or exclude zero! In any case, it is very MIS-LEADING to continue using “natural numbers” when referring to positive integers only.
In summary:
In this article, I have highlighted that “Natural Numbers” can include zero. Students and teachers need to be aware of this. For those who are taking examinations, it is best to refer to the syllabuses of the respective examination bodies for its definition on this term.
Schools in Singapore should dis-continue teaching this term. The issue, as mathematics continues to develop, will be clear in one day in the future. During this transition, we just have to be more aware of this issue.
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