ریاضی
What Is a Natural Number?
Natural numbers are the counting numbers 1, 2, 3, 4 and onwards. Whether zero belongs is a convention, not a fact — and here is which convention uses which.
Natural number
A natural number is one of the counting numbers — 1, 2, 3, 4 and onwards — though some conventions, including the standard ISO 80000-2, begin the list at 0.
- دیگر نام
- counting number
- طلبہ کہاں پڑھتے ہیں
- Grade 5 and 6 as the plain counting numbers, and again at IGCSE, where a question that says n is a natural number expects you to use the term precisely rather than loosely.
مختصر جواب
Natural numbers are the counting numbers: 1, 2, 3, 4 and onwards, with no negatives, fractions or decimals. Whether zero belongs is a matter of convention rather than fact. School textbooks in the UK usually start the list at 1, while ISO 80000-2 and university set theory start it at 0.
ایک مثال
ℕ = {1, 2, 3, 4, …} or ℕ = {0, 1, 2, 3, 4, …}
Both lines appear in print, and neither is a mistake. A textbook that starts at 1 can use natural number and positive integer interchangeably; one that starts at 0 cannot, and writes ℤ⁺ or ℕ* when it needs the positives alone. Check which convention your book uses once, at the start of the course, and you will not have to think about it again.
The first set of numbers, and the reason there are others
Natural numbers are what counting produces: three sheep, seventeen marks, one demo class. Nothing negative, nothing fractional, nothing with a decimal point. It is the smallest collection of numbers a student meets, and everything that comes afterwards exists because of something the naturals could not do.
That is worth spelling out, because it turns a list of definitions into a sequence with a reason. 3 − 5 has no natural answer, so the integers were needed. 3 ÷ 5 has no integer answer, so the rationals were needed. The side of a square of area 2 is not rational, so the irrationals were needed. Each set is the previous one plus whatever had been missing.
Does zero count? Two conventions, both in print
This is the disagreement students run into within a week of meeting the term, usually because a teacher and a website say different things. Neither of them is wrong, and neither is being careless.
Whether to include zero is a labelling decision taken before any mathematics happens, and it changes no arithmetic whatever. The two conventions in circulation are these:
- Starting at 1 — the usual convention in UK school textbooks. ℕ = {1, 2, 3, …}, and natural number and positive integer then mean the same thing.
- Starting at 0 — the convention of ISO 80000-2, of set theory and of most computer science. ℕ = {0, 1, 2, 3, …}, and the positive ones are written ℤ⁺ or ℕ* when they are needed separately.
When the disagreement actually bites
Almost never, in an exam. Questions are written so that the answer does not turn on the convention, and if one ever did, the question would be at fault rather than the candidate. The safe move is to say which reading you took: write n ≥ 1 in your working and the ambiguity disappears.
It does bite in three places outside the exam hall. Asked for the smallest natural number, you have to know which list you are on. Sequences may be indexed from n = 1 or from n = 0, and the formulae differ accordingly. And in programming, where array positions start at 0, the 0-inclusive convention is the only one that makes the code read sensibly.
عام سوالات
Is 0 a natural number?
Under the convention used in most UK school textbooks, no — the list starts at 1. Under ISO 80000-2 and standard set theory, yes. Both are in use in print. If a question appears to depend on it, state the convention you are using in your working; a well-set question will not depend on it.
Is a natural number the same as a positive integer?
They match exactly if your textbook starts the naturals at 1. If it starts at 0 they do not, because 0 is not positive. This is the clearest practical consequence of the disagreement, and it is why some books avoid natural number altogether and write positive integer, which has no ambiguity at all.
Are negative numbers or fractions natural numbers?
No. Counting cannot produce −3 or 5/8, and the natural numbers are exactly what counting produces. −3 is an integer, 5/8 is a rational number, and both are perfectly ordinary numbers — they simply sit in larger sets than this one.
Is 7.0 a natural number?
Yes, because 7.0 is 7. The decimal point changes how the number is written and nothing else. The same reasoning makes 14/2 a natural number: work the value out, then classify it, rather than judging by the way it appears on the page.
حوالہ جات
- ISO 80000-2:2019 Quantities and units — Part 2: Mathematics — International Organization for Standardization
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