1.1: The Set of Natural Numbers

The first mathematics we learned as children was how to count: “One, two, three. . .” (or “Un, deux, trois. . .” or what have you). Only later did we learn about other kinds of numbers: fractions and negative numbers, for example. The counting numbers are thus the most “natural”, and the set \{1, 2, 3, \ldots\} is sometimes called the set of natural numbers. However, for certain technical reasons it’s convenient to include zero and so we define the set of natural numbers as {\Bbb N} = \{ 0, 1, 2, \ldots \}.

The set brackets “\{” and “\}” tell us to consider the collection of things between them as one object—namely a set. The symbol “{\Bbb N}” is just an abbreviation to allow us to refer to this object concisely. The elements of a set are the “things” in the collection. The symbol “\in” is pronounced “is an element of”, so for example 37 \in {\Bbb N} is an abbreviation for the sentence “Thirty-seven is an element of the set of natural numbers.”.

The ellipses ( … ) in the definition of {\Bbb N} tell us to continue the pattern displayed. The result is an infinite set—one with no last element. Ellipses are also sometimes used in defining finite sets, for example the familiar A = \{ a, b, c, \ldots , z \}.

The usual convention is to use capital letters to refer to sets. To read the definition {\Bbb N} = \{0, 1, 2, \ldots \} out loud, you could say, “{\Bbb N} is the set whose elements are zero, one, two, and so on forever.” The formal way to pronounce the symbol {\Bbb N} is “the set of natural numbers”. Usually, unless there is some need to be precise, one simply says “en”.

This can be trickier than it might appear because it’s common to use “n” as a variable. The convention here is that lower-case letters refer to numbers. So one can say, for example, “If n \in {\Bbb N}, then n must be either even or odd”. This should be pronounced “If en is a natural number, then …”.

The slash “/” is used to denote negation, as in 1 \not= 2. This idea is common even outside of mathematics, where we find “\oslash” (meaning “no”) on street signs, for example. So it shouldn’t come as any surprise that {1 \over 2} \not\in {\Bbb N} stands for the (true) sentence “One-half is not an element of the set of natural numbers.” (or just “one-half isn’t in {\Bbb N}” or some such variation).

There are several other common ways to say “{1 \over 2} \not\in {\Bbb N}”: for example, “Two doesn’t go into one” or “One is not a multiple of two”. Here is another way: 2 \not | 1. This is pronounced “Two does not divide one”. In general, whenever we have natural numbers n and m we can say that n divides m, and write n \mid m, whenever {m \over n} \in {\Bbb N}. We can also say in this case that n is a factor of m, or that n is a divisor of m.

Thus, for example, 11 \mid 209, because {209 \over 11} = 19 and 19 \in {\Bbb N}. This can be rephrased slightly as “because 209 = 11 \cdot 19 and 19 is a natural number”. But 11 \not | 73 because there is no natural number j with 73 = 11 \cdot j. In other words, because {73 \over 11} \not\in {\Bbb N}.

Two special cases involve zero. Suppose we have n \in {\Bbb N} and n \not= 0. Then 0 \not | n and n | 0. To confirm these facts, look at the definition of “|”: since {0 \over n} = 0 and 0 \in {\Bbb N}, we get n|0 as advertised. But {n \over 0} \not\in {\Bbb N}. In fact {n \over 0} isn’t a number at all! Be careful when working with 0 to avoid thinking of the wrong special case.

In the original version [~1995] of this piece, several exercises follow (of course!). As of this moment, they’re too much trouble to reproduce here. Exercise: find an expert user of the various notations and discuss.

  1. Hypatia

    I hesitate to write because I’m delighted that you are posting something other than links. I’ve always enjoyed your posts and your writing. And also because I’ve not consulted an expert user – I’m certainly not one. However, I would like to make a comment and let the expert users comment.

    N = {1, 2, 3, …} does represent the natural numbers (sometimes called ‘counting’ numbers’ for reasons probably referring to your intro. However, I always thought that when zero was added to the set it had a new name, the whole numbers — W = {0, 1, 2,…} Perhaps this wasn’t always so. But it makes a great deal of sense. Zero is not a counting number. Young children, and perhaps their teachers, when counting objects, use ‘0’ to refer to no objects. This later leads to a very common beginning algebra error when solving and x = 0. Many students will express the solution set as the ‘null’ set instead of {0}.

  2. Hypatia

    Here is a possible explanation of my puzzlement. Obviously I should have checked before writing!

    From Wikipedia, the free encyclopedia

    In mathematics, a natural number (also called counting number) can mean either an element of the set {1, 2, 3, …} (the positive integers) or an element of the set {0, 1, 2, 3, …} (the non-negative integers). The former is generally used in number theory, while the latter is preferred in mathematical logic, set theory, and computer science.

  3. thanks for delurking.
    “hypatia” is a very cool name!

    everybody in the department i came up in
    seems to’ve called {1, 2, 3 …}
    “the positive integers” (& quite a few of ’em
    abbreviated it {\Bbb Z}^+);
    i always sort of figured this whole
    “whole number” thing was the invention
    of misguided pedagogues (like “ln”
    instead of “log” for the natural logarithm).

    can’t agree with w’edia at all–
    “counting numbers” refers to \Bbb Z^+
    *regardless* of how one defines \Bbb N!
    (computer-heads are often said to begin
    counting with zero but this is mostly in jest.)

  4. your comment about the solution set
    for x=0 is also very interesting.

    one day during a vent-the-pet-peeve
    session with a colleague (for instance,
    he’s strongly anti-“FOIL” [cf jonathan];
    i can take it or leave it).
    then he says something to the effect
    “they keep saying ‘no solution’
    whenever the solution is the empty set’.
    but waitaminute, mister anonymous!
    those students are righter than you are!
    “solution” is a technical term
    (a value of the variable that makes
    a given equation *true* is called
    a *solution* to the equation);
    there’s an *answer* to the problem
    (namely “no solution”) but there sure
    as hell isn’t any solution to the equation
    (x = x + 1, say, for concreteness)!
    indeed, that’s what it *means* to say
    the *solution set* (for pity sake) is empty!
    (let’s try to get with the program here.)
    i was of course much more polite about
    correcting him than this imaginary dialogue
    would make it appear … but he just blew me off
    with “i disagree” (or some such) as if this
    were merely a matter of opinion. and i let it go.
    if i’d’ve loved the guy like i should
    i would’ve risked ticking him off for life
    by arguing about it. but to heck with it.

  5. Hypatia

    You’re lucky it was just one colleague. We have whole departmental meetings of ‘vent the pet peeve’ . So as not to offend, everyone’s pet peeves are adopted as ‘official department policy” after two or three hours of discussion. When the adjunct faculty pool alters significantly, we start all over again.

  6. My question is that whats “about base 10 for natural numbers”?

  7. i don’t think i understand your question.
    maybe you should point google (or your
    favorite search engine) at “number bases

  8. Anonymous


  1. 1 cut & paste | the livingston review

    […] july07/16/08Section 1.1: Natural Numbers (1995) much my most-read post07/17/08 Section 1.2: Modular Equivalences (1995) more from the same lecture notes […]

  2. 2 Vlorbik On Math Ed | the livingston review

    […] 07/16/08Section 1.1: Natural Numbers (1995) much my most-read post 07/17/08 Section 1.2: Modular Equivalences (1995) more from the same […]

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