Number Sets: A Definition of Natural, Integer, Rational, and Real Numbers

What are number sets

OpenStax’s College Algebra builds up the number sets one layer at a time: start with the natural numbers, add zero to get the whole numbers, add negatives to get the integers, add fractions to get the rational numbers, and finally add the irrationals to get the real numbers. Each set sits entirely inside the next one, the way a smaller box fits inside a bigger one.

Not every calculation works inside a small set. Subtraction becomes fully reliable once you reach the integers, division once you reach the rational numbers, and square roots — along with a genuinely continuous number line — once you reach the real numbers.

Natural numbers and whole numbers

The natural numbers are the ones everyone learns first: the numbers used to count objects, days, people. OpenStax’s College Algebra puts it plainly — “the numbers we use for counting… are the natural numbers: 1, 2, 3, 4, 5, and so on” — which leaves zero out of the set. Zero gets its own step instead, the whole numbers, defined as the natural numbers plus zero: {0, 1, 2, 3, …}.

That’s a different convention from the one used in Italy and in Giuseppe Peano’s original axioms, where zero counts as a natural number from the outset; a reader should check which convention a given textbook has adopted before comparing definitions across sources. The operations performed on these numbers, addition, subtraction, multiplication, sit at the core of arithmetic, the branch of math that studies them.

Integers: subtraction that always works

Inside the whole numbers, subtraction can get stuck: 3 minus 5 isn’t a whole number. The integers solve that by adding the negative counterparts of the whole numbers: {…, −3, −2, −1, 0, 1, 2, 3, …}. With the integers, subtraction always produces an answer, no matter which two numbers you start with.

An equation like x + 5 = 2 has no solution among the whole numbers, since it would require a negative number. That’s exactly the kind of wall that equations run into the moment they move past simple counting — and the practical reason the integers exist.

Rational numbers: division that always works

Even among the integers, one operation still gets stuck: division. 7 divided by 2 isn’t an integer. The rational numbers fix that by adding fractions: OpenStax defines the set as every number that can be written as m/n, with m and n integers and n never zero. With the rational numbers, division always produces an answer, except when the denominator is zero.

Rational numbers always have a decimal expansion that either terminates or repeats. OpenStax uses 15/8, which comes out to 1.875 (a terminating decimal), and 4/11, which comes out to 0.3636… with the digits “36” repeating forever (a repeating decimal). Writing the rule as m/n, with letters standing in for specific numbers, is the same trick algebra uses to state a rule that holds for infinitely many cases at once.

Example: The number −5 belongs to the integers but not to the whole numbers, because whole numbers don’t go negative. The number 3/4 belongs to the rational numbers but not to the integers, because it isn’t a whole number. The number √2, roughly 1.41421356…, belongs to the real numbers but not to the rational numbers: its decimal digits never settle into a repeating pattern, so it can’t be written as a fraction.

Irrational numbers and the real number system

The real number system, according to OpenStax, is the union of the rational numbers and the irrational numbers: every number that is either one or the other. Irrational numbers are the ones with a decimal expansion that neither terminates nor repeats — √2 and π are the two most familiar examples.

With the real numbers, square roots work for every positive number, and the number line becomes continuous, without the gaps that the rational numbers alone would leave behind. Studying what happens near a single point on that gapless line is the territory calculus works in.

The rational numbers are countable: according to Treccani’s Enciclopedia della Matematica, they have exactly as many elements as the natural numbers, despite being dense on the number line (there’s always another rational number between any two you pick). The real numbers, by contrast, are not countable.

A short history: from the Pythagoreans to today’s symbols

The discovery that some quantities can’t be expressed as a ratio — what mathematicians call incommensurability — dates back to the Pythagoreans, according to Treccani. But a long stretch separates that discovery from irrational numbers gaining full standing as numbers: Treccani places that acceptance in the seventeenth and eighteenth centuries, many centuries later still. The rigorous construction didn’t arrive until the nineteenth century: in 1872, Dedekind published Stetigkeit und irrationale Zahlen (“Continuity and Irrational Numbers”), where he used R for the rationals and blackletter R for the reals.

The other symbols have their own history too, traced by the MacTutor History of Mathematics Archive at the University of St Andrews. In a different 1888 work, Was ist und was sollen die Zahlen, Dedekind denoted the natural numbers with N; Peano adopted the same symbol the following year. Before Z and Q became standard, other mathematicians used their own notation: in the 1930s, Landau wrote the integers with a fraktur Z topped by a bar, while van der Waerden used C for the integers and the Greek letter Γ for the rationals. Today’s familiar form of Z and Q first appeared in 1947, in the first volume of Algèbre by the group of mathematicians known as Bourbaki.

A summary table

Set	Symbol	Example	What it adds
Natural	N	1, 7, 152	the starting point, the counting numbers
Whole	W	0, 1, 7	zero
Integers	Z	−5, 0, 12	negative numbers, subtraction that always works
Rational	Q	3/4, −2/5, 1.875	fractions, division that always works
Real	R	√2, π, 3/4	irrational numbers, roots, a continuous number line

Every row of the table sits entirely inside the rows below it: a natural number is also a whole number, an integer, a rational number, and a real number, while the reverse almost never holds. Knowing which set a number belongs to, before doing any arithmetic at all, is what tells you which operations are safe to use on it.