Real Numbers — Properties

Pick any two rational numbers — say 1 and 2. Is there a rational number between them? Of course: 1.5. Is there one between 1 and 1.5? Yes: 1.25. Between 1 and 1.25? Yes: 1.125. You can keep halving the gap forever, and every midpoint you land on is still rational. The rationals seem to fill the line completely.

And yet they don't. There is a point on the number line, sitting exactly at a distance of \sqrt{2} from the origin, and no rational number occupies that point. You proved this in Number Systems: if \sqrt{2} were rational, you could write it as p/q in lowest terms and derive a contradiction. So the rationals, despite being packed infinitely tightly, leave holes — infinitely many of them.

The real numbers are what you get when you fill every hole. The result is a number line with no gaps at all: every point on the geometric line corresponds to exactly one real number, and every real number corresponds to exactly one point. This article is about the properties that make that gap-free line work — why the rationals fail, what "no gaps" actually means in precise terms, and how the resulting structure supports everything from square roots to limits to calculus.

The Archimedean property: no infinitely large or infinitely small reals

Before worrying about gaps, start with a simpler question. Can a real number be "infinitely large"? Can a positive real number be so small that no matter how many times you add it to itself, you never reach 1?

The answer to both is no, and the reason has a name.

Take a = 0.001 and b = 5. Then n = 5001 works: 5001 \times 0.001 = 5.001 > 5. Take a = 10^{-100} and b = 10^{100}. Then n = 10^{201} works. The numbers can be absurdly far apart, but a finite number of copies of the smaller one will always overtake the larger one.

This rules out "infinitesimals" — positive numbers smaller than every fraction 1/n. If such a number \varepsilon existed, then n\varepsilon < 1 for every positive integer n, contradicting the Archimedean property. The real number line has no infinitely small positive elements and no infinitely large elements. Every real number, no matter how large, is exceeded by some integer; every positive real number, no matter how small, can be stacked to exceed any target.

The Archimedean property has an immediate corollary that matters for everything below. For any real number x, there is an integer n with n \le x < n + 1. That integer is the floor of x, written \lfloor x \rfloor. Every real number sits between two consecutive integers — there is no real number floating "above" all the integers or "between" two integers with no integer nearby. The integers are spaced evenly, and every real number is within distance 1 of some integer.

Density of the rationals

Now for the first density result. You might expect gaps between the rationals — after all, they skip over \sqrt{2}, \pi, e, and infinitely many other irrationals. But the gaps are not visible as empty stretches of the line.

The proof uses the Archimedean property directly. Since b - a > 0, the Archimedean property gives a positive integer n with n(b - a) > 1, which means nb - na > 1. When two numbers are more than 1 apart, there is an integer between them — call it m. Then na < m < nb, so a < m/n < b. The number m/n is rational, and it sits strictly between a and b.

This works for any a < b, no matter how close together they are. Between 3.14159 and 3.14160? There is a rational there. Between \sqrt{2} and \sqrt{2} + 10^{-100}? Still a rational there. You can shrink the interval as small as you like and a rational will always fit inside it.

Density of the irrationals

The mirror result is just as true and just as striking.

The proof is a clever reuse of the previous result. By density of rationals, there is a rational q with a < q < b. Now consider the number q + \frac{\sqrt{2}}{n} for large enough n — specifically, pick n large enough that q + \frac{\sqrt{2}}{n} < b (the Archimedean property guarantees such n exists). Since q is rational and \frac{\sqrt{2}}{n} is irrational, their sum is irrational. And this irrational number sits between a and b.

So between any two reals, there is always a rational and always an irrational. The two types of number are interleaved so finely that you cannot find any stretch of the line — however short — that contains only one type. They are woven together at every scale.

Completeness: the property that fills the gaps

Density says the rationals are everywhere. So why are they not enough? Because density is about individual points, not about limits of sequences. The rationals are dense, but they have holes that only show up when you try to take limits.

Here is the classic example. Consider the sequence of rational approximations to \sqrt{2}:

Every term in this sequence is rational. The terms get closer and closer together — they are "trying" to converge. But the number they are converging toward is \sqrt{2}, which is not in \mathbb{Q}. If you lived entirely inside the rationals, this sequence would have no limit. It would approach a hole and never arrive.

The real numbers are the number system where this cannot happen. The property that prevents it is called completeness, and there are several equivalent ways to state it. The most useful one for school mathematics is the least upper bound property.

To unpack this, you need two definitions.

An upper bound of a set S is any number M such that x \le M for every x in S. For example, 2 is an upper bound of the set \{1, 1.4, 1.41, 1.414, \ldots\}, because every term is at most 2. So is 1.5. So is 1.42. There are many upper bounds.

The least upper bound (or supremum, written \sup S) is the smallest upper bound — the tightest ceiling you can place over the set. It is the number L such that (i) L is an upper bound of S, and (ii) no number smaller than L is also an upper bound of S.

For the sequence above, the supremum of the set of all those rational approximations is \sqrt{2}. The completeness property says that this supremum exists as a real number. In the rationals it wouldn't — \sqrt{2} is not rational, so the set would have no least upper bound inside \mathbb{Q}. In \mathbb{R} it does.

Supremum and infimum

The least-upper-bound idea has a mirror image.

A lower bound of a set S is any number m with m \le x for every x in S. The greatest lower bound (or infimum, written \inf S) is the largest lower bound — the tightest floor you can place under the set.

Completeness also guarantees that every non-empty subset of \mathbb{R} that is bounded below has an infimum in \mathbb{R}. (This follows from the least upper bound property applied to the negated set.)

Here are some examples to build intuition.

The last row is the critical one. If you restrict yourself to \mathbb{Q}, this set has no least upper bound — \sqrt{2} is not rational. In \mathbb{R}, the supremum exists. This single example is the reason the real numbers were invented: to be a number system where the least upper bound property holds.

A key subtlety: the supremum of a set does not have to belong to the set. The supremum of (0, 1) is 1, but 1 is not in the open interval (0, 1). Similarly, \inf S does not have to belong to S. When \sup S does belong to S, it is also the maximum of S. When it does not, S has a supremum but no maximum — the set gets arbitrarily close to a ceiling it never touches.

Why completeness matters

You might wonder why this property deserves so much attention. Three reasons.

First, \sqrt{2} exists because of completeness. Consider the set S = \{x \in \mathbb{R} : x \ge 0 \text{ and } x^2 < 2\}. This set is non-empty (1 is in it) and bounded above (2 is an upper bound). By completeness, \sup S exists — call it \alpha. A careful argument shows \alpha^2 = 2: if \alpha^2 < 2, you could nudge \alpha up slightly and still have (\alpha + \varepsilon)^2 < 2, contradicting the supremum; if \alpha^2 > 2, you could pull \alpha down and still be an upper bound, contradicting leastness. So \alpha^2 = 2 exactly, and \alpha = \sqrt{2}. Completeness is what guarantees the square root exists as a real number.

Second, limits of convergent sequences exist because of completeness. A sequence that "should" converge — whose terms get closer and closer together (a Cauchy sequence) — always has a limit in \mathbb{R}. In \mathbb{Q} it doesn't: the sequence 1, 1.4, 1.41, \ldots is Cauchy but has no rational limit. Completeness closes that door.

Third, the intermediate value theorem — the most intuitive theorem in mathematics — needs completeness. If a continuous function goes from negative to positive, it must cross zero somewhere. That "somewhere" is a real number guaranteed to exist by completeness. On the rationals, the function f(x) = x^2 - 2 goes from f(1) = -1 to f(2) = 2 without ever equalling zero, because the zero would be at \sqrt{2} and that point is missing.

Two worked examples

Common confusions

• "The rationals have gaps because they are countable." Countability alone does not create gaps. The rationals are countable and dense — they fill every interval — yet they are incomplete. The issue is not how many rationals there are, but whether limits of rational sequences stay rational. Some don't, and those are the gaps.

• "Irrational numbers are rare — most numbers on the number line are rational." The opposite is true. Between any two rationals there are uncountably many irrationals. The irrationals vastly outnumber the rationals, even though the rationals are dense. (This is a consequence of the rationals being countable and the reals being uncountable — a topic you meet in Sets — Introduction.)

• "\sup S must be in S." Not necessarily. The supremum of the open interval (0, 1) is 1, which is not in (0, 1). When the supremum is in the set, it is called the maximum. Many important sets have a supremum but no maximum.

• "Completeness means the number line is infinitely long." No. Completeness is about vertical gaps (missing points), not horizontal extent. The line is indeed infinite in both directions, but that property is the Archimedean property, not completeness. Completeness says there are no holes — every bounded sequence that "should" converge actually does.

• "The Archimedean property is obvious and does not need to be stated." It feels obvious for the reals, but there are number systems where it fails. The hyperreal numbers, used in non-standard analysis, contain infinitesimals — positive numbers smaller than every 1/n. Those systems are deliberately non-Archimedean. Stating the property explicitly is what pins down the reals as the specific number system where it holds.

Going deeper

Dedekind cuts — building \mathbb{R} from \mathbb{Q}

The completeness property is not something you prove about the reals — it is part of how you define them. One clean construction, due to Dedekind, works like this: a Dedekind cut is a partition of \mathbb{Q} into two non-empty sets L and R such that every element of L is less than every element of R, and L has no greatest element. Each such cut corresponds to a real number. If R has a least element, that element is a rational and the cut corresponds to it. If R has no least element, the cut defines a new number — an irrational — that sits in the "gap" between L and R.

For example, the cut L = \{q \in \mathbb{Q} : q < 0 \text{ or } q^2 < 2\}, R = \{q \in \mathbb{Q} : q > 0 \text{ and } q^2 \ge 2\} defines \sqrt{2}. Neither side contains a boundary element, so the cut defines a genuinely new number that fills the gap between the two halves.

This construction guarantees completeness by design: if S is any bounded set of cuts, the "union of all the L-halves" of the cuts in S is itself a valid cut, and it is the least upper bound. The construction builds the gap-filling property into the foundations.

The uniqueness theorem

Any ordered field that satisfies the least upper bound property is isomorphic to \mathbb{R} — meaning it is the same number system, just possibly written in different notation. So the reals are the unique complete ordered field. This is a deep result: it means that completeness, together with the field axioms and the order axioms, pins down the entire structure of the number line with no wiggle room. Every property of the reals — the intermediate value theorem, the existence of limits, the convergence of bounded monotone sequences — follows from that single package.

Completeness and calculus

Every major theorem in single-variable calculus rests on completeness. The Bolzano–Weierstrass theorem (every bounded sequence has a convergent subsequence), the extreme value theorem (a continuous function on a closed interval achieves its maximum and minimum), and the intermediate value theorem all fail on \mathbb{Q}. They hold on \mathbb{R} because \mathbb{R} is complete. When you take a course in real analysis, the entire first chapter is dedicated to proving these theorems from the least upper bound property.

Where this leads next

The properties on this page — the Archimedean property, density, completeness, and the supremum/infimum machinery — are the foundation for everything that involves limits, continuity, and convergence.

• Number Systems — the full tower of number systems, from \mathbb{N} to \mathbb{R}, and how each layer was built to fix what the previous one lacked.
• Operations and Properties — the field axioms (closure, commutativity, associativity, distributivity, identity, inverse) that the reals satisfy alongside completeness.
• Intervals and Inequalities Preview — the notation for subsets of \mathbb{R} like (a, b) and [a, b], and the technique for solving inequalities on the real line.
• Sets — Introduction — the formal language of sets, subsets, and cardinality that makes precise statements like "the rationals are countable but the reals are uncountable."
• Roots and Radicals — where the completeness-guaranteed existence of \sqrt{2} becomes a working tool for computation.