Mathematics.

numbers

Real Numbers

Foundations30 minDifficulty3 out of 10

You should know: natural numbers

Overview

A real number is any number that measures a continuous quantity — a length, a duration, a temperature. The real numbers include every rational number (fractions like 3/4) and every irrational number (like √2 and π), and together they fill the number line completely, with no gaps. Formally, the reals are the unique complete ordered field: they obey the ordinary rules of arithmetic (a field), come in a consistent order (ordered), and have no missing points (complete).

Intuition

Picture an infinite ruler. The whole numbers are evenly spaced tick marks; the fractions squeeze in between them, densely. But even the fractions leave microscopic holes — √2 is not a fraction, yet it is clearly a real length (the diagonal of a unit square). The real numbers are what you get when you fill in every last hole, so that every point on the line is a number and every number is a point. That 'no gaps' property (completeness) is exactly what makes calculus possible.

Formal Definition

Definition

The real numbers form a complete ordered field: a set with addition and multiplication satisfying the field axioms (associativity, commutativity, distributivity, identities, inverses), a compatible total order, and the completeness axiom — every non-empty set bounded above has a least upper bound.

(a+b)+c=a+(b+c)(a + b) + c = a + (b + c)

Grouping doesn't affect a sum

Associativity of +
(ab)c=a(bc)(ab)c = a(bc)

Grouping doesn't affect a product

Associativity of ×
a(b+c)=ab+aca(b + c) = ab + ac

Multiplication distributes over addition

Distributivity

Notation

NotationMeaning
R\mathbb{R}The set of all real numbers
Q\mathbb{Q}The rational numbers (fractions) — a subset of ℝ
R∖Q\mathbb{R}\setminus\mathbb{Q}The irrational numbers
[a,b][a,b]The closed interval of reals from a to b inclusive

Properties

Commutativity of addition

a+b=b+aa + b = b + a

Commutativity of multiplication

ab=baab = ba

Distributivity

a(b+c)=ab+aca(b + c) = ab + ac

Additive identity

a+0=aa + 0 = a

Multiplicative identity

a⋅1=aa \cdot 1 = a

Applications

Every physical measurement — position, time, mass, charge — is modeled as a real number, because real quantities vary continuously.

Worked Examples

  1. √9 = 3, a whole number, hence rational. √2 has no exact fraction representation — its decimal never repeats — so it is irrational.

    9=3∈Q,2∉Q\sqrt{9} = 3 \in \mathbb{Q}, \quad \sqrt{2} \notin \mathbb{Q}

Answer: √9 is rational (= 3); √2 is irrational.

Practice Problems

Difficulty 2/10

Which of these is irrational?

Difficulty 3/10

Is the sum of two irrational numbers always irrational? Give an example.

Common Mistakes

Common Mistake

Believing every number with a decimal point is 'not a whole number' and therefore irrational.

Irrational means 'not a ratio of integers.' 0.5 = 1/2 is rational despite the decimal. Irrationality is about non-repeating, non-terminating expansions, like π or √2.

Common Mistake

Thinking ℝ has a 'next' number after each value.

The reals are dense and complete — between any two reals there are infinitely many more. There is no immediate successor, unlike the integers.

Quiz

Which property does a(b+c) = ab + ac express?
What makes ℝ 'complete'?

Flashcards

1 / 3

Summary

  • The real numbers ℝ measure continuous quantities and fill the number line with no gaps.
  • ℝ is the unique complete ordered field: field axioms + total order + least-upper-bound property.
  • ℝ contains the rationals (fractions) and the irrationals (√2, π, e).
  • Completeness is what makes limits, continuity, and calculus work.

References

  1. BookRudin, W. Principles of Mathematical Analysis, 3rd ed. Ch. 1.
  2. WebsiteWikipedia — Real number