Addition and Multiplication of Real Numbers

The rationals have holes. Plot $\dfrac{1}{2}, \dfrac{2}{3}, \dfrac{3}{4}, \ldots$ on the number line: you build up density, but no rational equals $\sqrt{2}$. There IS a point exactly $\sqrt{2}$ steps from zero — you can prove it geometrically with a unit square's diagonal — but that point is invisible to $\mathbb{Q}$. The real numbers $\mathbb{R}$ are the smallest extension that fills every such hole.

The four laws keep working

The real numbers satisfy all the rules we built up for integers and rationals:

Addition: commutative, associative, identity $0$, every real has an additive inverse $-a$.

Multiplication: commutative, associative, identity $1$, every nonzero real has a multiplicative inverse $a^{-1}$.

Distributive: $a(b + c) = ab + ac$ ties the two operations together.

A set with these properties is called a field. $\mathbb{R}$ is a field, so is $\mathbb{Q}$. The point of moving from $\mathbb{Q}$ to $\mathbb{R}$ is the extra property the rationals do not have.

Irrational numbers: the new arrivals

An irrational number is a real number that is NOT a ratio of two integers. The classic example is $\sqrt{2}$. The Pythagoreans proved this by contradiction around 500 BCE: suppose $\sqrt{2} = \dfrac{a}{b}$ in lowest terms. Then $2b^2 = a^2$, so $a^2$ is even, so $a$ is even, so $a = 2k$. Substitute: $2b^2 = 4k^2$, so $b^2 = 2k^2$, so $b$ is also even — contradicting "lowest terms." Other famous irrationals: $\pi, e, \sqrt[3]{2}, \log_2 3$.

(On the number line, the rationals are dense — arbitrarily close to every point. Irrationals slip in between them. You cannot see the holes by eye, but they are there.)

Completeness: the property that defines $\mathbb{R}$

Here is what makes $\mathbb{R}$ different from $\mathbb{Q}$: every nonempty set of real numbers that is bounded above has a least upper bound (a supremum) in $\mathbb{R}$. The set of rationals whose square is less than $2$ is bounded above by $2$ but has no least upper bound inside $\mathbb{Q}$ — the least upper bound IS $\sqrt{2}$, which is missing. In $\mathbb{R}$, the least upper bound always exists.

The completeness property is what makes calculus possible. Without it, you cannot prove that a continuous function on a closed interval attains its maximum, that limits of bounded monotone sequences exist, or that the intermediate value theorem holds.

Try it

• Show that $\sqrt{3}$ is irrational using the same parity-style argument as for $\sqrt{2}$. (Hint: assume $\sqrt{3} = a/b$ in lowest terms and derive $3 \mid a$ and $3 \mid b$.)
• Compute $\sqrt{2} \cdot \sqrt{8}$. The result is rational — surprising! What does this tell you about "closure" of the irrationals under multiplication?
• Find a rational number strictly between $\dfrac{1}{3}$ and $\dfrac{1}{2}$. (Hint: take the average.)
• Is $\pi + (-\pi)$ rational or irrational? Why does that not contradict "rational + irrational = irrational"?

A trap to watch for

Beginners often write $\sqrt{a + b} = \sqrt{a} + \sqrt{b}$. That is false. Try $a = 9, b = 16$: the left side is $\sqrt{25} = 5$, the right side is $3 + 4 = 7$. Square roots distribute over multiplication, not addition: $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$ for $a, b \geq 0$, but never $\sqrt{a + b}$. The fix: any time you write a radical, ask "am I splitting across a product or a sum?" If a sum, you cannot split.

What you now know

You can recognise the field axioms in action on $\mathbb{R}$, prove specific square roots irrational, and articulate why completeness is the property that makes the reals the "right" setting for analysis. The next section adds the ingredient that makes algebra geometric: positivity, the order relation that turns $\mathbb{R}$ into an ordered field.

Quick check

References

• Lang, S. (1971). Basic Mathematics. Springer. Chapter 3, §1 — the field axioms for $\mathbb{R}$.
• Spivak, M. (2008). Calculus, 4th ed. Publish or Perish. Chapter 1 — the field, order, and completeness axioms as the entire foundation of analysis.
• Rudin, W. (1976). Principles of Mathematical Analysis, 3rd ed. McGraw-Hill. Chapter 1 — $\mathbb{R}$ as the unique complete ordered field, constructed from $\mathbb{Q}$ via Dedekind cuts.