Rational Numbers

Division has a problem in $\mathbb{Z}$. You can add, subtract, and multiply any two integers and stay inside the set, but $7 \div 2$ falls off the edge: there is no integer that solves it. The way out is exactly parallel to how the integers solved subtraction's problem — we invent the smallest number system that contains $\mathbb{Z}$ AND lets division (by anything nonzero) close up. That system is $\mathbb{Q}$.

The rationals, defined cleanly

A rational number is anything expressible as $\dfrac{a}{b}$ with $a, b \in \mathbb{Z}$ and $b \neq 0$. The set:

The letter $\mathbb{Q}$ stands for quotient. Every integer $n$ sits inside $\mathbb{Q}$ as $\dfrac{n}{1}$, so we did not lose anything — we strictly extended.

The same fraction wears many faces

The expressions $\dfrac{1}{2}$, $\dfrac{2}{4}$, $\dfrac{50}{100}$ all name the same rational number. The rule is: $\dfrac{a}{b} = \dfrac{ac}{bc}$ for any nonzero $c$. A fraction is in lowest terms when $\gcd(|a|, |b|) = 1$ — you have stripped every common factor.

(Set A to 2/4 and B to 1/2 in the widget. The bars look identical, because they ARE identical — same rational, two faces. Now set A to 3/6 and see another face of the same number.)

The four arithmetic operations

Addition. $\dfrac{a}{b} + \dfrac{c}{d} = \dfrac{ad + bc}{bd}$. Find a common denominator first, then add numerators.

Multiplication. $\dfrac{a}{b} \cdot \dfrac{c}{d} = \dfrac{ac}{bd}$. Numerators times numerators, denominators times denominators.

Division. $\dfrac{a}{b} \div \dfrac{c}{d} = \dfrac{a}{b} \cdot \dfrac{d}{c} = \dfrac{ad}{bc}$. Dividing by a fraction is multiplying by its reciprocal. (You will see why in the next section.)

Decimal expansions reveal the rationals

Every rational number has a decimal expansion that either terminates (like $\tfrac{1}{4} = 0.25$) or eventually repeats (like $\tfrac{1}{3} = 0.\overline{3}$ or $\tfrac{1}{7} = 0.\overline{142857}$). This is exactly the property that distinguishes rationals from irrationals — numbers like $\pi$ whose decimal never terminates and never settles into a pattern.

Try it

• Before touching the widget: between $A = 3/4$ and $B = 2/3$, which is bigger? Now set the sliders and confirm with cross-multiplication: compare $3 \times 3$ to $4 \times 2$.
• Compute $\dfrac{2}{3} + \dfrac{5}{4}$ using a common denominator. Predict the decimal value before checking.
• Find three different fraction faces of $\dfrac{3}{5}$ that all reduce to the same lowest-terms form.
• Convert $0.\overline{36}$ to a fraction. (Hint: let $x = 0.\overline{36}$, multiply by 100, subtract $x$.)

Pause: every integer is a rational number, but not every rational is an integer. Can you name three rationals that are NOT integers? Can you name a real number that is not rational?

Try it in code

A trap to watch for

The most common beginner error is writing $\dfrac{a}{b} + \dfrac{c}{d} = \dfrac{a + c}{b + d}$. This is wrong. Try $\dfrac{1}{2} + \dfrac{1}{2}$: the correct answer is $1$, but the buggy rule gives $\dfrac{2}{4} = \dfrac{1}{2}$. The reason fractions do not add "component-wise" is the same reason you cannot add "3 apples + 4 oranges" without converting to a common unit. The fix: always find a common denominator first.

What you now know

The rationals $\mathbb{Q}$ are the smallest extension of $\mathbb{Z}$ in which you can divide by any nonzero number. They share the four laws of the integers AND add a new one: every nonzero rational has a multiplicative inverse. That last property is so important it gets its own section. That is where we are headed next.

Quick check

References

• Lang, S. (1971). Basic Mathematics. Springer. Chapter 1, §5 — constructing $\mathbb{Q}$ from $\mathbb{Z}$ via equivalence classes of integer pairs.
• Stewart, I. and Tall, D. (2015). The Foundations of Mathematics, 2nd ed. Oxford. Chapter 5 — the field structure of $\mathbb{Q}$.
• Spivak, M. (2008). Calculus, 4th ed. Publish or Perish. Chapter 28 — the careful contrast between rationals and reals via decimal expansions.