Foundations — Rational numbers — definition, decimal expansion (terminating - repeating)

Before we can talk about terminating and repeating decimals, we must earn every symbol the parent note throws at us. This page builds them one at a time, from absolute zero. Nothing appears here until you can see it.

The number-line picture (our home base)

Every symbol below lives somewhere on a single horizontal line. Pin this image in your mind first — everything else hangs off it.

1. Counting numbers and the symbol $\mathbb{Z}$

The very first numbers you ever met were the counting numbers: $1,2,3,\dots$ — one apple, two apples. Add zero and the negatives (one step left of zero for each step right) and you get the integers.

The picture: the evenly-spaced fence posts on the number line. No posts between them.

Why the topic needs it: a rational number is built out of two integers. If you can't name the ingredients, you can't build the dish.

The symbol $\in$ means "is an element of" — it points into a collection. So $p\in \mathbb{Z}$ reads "$p$ is one of the integers." Picture an arrow landing on one of the fence posts.

2. Building a fraction: the symbols $p$, $q$, and $\frac{p}{q}$

Now take two integers and stack one over the other.

The picture: cut a bar (or a pizza) into $q$ equal slices, then shade $p$ of them. The fraction is the shaded amount.

Why the topic needs it: this is the literal definition of "rational." The letters $p$ and $q$ are just placeholders — actors playing the role of "some integer." When you see $\frac{p}{q}$, hear "some whole number of slices out of some whole number of equal parts."

3. The symbol $\mathbb{Q}$ — the whole family of fractions

Collect every fraction you could ever build this way, and you get the rationals.

The picture: the number line, now filled in with dots at every fraction — packed so tightly you can't see gaps (though gaps secretly exist; that's the story of Irrational numbers and Dense sets).

Because any integer $n$ equals $\frac{n}{1}$ (one whole, undivided), every integer is also a rational. In symbols $\mathbb{Z}\subset \mathbb{Q}$, where $\subset$ means "is contained inside." Picture the fence posts sitting among the denser fraction-dots.

4. Lowest terms and the symbol $\gcd$

The same position on the line can be named many ways: $\frac{2}{4}$, $\frac{3}{6}$, $\frac{1}{2}$ all point to the same dot. We pick the cleanest name.

The picture: $\frac{6}{8}$ — six shaded slices out of eight — but you can group them in pairs and re-see the same shading as $\frac{3}{4}$. Once no regrouping helps, you're in lowest terms.

Why the topic needs it: the terminating/repeating test only works on the reduced fraction. A stray factor of 2 or 5 hidden in an unreduced denominator would give a wrong answer. Finding $\gcd$ relies on breaking numbers into primes — see Prime factorization.

5. Prime factorization and the symbols $2^a\cdot 5^b$

To test a denominator, we crack it open into primes.

The picture: think of a number as a tower of LEGO bricks, where each brick is a prime. $8=2\cdot 2\cdot 2$ is three "2-bricks"; $12=2\cdot 2\cdot 3$ is two "2-bricks" and one "3-brick."

The exponent notation: writing the same brick many times is tedious, so $2\cdot 2\cdot 2=2^3$. The small raised number (the exponent) counts how many copies. So $8=2^3\cdot 5^0$, where $5^0=1$ means "no 5-bricks at all."

Why the topic needs it: the terminating rule says $q=2^a\cdot 5^b$ — the denominator's tower is built only from 2-bricks and 5-bricks. Any other brick (a 3, a 7, …) forces the decimal to repeat.

6. Base ten and why $10=2\times 5$ is the hero

Our whole decimal system counts in groups of ten. Each place is ten times the one to its right.

The picture: columns labelled $\dots ,100,10,1,\frac{1}{10},\frac{1}{100},\dots$ — each a power of ten. A decimal like $0.875$ means $\frac{8}{10}+\frac{7}{100}+\frac{5}{1000}$.

The key fact: $10=2\times 5$. So $10^n=2^n\cdot 5^n$. A denominator made only of 2-bricks and 5-bricks can always be padded up into a pure power of ten — and then the fraction is a decimal that stops. That is the entire secret behind the terminating rule.

7. Long division and the idea of a remainder

Turning $\frac{p}{q}$ into a decimal is long division carried past the decimal point.

The picture: a row of only $q$ mailboxes labelled $0$ to $q-1$. Each step of the division drops a letter into one mailbox.

Why the topic needs it — the pigeonhole punchline: because only $q$ mailboxes exist, after at most $q$ division steps a remainder you've already seen must reappear. The moment it does, the whole pattern loops. That is precisely why non-terminating rationals repeat. (Study of remainders is Modular arithmetic.)

8. Euler's totient $\varphi (q)$ — the repetend's speed limit

The repeating block (the repetend) has a length, and that length obeys a rule.

The picture: line up dots $1,2,\dots ,q$; keep only those not sharing a brick with $q$; $\varphi (q)$ is how many survive. For a prime $p$, all of $1\dots p-1$ survive, so $\varphi (p)=p-1$.

Why the topic needs it: the length of the repeating block always divides $\varphi (q)$. For $\frac{1}{7}$, $\varphi (7)=6$ and the repetend "142857" is exactly 6 long. This is a bound, not an equality — a trap the parent note warns about.

9. The overline $\overline{abc}$ and the "such that" bar

Two tiny notations tidy everything up.

How it all fits together

Read it top-down: integers give fractions, fractions give $\mathbb{Q}$; prime bricks + base ten decide terminating; long division's limited remainders force repeating; the totient bounds how long the repeat can be.

Equipment checklist

Test yourself — cover the right side and answer aloud.