Foundations — Fractions — proper, improper, mixed numbers

This page is the toolbox. Before you touch the parent topic, we build every symbol it uses from nothing: the counting numbers, the fraction bar, the two numbers, the equals sign, division with a leftover, mixed-number notation, negative numbers, and the number line. No symbol appears here until it has plain words and a picture attached. We introduce them in an order where each one only uses symbols already built.

0. The idea before any symbol: "equal pieces"

Forget notation for a moment. Take a single strawberry tart. Cut it into pieces that are all exactly the same size. That word equal is the whole game.

Look at the picture. The left tart is cut into fair pieces — all the same. The right tart is cut into unfair pieces. Fractions only ever describe the left kind.

1. The counting numbers $0,1,2,3,\dots$

Why the topic needs them: a fraction has a count on top (how many pieces I took) and a count on the bottom (how many pieces make a whole). Both of those are counting numbers. You cannot take "two-and-a-half pieces of a piece" — you count pieces in whole steps.

2. Letters that stand for a number: $a$ and $b$

Before we draw any fraction, we need a way to talk about any fraction at once. We do that with letters.

Why introduce this first? Rules must work for every fraction, not just one example. Letters let us state one rule that covers all cases — but we can only use them once we know what numbers they are allowed to be. That is the next box.

3. The fraction bar and the two numbers

Now that $a$ and $b$ are declared placeholders, we can stack them. The horizontal line is the fraction bar. It has a number above and a number below.

Read it as a sentence. "Three over four" = "cut a whole into $4$ equal pieces, take $3$ of them."

4. The forbidden case, and why the bottom must be positive

Why the denominator can never be $0$. The denominator sets the piece size by cutting the whole into $b$ equal parts. Cutting into zero parts is meaningless — there is no piece to count, no size to speak of. Picture trying to slice a tart into "0 slices": you either haven't cut it, or you've destroyed the question. So a fraction with $0$ on the bottom is undefined.

Why the denominator must be positive (not just non-zero). "Cut the tart into $b$ equal pieces" only makes sense when $b$ is a genuine count of pieces — $1$ piece, $2$ pieces, $3$ pieces. There is no such thing as cutting into $-4$ pieces, so at foundation level we require $b>0$, i.e. $b\in {\mathbb{N}}^{+}$. (Later maths lets the whole fraction be negative — but by putting the minus sign out front, on the numerator, never on this piece-count. See §8.)

5. The equals sign $=$ and mixed-number notation

We are about to write two things that mean the same amount but look different. First we need the symbol that says "same amount," and the shorthand for writing a whole plus a fraction.

So when we later write that "seven-quarters" $=$ "one and three-quarters," the $=$ promises these are one identical quantity, and the mixed form is read as "one whole plus three-quarters."

6. Division with a leftover, and why the bar is division

Division with a remainder

The parent's main engine is splitting a whole number into "so many whole groups, plus a leftover."

Read the picture. $17$ dots, grouped in $5$s: you get $3$ full groups ($q=3$) and $2$ dots left over ($r=2$). So $17=3\cdot 5+2$.

This is the whole machine behind improper → mixed in the parent note. See Division with remainder for more.

7. Multiplication as "groups of" — the $\times$ and $\cdot$ signs

Why the topic needs it: to turn wholes back into pieces. "$4$ wholes, each worth $8$ eighths" is $4\times 8=32$ eighths — that is the mixed→improper rule. Multiplication is how you re-express a whole number in the small piece size.

8. The minus sign and negative numbers

So far every number has counted a real quantity of pieces. But amounts can also point the other way from zero — a debt, a step left instead of right.

Why the topic needs it. The parent note says a proper fraction's value lies between $-1$ and $1$. That $-1$ only makes sense once you know what a negative fraction is: proper fractions can point either side of $0$ but never reach a full whole in either direction.

9. Comparing sizes: $<$ and $\ge$

Why the topic needs them. The whole "three costumes" classification is a comparison of the two numbers:

• numerator $<$ denominator → proper (less than one whole),
• numerator $\ge$ denominator → improper (one whole or more).

The $\ge$ is deliberate: "nine over nine" has top equal to bottom, and that already counts as "improper" because it makes exactly one whole.

10. Where fractions live: the number line

A fraction is not just pizza — it is a point on a ruler.

Read the picture:

• "Three-quarters" sits between $0$ and $1$ — that is what proper looks like.
• "Seven-quarters" lands past $1$ — that is what improper looks like.
• "One and three-quarters" points at the exact same spot as seven-quarters: one full step of $1$, then $3$ more quarter-steps. Same point, two names — that is what the $=$ sign guarantees.
• "Minus three-quarters" sits the same distance the other side of $0$ — a proper fraction pointing left.

How these foundations feed the topic

Each block below is used by the one after it. Read the arrows as "is needed for."

How to read this map instructionally: start at the top-left with equal parts — nothing works without it. Follow any path downward and you never meet a symbol before its own box. Notice three separate streams all pouring into Fractions topic at the bottom: the conversion stream (division + multiplication + equals/mixed), the classification stream (comparing top and bottom), and the placement stream (negatives + number line). The parent note is exactly where those three streams meet.

Equipment checklist

Test yourself — read the question, answer aloud, then reveal.