Fraction

The parts of one

$$\frac{3}{4}$$

The denominator $4$ says the whole was cut into four equal parts. The numerator $3$ says you have three of them.

The word “equal” is doing real work. Four pieces of different sizes are not quarters.

Three things one fraction means

Reading	Example
parts of a whole	three of four equal slices
a point on a number line	the third tick of four between $0$ and $1$
a division	$3 \div 4 = 0.75$

All three describe the same number. The division reading is the one that turns a fraction into a decimal, and the number-line reading is the one that makes it a number rather than a shape. See fractions on a number line.

Names by size

• Proper: numerator smaller than denominator, so less than $1$ — $\tfrac{3}{4}$.
• Improper: numerator at least as big, so $1$ or more — $\tfrac{7}{4}$.
• Unit fraction: numerator of $1$ — $\tfrac{1}{4}$. These are the building blocks, since $\tfrac{3}{4}$ is three of them.

Many names for one amount

Any fraction can be written endlessly many ways.

$$\frac{1}{2} = \frac{2}{4} = \frac{50}{100}$$

That flexibility is what lets you give two fractions a shared denominator before adding them. See equivalent fractions.