Fractional notation is the way we write fractions: one number stacked above another, separated by a line. The top number is called the numerator, and it represents how many parts you have. The bottom number is the denominator, and it tells you how many equal parts make up the whole. So 3/4 means you have 3 out of 4 equal parts.
At its core, a fraction is just a division problem waiting to happen. The numerator divided by the denominator gives you the decimal value: 3 ÷ 4 = 0.75. But fractional notation keeps that relationship intact, which makes it easier to work with in many everyday and mathematical situations.
The Parts of a Fraction
Every fraction written in standard notation has three components: the numerator on top, the denominator on the bottom, and the line between them. That horizontal line has a formal name, the vinculum, though you’ll rarely hear anyone call it that outside a math history class. When fractions are written inline with a diagonal slash (like 3/4), that slash is called a solidus.
The numerator tells you the count. If a pizza is cut into 8 slices and you eat 3, your share is written as 3/8. The 3 is your count of slices. The denominator tells you the size of each piece. Eighths are smaller than quarters, which is why 3/8 is less than 3/4 even though the numerator is the same.
Types of Fractions
Fractions fall into a few categories depending on the relationship between the numerator and denominator.
A proper fraction has a numerator smaller than the denominator. Examples include 3/5, 1/4, and 7/8. These always represent a value less than one whole.
An improper fraction has a numerator larger than the denominator, like 9/4 or 7/3. These represent more than one whole. Nine-fourths means you have nine quarter-pieces, which is more than two complete wholes.
A mixed number combines a whole number with a proper fraction. The improper fraction 9/4 can be rewritten as 2 1/4, meaning two wholes and one quarter left over. To convert between the two, multiply the whole number by the denominator, add the numerator, and place the result over the original denominator. So 4 2/3 becomes (4 × 3 + 2) / 3 = 14/3.
Equivalent Fractions and Simplifying
Different-looking fractions can represent the exact same value. The fractions 2/3 and 4/6 are equivalent because they occupy the same point on a number line. You can create equivalent fractions by multiplying or dividing both the numerator and denominator by the same number. Multiply the top and bottom of 2/3 by 2, and you get 4/6. Nothing about the value changes.
Simplifying (or reducing) a fraction reverses this process. You divide both the numerator and denominator by their largest shared factor until no number except 1 divides evenly into both. Take 9/15: both 9 and 15 are divisible by 3, so dividing each by 3 gives you 3/5. That’s the fraction in its lowest terms. One reliable method is to break both numbers into their prime factors and cancel out everything they share.
Converting Fractions to Decimals and Percentages
To turn any fraction into a decimal, divide the numerator by the denominator. The fraction 3/8 becomes 3 ÷ 8 = 0.375. Some fractions produce clean decimals (1/4 = 0.25), while others repeat forever (1/3 = 0.333…).
To convert a fraction to a percentage, the goal is to rewrite it with a denominator of 100. For 3/5, you multiply both top and bottom by 20 (since 5 × 20 = 100), giving you 60/100, which is 60%. Going the other direction, a percentage like 15% becomes 15/100, which simplifies to 3/20 when you divide both numbers by 5.
When Fractions Work Better Than Decimals
Fractions and decimals express the same values, but each format has situations where it fits more naturally. Decimals tend to be better when precision matters, like in scientific measurements or financial calculations. Fractions shine when exact ratios matter or when the numbers divide unevenly.
Cooking is the classic example. Recipes call for 3/4 cup of bread crumbs or 1/2 teaspoon of salt because measuring cups and spoons are designed around fractions. Telling someone to use 0.75 cups of flour communicates the same amount but doesn’t match the markings on their tools. Construction and woodworking use fractional inches for the same reason: a drill bit labeled 5/16″ is far more practical than one labeled 0.3125″.
Fractions also preserve exactness in ways decimals sometimes can’t. The fraction 1/3 is precise. Its decimal equivalent, 0.333…, is an infinite repeating decimal that can only ever be rounded.
Finding Common Denominators
Before you can add or compare fractions with different denominators, you need a common denominator, a shared bottom number. The most efficient choice is the least common denominator (LCD), which is the smallest number that both denominators divide into evenly.
To find it, break each denominator into its prime factors, then take the highest power of every prime that appears. For 4 (2 × 2) and 6 (2 × 3), the LCD is 2² × 3 = 12. Once both fractions share a denominator, you can add or subtract the numerators directly. So 1/4 + 1/6 becomes 3/12 + 2/12 = 5/12.
Where Fractional Notation Came From
People have been writing fractions for thousands of years, though the notation looked very different at first. As early as 1800 BC, the Egyptians wrote fractions, but they restricted themselves to unit fractions, those with 1 as the numerator. To express something like 3/4, they would write it as a sum of unit fractions: 1/2 + 1/4. Repeating a unit fraction in the sum wasn’t allowed, which made some representations surprisingly complex.
The format we use today traces back to Indian mathematicians around 500 AD. They wrote the numerator above the denominator, just as we do now, but without a line between them. It was Arab mathematicians who later added the horizontal or slanted line separating the two numbers, giving us the notation that has persisted largely unchanged for over a thousand years.