Concept · Fractions
What Is a Fraction? Numerator and Denominator
A fraction is two numbers doing two different jobs. Once you know which job is which, most of the rules stop needing to be memorised.
The two numbers, and what each one does
The short answer to what is a fraction: it is a way of writing a quantity that is not a whole number, using two numbers separated by a fraction bar. The two numbers are not interchangeable — each has a distinct job.
- The denominator, on the bottom, says how many equal parts the whole has been cut into. It sets the size of each part.
- The numerator, on top, says how many of those parts you have. It is a count.
So 3/4 means the whole was cut into four equal parts and you have three of them. The word denominator shares a root with 'denomination' — it names the currency you are counting in, exactly as 'five twenties' names both a size and a count.
The three forms a fraction takes
Any fractional quantity can be written three ways, and questions move between them constantly.
| Form | Example | When it is used |
|---|---|---|
| Proper | 3/4 | numerator smaller than denominator — less than one whole |
| Improper | 11/4 | numerator larger — easier to calculate with |
| Mixed number | 2¾ | whole part plus a fraction — easier to picture |
Improper is not a criticism. An improper fraction is usually the more convenient form for arithmetic, which is why most methods start by converting a mixed number into one.
- 2 ¾the mixed number
- 2 × 4 = 8the whole part is 8 quarters
- 8 + 3 = 11add the 3 quarters already there
- 11⁄4eleven quarters in total
A fraction is also a division
3/4 and 3 ÷ 4 are the same statement. The fraction bar is a division sign that has been turned on its side, which is why dividing the numerator by the denominator gives the decimal: 3 ÷ 4 = 0.75.
Reading it that way makes two things obvious. A denominator of zero is meaningless, because dividing by zero is meaningless. And any whole number is already a fraction — 5 is 5/1, cut into parts of size one.
Equivalent fractions
Multiplying the numerator and the denominator by the same number changes how a fraction is written but not what it is worth. Cutting a cake into eight pieces and taking two is the same as cutting it into four and taking one.
- 1⁄4one part of four
- (1 × 2)⁄(4 × 2)double both the size count and the part count
- 2⁄8twice as many pieces, each half the size
Equivalent fractions are the machinery behind almost everything else. Finding a common denominator is finding equivalent fractions that happen to match; simplifying is the same move run backwards, dividing both parts by a shared factor to reach lowest terms.
Unit fractions, and why they are the building blocks
A unit fraction has 1 on top: 1/2, 1/3, 1/8. Every other fraction is a count of unit fractions — 5/8 is simply five copies of 1/8.
Thinking this way removes the mystery from adding. Five eighths plus two eighths is seven eighths for the same reason five apples plus two apples is seven apples, and it immediately shows why 1/2 + 1/3 cannot be added until both are rewritten as sixths.
Fractions, decimals and percentages
These are three notations for one idea, and each has a situation it suits.
| Fraction | Decimal | Percentage | Best for |
|---|---|---|---|
| 1/2 | 0.5 | 50% | any of the three |
| 1/3 | 0.333… | 33.3% | the fraction — the decimal never ends |
| 3/4 | 0.75 | 75% | any of the three |
| 1/8 | 0.125 | 12.5% | the fraction or the decimal |
The middle row is the important one. One third is exact as a fraction and inexact as a decimal, no matter how many digits you write. That is the main practical reason to keep working in fractions rather than converting early.
Questions about what is a fraction
What do the numerator and denominator mean?
The denominator, on the bottom, is how many equal parts the whole is divided into — it sets the size of each part. The numerator, on top, is how many of those parts you have.
Is an improper fraction wrong?
No. An improper fraction simply has a numerator larger than its denominator, which means it is worth more than one whole. It is usually the easier form to calculate with.
Can a denominator be zero?
No. A fraction is a division, and dividing by zero has no meaning, so a fraction with zero on the bottom is undefined.
Is every whole number a fraction?
Yes. Any whole number n can be written n/1 — the whole cut into parts of size one, with n of them taken.