Fraction

A fraction comes from the Latin word fractus, meaning "broken." It helps us describe a part of a whole. For example, when we talk about one-half or three-quarters, we are using fractions.

A simple fraction, like ⁠1/2⁠ or ⁠17/3⁠, has two parts: the top number is called the numerator, and the bottom number is the denominator. The numerator tells us how many parts we have, and the denominator tells us how many parts make up a whole. So, in ⁠3/4⁠, we have 3 parts out of 4 equal parts that make a whole.

Fractions are useful because they can show ratios and help us do division. For instance, the fraction ⁠3/4⁠ can mean the ratio of 3 to 4 or the result of dividing 3 by 4.

In math, a special kind of number called a rational number can always be written as a fraction where the top and bottom are whole numbers and the bottom is not zero. All these fractions help us understand numbers better and are part of a bigger world of math called Q, which stands for quotient. Fractions are important because they help us measure, share, and solve many real-life problems.

Vocabulary

See also: Numeral (linguistics) § Fractional numbers, English numerals § Fractions and decimals, and Unicode subscripts and superscripts § Fraction slash

A fraction tells us how many parts of something we have, out of the total parts that make up a whole. The top number is called the numerator. It counts the parts we have. The bottom number is the denominator. It shows what kind of parts we are talking about — like halves, thirds, or quarters.

For example, in the fraction ⁠8/5⁠, the numerator is 8. This means we have eight parts. The denominator is 5, telling us each part is a "fifth" of the whole. We can read fractions in different ways: ⁠1/2⁠ can be "one-half," "one over two," or just "half" if the numerator is 1. When we write fractions in words, we usually connect the numbers with a hyphen, like "two-fifths."

Forms of fractions

Fractions are ways to show parts of a whole. A simple fraction, also called a common fraction or vulgar fraction, is written as one number over another, like ⁠1/2⁠ or ⁠17/3___ The top number is called the numerator, and the bottom number is the denominator. The denominator can never be zero.

Fractions can be proper or improper. A proper fraction is smaller than 1, like ⁠2/3___ or ⁠4/9___ An improper fraction is larger than or equal to 1, like ⁠9/4___ or ⁠3/3∤. Every fraction can also be written as a mixed number, which combines a whole number and a fraction, such as ⁠2 3/4∤.

Arithmetic with fractions

Like whole numbers, fractions follow special rules called the commutative, associative, and distributive laws. We also cannot divide by zero, as explained in the rule about division by zero.

Mixed numbers, which combine whole numbers and fractions, can be worked with by turning them into improper fractions or by handling the whole number and fraction parts separately.

Equivalent fractions

Multiplying the top and bottom of a fraction by the same number (that is not zero) gives a fraction that is the same as the original. This works because multiplying by a number over itself (like 2/2) is the same as multiplying by one, which does not change the value.

For example, start with the fraction ⁠1/2⁠. If we multiply both the top and bottom by 2, we get ⁠2/4⁠. Both ⁠1/2⁠ and ⁠2/4⁠ equal 0.5. Imagine a cake cut into four pieces; two pieces (⁠2/4⁠) are the same as half the cake (⁠1/2⁠).

Simplifying (reducing) fractions

Dividing the top and bottom of a fraction by the same number (that is not zero) gives an equivalent fraction. If the top and bottom can both be divided by a number larger than 1, we can make the fraction simpler.

For example, if both the top and bottom of a fraction can be divided by a number called a factor, we can reduce it. If a fraction’s top and bottom share a factor larger than 1, we can divide both by that factor to get a simpler fraction.

If the top and bottom do not share any factors larger than 1, the fraction is already in its lowest terms. For example, 3/9 can be simplified because both numbers can be divided by 3. But 3/8 is already in lowest terms because only 1 divides into both numbers evenly.

Using these ideas, we can see that ⁠5/10⁠ = ⁠1/2⁠ = ⁠10/20⁠ = ⁠50/100⁠.

Comparing fractions

When fractions have the same bottom number (denominator) and that number is positive, we compare them by looking at the top numbers (numerators).

For example, 3/4 is bigger than 2/4 because 3 is bigger than 2, and both have the same denominator, 4.

If the denominators are negative, the result flips: 4/6 is bigger than 3/6.

For harder comparisons like 5/18 and 4/17, we can give both fractions the same denominator. Multiply the top and bottom of each fraction by the other fraction’s denominator. Then compare the new top numbers. Since 5×17 (= 85) is bigger than 4×18 (= 72), we know that ⁠5/18⁠ is bigger than ⁠4/17⁠.

Addition

We can only add amounts that are the same kind. For example, we can add quarters to quarters. If we want to add different kinds, like thirds to quarters, we must first change them to the same kind.

Imagine two pockets: one has two quarters, and the other has three quarters. Together, there are five quarters. Since four quarters equal one dollar, we can write this as:

2/4 + 3/4 = 5/4 = 1 1/4.

Adding unlike quantities

To add fractions that are different, we need to change them to the same kind. We do this by multiplying the two denominators together.

For example, to add quarters to thirds, we change both to twelfths:

1/4 + 1/3 = 3/12 + 4/12 = 7/12.

Subtraction

Subtracting fractions is very similar to adding them. We find a common denominator, change each fraction to one with that denominator, and then subtract the top numbers.

For example:

2/3 − 1/2 = 4/6 − 3/6 = 1/6.

Multiplication

Multiplying a fraction by another fraction

To multiply fractions, multiply the top numbers together and the bottom numbers together.

For example:

2/3 × 3/4 = 6/12.

Multiplying a fraction by a whole number

A whole number can be thought of as that number divided by 1. We can still use the same multiplication rules.

For example:

6 × 3/4 = 6/1 × 3/4 = 18/4.

Multiplying mixed numbers

To multiply mixed numbers, change each to an improper fraction first.

For example:

3 × 2 3/4 = 3/1 × 11/4 = 33/4 = 8 1/4.

Division

To divide a fraction by a whole number, either divide the top number if it fits evenly, or multiply the bottom number by the whole number. For example, 10/3 ÷ 5 equals 2/3 and also equals 10/15, which simplifies to 2/3. To divide a number by a fraction, multiply that number by the reciprocal of the fraction. For example, 1/2 ÷ 3/4 = 1/2 × 4/3 = 2/3.

Converting between fractions and decimal notation

To turn a fraction into a decimal, divide the top number by the bottom number. For example, to change ⁠1/4⁠ to a decimal, divide 1 by 4 to get 0.25. To change ⁠1/3⁠ to a decimal, divide 1 by 3 and stop when you have enough decimal places, such as 0.3333 for four places. The fraction ⁠1/4⁠ is exact after two decimal places, but ⁠1/3⁠ goes on forever.

To turn a decimal into a fraction, remove the decimal point, use that number as the top, and use 1 followed by zeroes (as many as the decimal places) as the bottom. For example, 1.23 becomes 123/100.

Converting repeating digits in decimal notation to fractions

Sometimes decimals repeat, like 0.789789789.... We can turn these repeating decimals into fractions.

For simple repeating decimals like 0.5, we use 5/9. For 0.62, we use 62/99. For 0.264, we use 264/999. For 0.6291, we use 6291/9999.

If there are zeros before the repeating part, like 0.05, we use 5/90. For 0.000392, we use 392/999000. For 0.0012, we use 12/9900.

If there are numbers before the repeating part, like 0.1523987, we can split it into two parts: 0.1523 and 0.0000987. We turn each part into a fraction and add them together.

We can also use algebra. We let ⁠x⁠ be the repeating decimal, multiply to move the decimal point, subtract to remove the repeating part, and then divide to get the fraction. For example, for 0.1523987:

1. Let ⁠x⁠ = 0.1523987
2. Multiply by 10,000 to move the decimal: 10,000⁠x⁠ = 1,523.987
3. Multiply by 10,000 again to shift more: 10,000,000⁠x⁠ = 1,523,987.987
4. Subtract: 9,990,000⁠x⁠ = 1,522,464
5. Divide: ⁠x⁠ = ⁠1522464/9990000⁠

Fractions in abstract mathematics

Fractions are important for everyday use and for mathematicians who study them. Mathematicians think of fractions as pairs of numbers (a, b), where a and b are whole numbers and b is not zero. They study how to add, subtract, multiply, and divide these pairs using special rules.

These rules make sure fractions work the same way, no matter how they look. For example, the fraction 1/2 is the same as 2/4, and both follow the same rules in calculations. This helps mathematicians learn more about numbers and use fractions in many different ways.

Algebraic fractions

Main article: Algebraic fraction

An algebraic fraction is made from two algebraic expressions divided by each other. The bottom part (denominator) can never be zero. For example, you might see 3x divided by (x squared plus 2x minus 3).

When both the top (numerator) and bottom (denominator) are polynomials, we call it a rational fraction. If the fraction includes roots or other non-polynomial parts, it’s called an irrational fraction. These fractions follow the same basic rules as the fractions you already know.

Radical expressions

Main articles: Nth root and Rationalization (mathematics)

Sometimes, fractions can have special numbers called radicals in the top or bottom part. When radicals are in the bottom part, we can make the fraction easier to use by a process called rationalization. This changes the fraction so the bottom part no longer has radicals. This makes adding, comparing, or dividing the fractions simpler.

For example, if the bottom part is a simple square root, like √7, you can multiply both the top and bottom of the fraction by that same square root. This makes the radicals in the bottom disappear, and the fraction becomes easier to work with. Even if the top part ends up with a radical after this, it is still helpful because the bottom is now a regular number.

Typographical variations

See also: Slash § Encoding

In books and on computers, fractions can look different. Sometimes, a fraction like ½ (one half) is printed as one special symbol. This is often used in everyday writing.

There are four main ways to write fractions. One way uses symbols like ¼ or ¾. Another way uses a horizontal line, like ⁠1/2⁠. A third way uses a slash, like 1/2. The last way stacks numbers, like 1⁄2, and works well for bigger fractions.

History

The earliest fractions were simple parts of whole numbers, like one part of two or one part of three. Ancient Egyptians used fractions around 1000 BC, dividing numbers in ways similar to today. They had special ways to write fractions for weights and measurements.

Later, Greeks also used fractions, and mathematicians in India developed new ways to write them. Over time, different cultures, including Muslim mathematicians from Morocco and European scholars, created the fraction bars we use today to show numbers like one-half or three-quarters.

In formal education

In primary schools, fractions are often taught using tools like Cuisenaire rods, fraction bars, pattern blocks, and paper for folding or cutting. These hands-on materials help students see parts of a whole.

Several states in the United States follow guidelines from the Common Core State Standards Initiative for teaching fractions. These guidelines explain that a fraction is a number that can be written in the form a⁄b, where a is a whole number and b is a positive whole number. This helps teachers plan how to introduce and build students' understanding of fractions step by step.