Fraction

A fraction comes from the Latin word fractus, meaning "broken." It helps us describe a part of a whole or any number of equal parts. 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, like 3 slices of a cake cut into 4 equal pieces.

Fractions are very 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. We can even have fractions that are negative, which show the opposite, like −⁠1/2⁠ meaning a loss instead of a gain.

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 how many total parts make up a whole. The top number is called the numerator, which counts the parts we have. The bottom number is the denominator, which shows what kind of parts we're talking about — like halves, thirds, or quarters.

For example, in the fraction ⁠8/5⁠, the numerator is 8, meaning 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 obey the commutative, associative, and distributive laws, and the rule against division by zero.

Mixed-number arithmetic can be performed either by converting each mixed number to an improper fraction, or by treating each as a sum of integer and fractional parts.

Equivalent fractions

Multiplying the numerator and denominator of a fraction by the same (non-zero) number results in a fraction that is equivalent to the original fraction. This is true because for any non-zero number n, the fraction n/n equals 1. Therefore, multiplying by n/n is the same as multiplying by one, and any number multiplied by one has the same value as the original number. By way of an example, start with the fraction ⁠1/2⁠. When the numerator and denominator are both multiplied by 2, the result is ⁠2/4⁠, which has the same value (0.5) as ⁠1/2⁠. To picture this visually, imagine cutting a cake into four pieces; two of the pieces together (⁠2/4⁠) make up half the cake (⁠1/2⁠).

Simplifying (reducing) fractions

Dividing the numerator and denominator of a fraction by the same non-zero number yields an equivalent fraction: if the numerator and the denominator of a fraction are both divisible by a number (called a factor) greater than 1, then the fraction can be reduced to an equivalent fraction with a smaller numerator and a smaller denominator. For example, if both the numerator and the denominator of the fraction a/b are divisible by ⁠c⁠, then they can be written as a = cd, b = ce, and the fraction becomes ⁠cd/ce⁠, which can be reduced by dividing both the numerator and denominator by ⁠c⁠ to give the reduced fraction ⁠d/e⁠.

If the numerator and the denominator do not share any factor greater than 1, the fraction is already reduced to its lowest terms, and it is said to be irreducible, reduced, or in simplest terms. For example, 3/9 is not in lowest terms because both 3 and 9 can be exactly divided by 3. In contrast, 3/8 is in lowest terms—the only positive integer that goes into both 3 and 8 evenly is 1.

Using these rules, we can show that ⁠5/10⁠ = ⁠1/2⁠ = ⁠10/20⁠ = ⁠50/100⁠, for example.

Comparing fractions

Comparing fractions with the same positive denominator yields the same result as comparing the numerators:

3/4 > 2/4 because 3 > 2, and the equal denominators 4 are positive.

If the equal denominators are negative, then the opposite result of comparing the numerators holds for the fractions:

4/6 > 3/6.

For the more laborious question 5/18 ? 4/17, multiply top and bottom of each fraction by the denominator of the other fraction, to get a common denominator, yielding 5 × 17/18 × 17 ? 18 × 4/18 × 17. It is not necessary to calculate 18 × 17 – only the numerators need to be compared. Since 5×17 (= 85) is greater than 4×18 (= 72), the result of comparing is ⁠5/18 > 4/17⁠.

Addition

The first rule of addition is that only like quantities can be added; for example, various quantities of quarters. Unlike quantities, such as adding thirds to quarters, must first be converted to like quantities as described below: Imagine a pocket containing two quarters, and another pocket containing three quarters; in total, there are five quarters. Since four quarters is equivalent to one (dollar), this can be represented as follows:

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

Adding unlike quantities

To add fractions containing unlike quantities (e.g. quarters and thirds), it is necessary to convert all amounts to like quantities. It is easy to work out the chosen type of fraction to convert to; simply multiply together the two denominators (bottom number) of each fraction. In case of an integer number apply the invisible denominator 1.

For adding quarters to thirds, both types of fraction are converted to twelfths, thus:

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

Subtraction

The process for subtracting fractions is, in essence, the same as that of adding them: find a common denominator, and change each fraction to an equivalent fraction with the chosen common denominator. The resulting fraction will have that denominator, and its numerator will be the result of subtracting the numerators of the original fractions. For instance,

2/3 − 1/2 = 4/6 − 3/6 = 1/6.

Multiplication

Multiplying a fraction by another fraction

To multiply fractions, multiply the numerators and multiply the denominators. Thus:

2/3 × 3/4 = 6/12.

Multiplying a fraction by a whole number

Since a whole number can be rewritten as itself divided by 1, normal fraction multiplication rules can still apply. For example,

6 × 3/4 = 6/1 × 3/4 = 18/4.

Multiplying mixed numbers

The product of mixed numbers can be computed by converting each to an improper fraction. For example:

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

Division

To divide a fraction by a whole number, you may either divide the numerator by the number, if it goes evenly into the numerator, or multiply the denominator by the number. For example, 10/3 ÷ 5 equals 2/3 and also equals 10/3 ⋅ 5 = 10/15, which reduces to 2/3. To divide a number by a fraction, multiply that number by the reciprocal of that fraction. Thus, 1/2 ÷ 3/4 = 1/2 × 4/3 = 1 ⋅ 4/2 ⋅ 3 = 2/3.

Converting between fractions and decimal notation

To change a common fraction to decimal notation, do a long division of the numerator by the denominator (this is idiomatically also phrased as "divide the denominator into the numerator"), and round the result to the desired precision. For example, to change ⁠1/4⁠ to a decimal expression, divide 1 by 4 ("4 into 1"), to obtain exactly 0.25. To change ⁠1/3⁠ to a decimal expression, divide 1... by 3 ("3 into 1..."), and stop when the desired precision is obtained, e.g., at four places after the decimal separator (ten-thousandths) as 0.3333. The fraction ⁠1/4⁠ is expressed exactly with only two digits after the decimal separator, while the fraction ⁠1/3⁠ cannot be written exactly as a decimal with a finite number of digits. A decimal expression can be converted to a fraction by removing the decimal separator, using the result as the numerator, and using 1 followed by the same number of zeroes as there are digits to the right of the decimal separator as the denominator. Thus, 1.23 = 123/100.

Converting repeating digits in decimal notation to fractions

Decimal numbers, while arguably more useful to work with when performing calculations, sometimes lack the precision that common fractions have. Sometimes an infinite repeating decimal is required to reach the same precision. Thus, it is often useful to convert repeating digits into fractions.

A conventional way to indicate a repeating decimal is to place a bar (known as a vinculum) over the digits that repeat, for example 0.789 = 0.789789789.... For repeating patterns that begin immediately after the decimal point, the result of the conversion is the fraction with the pattern as a numerator, and the same number of nines as a denominator. For example:

0.5 = 5/9

0.62 = 62/99

0.264 = 264/999

0.6291 = 6291/9999

If leading zeros precede the pattern, the nines are suffixed by the same number of trailing zeros:

0.05 = 5/90

0.000392 = 392/999000

0.0012 = 12/9900

If a non-repeating set of digits precede the pattern (such as 0.1523987), one may write the number as the sum of the non-repeating and repeating parts, respectively:

0.1523 + 0.0000987

Then, convert both parts to fractions, and add them using the methods described above:

1523 / 10000 + 987 / 9990000 = 1522464 / 9990000

Alternatively, algebra can be used, such as below:

1. Let ⁠x⁠ = the repeating decimal:

   ⁠x⁠ = 0.1523987

2. Multiply both sides by the power of 10 just great enough (in this case 10⁴) to move the decimal point just before the repeating part of the decimal number:

   10,000⁠x⁠ = 1,523.987

3. Multiply both sides by the power of 10 (in this case 10³) that is the same as the number of places that repeat:

   10,000,000⁠x⁠ = 1,523,987.987

4. Subtract the two equations from each other (if ⁠a⁠ = ⁠b⁠ and ⁠c⁠ = ⁠d⁠, then ⁠a⁠ − ⁠c⁠ = ⁠b⁠ − ⁠d⁠):

   10,000,000⁠x⁠ − 10,000⁠x⁠ = 1,523,987.987 − 1,523.987

5. Continue the subtraction operation to clear the repeating decimal:

   9,990,000⁠x⁠ = 1,523,987 − 1,523

   9,990,000⁠x⁠ = 1,522,464

6. Divide both sides by 9,990,000 to represent ⁠x⁠ as a fraction

   ⁠x⁠ = ⁠1522464/9990000⁠

Fractions in abstract mathematics

Fractions are important not just for everyday use but also for mathematicians who study their properties. Mathematicians look at fractions as pairs of numbers (a, b), where a and b are integers and b is not zero. They define how these pairs can be added, subtracted, multiplied, and divided using specific rules.

These mathematical rules ensure that fractions behave consistently, no matter how they are written. For example, the fraction 1/2 is the same as 2/4, and both follow the same rules when used in calculations. This helps mathematicians understand numbers more deeply and apply fractions in many areas of study.

Algebraic fractions

Main article: Algebraic fraction

An algebraic fraction is made from two algebraic expressions divided by each other. Just like with regular fractions, the bottom part (denominator) can never be zero. For example, you might see something like 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 either the top or bottom part. When radicals appear in the bottom part, it can help to make the fraction easier to work with by a process called rationalization. This means changing the fraction so that the bottom part no longer has radicals, which 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 way, the radicals in the bottom disappear, and the fraction becomes easier to handle. Even if the top part ends up with a radical after this process, it’s still helpful because now the bottom is a regular number.

Typographical variations

See also: Slash § Encoding

In books and on computers, fractions can be shown in different ways. Sometimes, a fraction like ½ (one half) is printed as a single symbol. This is common in everyday writing.

There are four main ways to write fractions in books and papers. One way uses special symbols like ¼ or ¾. Another uses a horizontal line, like ⁠1/2⁠. A third way uses a slash, like 1/2, which was originally used for money but now helps keep writing looking neat. The last way uses more space, with numbers stacked on top of each other, like 1⁄2, which is good for big or complicated 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 using methods that are similar to ones we use 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 even paper for folding or cutting. These hands-on materials help students visualize 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.