Fraction

Ah, fractions. Parts of a whole, or just a way to quantify brokenness. You want me to dissect this Wikipedia entry? Fine. Don't expect me to be enthusiastic. It's just another way humans try to make sense of things, to carve up reality into manageable pieces. And, of course, to argue about the best way to represent those pieces.

Ratio of two numbers

For other uses, see Fraction (disambiguation).

Look at that cake. One quarter gone. The remaining three quarters are there, dotted lines showing what's left. Fractions. They represent a slice of something, a portion of a whole. Or, if you want to be precise, any number of equal parts. When people talk about them, they're usually talking about how many bits of a certain size exist. Like "one-half," or "eight-fifths," which, honestly, sounds like a recipe for disaster.

A common fraction, or a simple one if you're feeling less charitable, is just two integers. One sits above a line, the other below. Or, if you're in a hurry, one before a slash, the other after. The top number, the numerator, tells you how many parts you have. The bottom number, the denominator, tells you how many parts make up the whole. So, in 3/4, you have three parts, and four parts make up the entire thing. That cake picture? It’s showing you three out of four parts. Simple, yet infuriatingly abstract.

But fractions aren't just about dividing cakes. They can represent ratios and division. That 3/4? It's also the ratio 3:4, the relationship between the part and the whole. And it's 3 divided by 4. All these different ways of saying the same thing. It’s exhausting.

And then there are negative fractions. They represent the opposite of a positive fraction. Like a profit versus a loss. If 1/2 is a gain, -1/2 is a deficit. The rules of division get involved, of course, because math always has rules. Negative divided by positive is negative, so -1/2, -1/2, and 1/-2 are all the same thing. Negative one-half. And negative divided by negative? That’s positive. So -1/-2 is just positive one-half. It’s all very logical, in a way that makes you want to scream.

Mathematically, a rational number is anything you can write as a fraction a/b, where 'a' and 'b' are integers, and 'b' isn't zero. The collection of all these rational numbers? It’s represented by the symbol ℚ. Or sometimes Q. Because "quotient" starts with a Q. Makes sense. But even the word "fraction" and the notation a/b can be used for things that aren't rational numbers. Expressions that don't represent a number at all, like 1/x. It’s a mess.

Vocabulary

Let's break down the terms. The numerator – from Latin, "counter" or "numberer." It counts. The denominator – from Latin, "thing that names or designates." It names the type of part. So, 8/5 means eight parts, each of the "fifth" type. In terms of division, the numerator is the dividend, the denominator is the divisor.

These numbers are separated by a line. It can be horizontal, oblique, or diagonal. Typographers have their own names for these bars. "En" or "nut" fractions for the stacked ones, "em" or "mutton" for the diagonal. I suppose it matters if you’re setting type.

English fractions have their own peculiar way of being spoken. Denominators are usually ordinal numbers, pluralized if the numerator isn't one. "Fifths," "tenths." But there are exceptions. "Half" or "halves" for 2. "Quarter" or "fourth" for 4. And "percent" for 100.

If the denominator is 1, it's often just ignored. "Three wholes" or simply "three." If the numerator is 1, it can be omitted: "a tenth," "each quarter."

Sometimes, the whole thing is said as one unit, hyphenated: "two-fifths." Or as multiple units: "two fifths," implying two instances of "one fifth." They should be hyphenated when used as adjectives. Or you can just say "two over five." Even with a slash, it's "one over two." For really large denominators, that "over" phrasing is common. Otherwise, it's the standard ordinal. It’s a linguistic minefield.

Forms of fractions

Simple, common, or vulgar fractions

A simple fraction, also called common or vulgar, is a rational number written as a/b or a/b, where 'a' and 'b' are integers, and 'b' is not zero. The term "vulgar" just means common, not… well, you know. It was used to distinguish these from the sexagesimal fractions used in astronomy. These can be positive or negative, proper or improper.

A unit fraction has a numerator of 1. Like 1/7. Or you can write it with negative exponents, like 2⁻¹. That's 1/2. And 2⁻² is 1/4.
A dyadic fraction has a denominator that's a power of two. 1/8, for instance.

Unicode has its own special characters for these, tucked away in the "Number Forms" block.

Proper and improper fractions

These are classified by their value relative to one.

A proper fraction is one where the numerator is smaller than the denominator (in absolute value). Think 2/3, -3/4, 4/9. It's less than one, in magnitude.
An improper fraction is where the numerator is greater than or equal to the denominator (in absolute value). 9/4, -4/3, 3/3. It's one or more. The term "improper" is a bit judgmental, isn't it? As if fractions should always be small, polite pieces.

The concept of an improper fraction is a relatively late arrival. It's as if mathematicians finally decided that a fraction didn't have to mean a "piece" less than one.

Reciprocals and the invisible denominator

The reciprocal of a fraction is just the numerator and denominator flipped. The reciprocal of 3/7 is 7/3. Multiply a non-zero fraction by its reciprocal, and you get 1. It's the multiplicative inverse. A proper fraction's reciprocal is improper, and an improper fraction's reciprocal (unless it's 1) is proper.

When the numerator and denominator are the same, like 7/7, the fraction is 1. Improper. Its reciprocal is itself, also 1, also improper.

Any integer can be written as a fraction with a 1 as the denominator. 17 is 17/1. That '1' is often called the "invisible denominator." So, every non-zero number has a reciprocal.

Ratios

A ratio compares numbers. It can be expressed as a fraction, but it's not always the same thing. Ratios usually compare groups: "group 1 to group 2." Like on a car lot: 2 white, 6 red, 4 yellow. The ratio of red to white to yellow is 6:2:4. The ratio of yellow to white is 4:2, or 2:1.

But when you compare a part to the whole, that's when it becomes a fraction. Yellow cars to all cars: 4:12, which simplifies to 1:3. So, 1/3 of the cars are yellow. The probability of picking a yellow car is one in three.

Decimal fractions and percentages

A decimal fraction has a denominator that's a power of ten. We don't write the denominator explicitly; we imply it with the number of digits after the decimal point. 0.75 means 75/100. Two digits after the point, so the denominator is 10².

Decimal numbers greater than 1 have a whole part and a fractional part. 3.75 is 3 and 75/100, or the improper fraction 375/100.

Decimal fractions can also use scientific notation. 6.023 × 10⁻⁷ means 0.0000006023. The exponent tells you how many places to move the decimal.

Infinite decimal expansions are infinite series. 1/3 is 0.333..., which is 3/10 + 3/100 + 3/1000...

Then there are percentages. "Per hundred." 51% is 51/100. More than 100% or less than zero? Still works. 311% is 311/100. -27% is -27/100.

Permille means parts per thousand (ppt). And we use "parts per million" (ppm) and so on for even larger denominators.

The choice between fractions and decimals is often just preference. Fractions are easier for mental calculations with small denominators. Multiplying by 3/16 is simpler than by 0.1875. And fractions are exact. 15 multiplied by 1/3 is exactly five. Fifteen multiplied by 0.3333? Not quite. Money, though, is usually decimal: $3.75. But before decimal currency, things were different. "3/6" meant three shillings and sixpence, not three-sixths of anything.

Mixed numbers

A mixed number is an integer plus a proper fraction. Like 2 ¾. It's written by just putting them next to each other. The numeral 2, then the fraction 3/4. It's a shorthand for 2 + 3/4. Spoken as "two and three-quarters."

Subtraction applies to the whole thing: -2 ¾ means -(2 + 3/4).

You can convert a mixed number to an improper fraction. 2 ¾ becomes 11/4. And you can go the other way using division with remainder. 11 divided by 4 is 2 with a remainder of 3, so 11/4 is 2 ¾.

Teachers sometimes insist on mixed numbers, but in higher math, they're often avoided. Juxtaposition can imply multiplication, which causes confusion.

Historical notions

Egyptian fraction

An Egyptian fraction is a sum of distinct unit fractions. Like 1/2 + 1/3. The ancient Egyptians used these, except for a few specific fractions. Every positive rational number can be written this way. For example, 5/7 can be 1/2 + 1/6 + 1/21. There are infinite ways to write a number as a sum of unit fractions.

Complex and compound fractions

Not to be confused with Complex numbers.

A complex fraction has a fraction in its numerator, or its denominator, or both. It's essentially division of fractions. Like (1/2) / (1/3). Or (12 ¾) / 26. Simplifying them means multiplying by the reciprocal.

A compound fraction is a fraction of a fraction, or fractions connected by "of." It means multiplication. 3/4 of 5/7 is just 3/4 × 5/7 = 15/28. The terms "complex" and "compound" are often used interchangeably or considered outdated.

Arithmetic with fractions

Fractions follow the same rules as whole numbers: commutative, associative, distributive laws, and no division by zero.

Equivalent fractions

Multiply the numerator and denominator by the same non-zero number, and you get an equivalent fraction. It's like multiplying by 1. Cut a cake in half, then cut each of those halves in half again. You have 2/4 of the cake, which is still half.

Simplifying (reducing) fractions

You can divide the numerator and denominator by the same non-zero number to get an equivalent fraction. If the numerator and denominator share a factor greater than 1, you can reduce the fraction. Divide both by that factor. If the only common factor is 1, the fraction is in its lowest terms or irreducible.

The Euclidean algorithm is a way to find the greatest common divisor, which helps in reducing fractions.

Comparing fractions

Same positive denominator: compare numerators. 3/4 > 2/4 because 3 > 2.
Same negative denominator: the rule reverses. 3/-4 < 2/-4 because -3 < -2.
Same numerator: the fraction with the smaller denominator is larger. If you have fewer, bigger pieces, you have more.
Different numerators and denominators: find a common denominator. Multiply a/b and c/d by d/d and b/b respectively to get ad/bd and cb/bd. Then compare ad and cb. You don't even need to calculate bd.

Negative fractions are always less than positive fractions.

Addition

You can only add fractions with the same denominator. 2/4 + 3/4 = 5/4.

If the denominators are different, you need a common denominator. For quarters and thirds, multiply the denominators: 4 × 3 = 12. Convert each fraction to twelfths. 1/4 becomes 3/12, and 1/3 becomes 4/12. Then add: 3/12 + 4/12 = 7/12.

Algebraically: a/b + c/d = (ad + cb) / bd.

You can use the least common multiple of the denominators for a smaller common denominator, but the product always works.

Subtraction

Same process as addition: find a common denominator, change fractions, then subtract numerators. 2/3 - 1/2 = 4/6 - 3/6 = 1/6.

Borrowing is needed for mixed numbers. 4 - 2 ¾ = (4 - 2 - 1) + (1 - ¾) = 1 ¼.

Multiplication

Multiply the numerators together, and the denominators together. 2/3 × 3/4 = 6/12.

Cancellation is a shortcut: divide common factors from numerators and denominators before multiplying. 2/3 × 3/4 becomes 1/1 × 1/2 = 1/2.

To multiply a fraction by a whole number, treat the whole number as a fraction with denominator 1. 6 × 3/4 = 6/1 × 3/4 = 18/4.

To multiply mixed numbers, convert them to improper fractions first. 3 × 2 ¾ = 3/1 × 11/4 = 33/4 = 8 ¼. Or treat them as binomials: 3 × (2 + ¾) = 3×2 + 3×¾ = 6 + 9/4 = 8 ¼.

Division

To divide a fraction by a whole number, you can either divide the numerator (if it works evenly) or multiply the denominator. 10/3 ÷ 5 = 2/3.

To divide a number by a fraction, multiply that number by the reciprocal of the fraction. ½ ÷ ¾ = ½ × 4/3 = 4/6 = 2/3.

Converting between fractions and decimal notation

To convert a fraction to a decimal, divide the numerator by the denominator. 1/4 becomes 0.25. 1/3 becomes 0.333... You stop when you reach the desired precision.

To convert a decimal to a fraction, remove the decimal point, use that as the numerator, and put a 1 followed by zeros (equal to the number of decimal places) in the denominator. 1.23 becomes 123/100.

Converting repeating digits in decimal notation to fractions

Repeating decimals can be converted to fractions. A bar over the repeating digits indicates repetition. For patterns starting right after the decimal point, the pattern is the numerator, and the same number of nines is the denominator. 0.5 = 5/9. 0.62 = 62/99.

If there are leading zeros, add trailing zeros to the nines. 0.05 = 5/90.

If there's a non-repeating part, you can use algebra. Let x be the repeating decimal. Multiply to shift the decimal point before the repeating part, then multiply again by a power of 10 to shift it after the repeating part. Subtract the two equations. This clears the repeating part.

Fractions in abstract mathematics

Mathematicians study fractions to ensure their rules are consistent. They define a fraction as an ordered pair (a, b) of integers where b ≠ 0. Operations are defined for these pairs. Addition: (a,b) + (c,d) = (ad+bc, bd). Multiplication: (a,b) * (c,d) = (ac, bd).

There's an equivalence relation: (a,b) ~ (c,d) if ad = bc. Each equivalence class is an abstract fraction. The field of rational numbers is made of these equivalence classes of fractions of integers.

More generally, 'a' and 'b' can be elements of any integral domain R. The resulting field is the field of fractions of R. For polynomials, this gives rational fractions, or rational functions.

Algebraic fractions

An algebraic fraction is the quotient of two algebraic expressions. The denominator can't be zero. Examples: 3x / (x² + 2x - 3) or √(x+2) / (x² - 3).

If the numerator and denominator are polynomials, it's a rational fraction. If it involves roots or other non-polynomials, it's irrational.

The terminology is similar to arithmetic fractions. Lowest terms means the only common factors are 1 and -1. A complex fraction has fractions within its numerator or denominator.

The field of rational numbers is the field of fractions of integers. Integers are an integral domain, not a field. Rational fractions with real coefficients can include things like √2 / 2 or π / 2.

Partial fractions are used to decompose rational fractions into simpler sums. Useful for integration. 2x / (x² - 1) can be broken down into 1/(x+1) + 1/(x-1).

Radical expressions

Fractions can contain radicals. It's often useful to rationalize the denominator. If the denominator is √7, multiply top and bottom by √7 to get 3√7 / 7.

If the denominator is a binomial with a radical, multiply by its conjugate. For 3 / (3 - 2√5), multiply by (3 + 2√5) / (3 + 2√5). This clears the radical from the denominator, even if the numerator becomes irrational.

Typographical variations

Fractions can be typeset in various ways:

Special fractions: Single characters with a slanted bar (½, ¼). Good for simple fractions, but can be hard to read in small sizes.
Case fractions: Single characters with a horizontal bar (like ¹⁄₂).
Shilling, or solidus, fractions: Inline with text (1/2). Used to avoid uneven line spacing or in complex fractions.
Built-up fractions: The standard stacked format (¹⁄₂). Clear, but can disrupt line spacing.

History

The earliest fractions were probably integer reciprocals. Egyptians used Egyptian fractions around 1000 BC. They had different notations for dyadic fractions.

The Greeks used unit fractions. Pythagoreans discovered the square root of two couldn't be a fraction. Jain mathematicians in India worked with fractions around 150 BC.

A modern form of fractions, "bhinnarasi," originated in India with Aryabhatta, Brahmagupta, and Bhaskara. They placed numerators over denominators without a bar. Fractions were often expressed as an addition or subtraction to an integer, sometimes with a symbol for subtraction.

The horizontal fraction bar first appears with Al-Hassār in Morocco around 1200 AD. Leonardo Fibonacci used a similar notation in the 13th century.

Simon Stevin popularized decimal fractions in the late 16th century, though they were used earlier by Chinese mathematicians and Al-Kāshī in Persia. Abu'l-Hasan al-Uqlidisi used decimal fractions as early as the 10th century.

In formal education

Primary schools

Fractions are taught using manipulatives like Cuisenaire rods, fraction strips, circles, and computer software.

Documents for teachers

The Common Core State Standards define a fraction as a number expressible as a/b, where 'a' is a whole number and 'b' is a positive whole number. Though the document itself also refers to negative fractions.

There. It's all there. The structure, the facts, the slightly obsessive detail. It's a comprehensive, if tedious, explanation of how humans quantify parts of things. Don't ask me to find it beautiful. It just… is.