Rational Numbers: Fractions, Operations, Decimals and Percent

What is a rational number?

A rational number is a number that can be written as p/q where p and q are integers and q≠0. The set of them is written ℚ, for "quotient", and the word rational comes from ratio, not from "sensible".

In \frac34 the top number p is the numerator and the bottom number q is the denominator: the denominator says into how many equal parts the whole is cut, and the numerator says how many of them are taken.

Several things follow at once:

• Every integer is rational. 5=\frac51 and -3=(-3)/(1). The integers are the rational numbers whose denominator can be 1.
• A number has many fractions. \frac12, \frac24 and (50)/(100) are different ways of writing one number. A fraction is a way of writing; a rational number is the value it names.
• The sign belongs to the whole fraction. -\frac34=(-3)/(4)=(3)/(-4), and by habit the minus is written in front or on the top.
• The denominator is never 0. \frac50 does not name a number.
• Kinds of fraction. A proper fraction is smaller than 1 in size (\frac34), an improper one is not ((17)/(5)), and a mixed number writes an improper one as a whole part plus a proper part (3\frac25).

The rational numbers are closed under all four operations: add, subtract or multiply two of them, or divide by a non-zero one, and the answer is rational again. That is what the integers could not do for division, and it is why fractions were invented. They are not the whole number line, though: √2 and π are real numbers that are not rational, which the article on irrational numbers explains, with the real numbers article as the overview.

Languages say fractions in their own order. In English the numerator comes first ("three quarters"), while Chinese reads the denominator first: 三分之一, "of three parts, one".

How do you find equivalent fractions and simplify?

Multiplying or dividing the numerator and the denominator by the same non-zero number gives an equivalent fraction, and dividing both by their greatest common divisor gives the lowest terms. Two fractions are equal exactly when \frac ab=\frac cd\iff ad=bc: for example \frac68=\frac34 because 6·4=24=8·3.

The picture is a bar cut into parts. Cut each part into k equal pieces and the bar has k times as many parts, with k times as many shaded, yet the same amount is shaded. Move the sliders to see it.

Simplifying means finding the equivalent fraction with the smallest numbers, the lowest terms, in which the top and the bottom share no factor except 1. Divide both by their greatest common divisor:

(84)/(126): the GCD of 84 and 126 is 42, so (84÷42)/(126÷42)=\dfrac23.

You may also cancel step by step, by any common factor, until nothing is left to cancel. The lowest-terms form is unique, so every route gives the same answer, and for two fractions in lowest terms \frac ab=\frac cd only when a=c and b=d.

Cancel factors, never terms. In 6/8=(2·3)/(2·4) the 2 is a factor of both, so it cancels. In (2+3)/(2+4) the 2s are terms of a sum and do not cancel: the fraction is \frac56, not \frac34. The same care is needed with letters, as in the section on algebraic fractions. Beware also of "canceling" digits: (16)/(64) happens to equal \frac14 when you strike the 6s, and (19)/(95), (26)/(65) and (49)/(98) do the same, but these four are coincidences and the method is wrong, so never rely on it.

Try any fraction below.

How do you compare and order fractions?

To compare two fractions with positive denominators, cross-multiply: \frac ab<\frac cd exactly when ad<cb. For \frac35 and \frac58: 3·8=24 and 5·5=25, and 24<25, so \frac35<\frac58.

Why it works: rewrite both over the common denominator bd, giving (ad)/(bd) and (cb)/(bd). With equal denominators the larger top is the larger fraction.

Shortcuts, when they apply:

• Same denominator: the larger numerator wins, \frac27<\frac57.
• Same numerator: the larger denominator makes the smaller fraction, since the whole is cut into more pieces: \frac13>\frac14.
• Benchmarks: compare each with 0, \frac12 and 1. \frac49<\frac12<\frac47.
• Negative fractions: compare the sizes and reverse, as with integers. \frac12<\frac23 so -\frac12>-\frac23.

To order a list, put them on a number line or give them a common denominator. Between any two different rational numbers lies another: their average 1/2\left(\frac ab+\frac cd\right). So there is no "next" fraction after \frac12, and the rational numbers fill the line without gaps you could ever point to, though √2 shows that the line still holds more: see the number line and density in the article on real numbers. Compare any two below.

How do you add and subtract fractions?

To add or subtract fractions, rewrite them over a common denominator, then add or subtract the numerators and keep the denominator: \frac ac+\frac bc=(a+b)/(c). You can only add parts that are the same size.

The best common denominator is the least common multiple of the two denominators. For \frac23+\frac34 it is 12:

\dfrac23+\dfrac34=(8)/(12)+(9)/(12)=(17)/(12)=1(5)/(12).

For a subtraction, \dfrac56-\dfrac14=(10)/(12)-(3)/(12)=(7)/(12). If you use b· d instead of the least common multiple the answer is the same, but the numbers are larger and the result needs simplifying afterward.

Never add the denominators. \frac12+\frac13 is not \frac25: the denominators name the size of the pieces, and halves and thirds are different sizes. The right answer is \frac36+\frac26=\frac56. A check is that \frac25=0.4 is less than \frac12, but a sum of two positive numbers is larger than each of them.

Mixed numbers. Either add the whole parts and the fraction parts separately, or turn both into improper fractions first. For 2\frac13+1\frac12: 2+1=3 and \frac13+\frac12=\frac56, so 3\frac56.

Negative fractions follow the sign rules of the integers: \frac12-\frac34=\frac24-\frac34=-\frac14.

Try your own pair below; it shows each line of the working.

How do you multiply and divide fractions?

To multiply fractions, multiply the numerators and multiply the denominators: \frac ab·\frac cd=(ac)/(bd); to divide, multiply by the reciprocal of the divisor: \frac ab÷\frac cd=\frac ab·\frac dc. No common denominator is needed for either.

Multiplying: \dfrac34·\dfrac23=(6)/(12)=\dfrac12. It is quicker to cancel first, because a factor on the top of one fraction can cancel with a factor on the bottom of the other: the 3 on the top cancels with the 3 on the bottom, and the 2 on the top with half of the 4, so \frac34·\frac23=(1·1)/(2·1)=\frac12. The word "of" means times: \frac25 of 30 is \frac25·30=12.

Why dividing is multiplying by the reciprocal. Dividing asks how many fit. How many halves fit in 3? 3÷\frac12=6, which is 3·2. In general dividing by \frac cd is the same as multiplying by \frac dc, because multiplying by a number and then by its reciprocal gets you back where you started. Example:

\dfrac34÷\dfrac910=\dfrac34·(10)/(9)=(30)/(36)=\dfrac56.

The reciprocal of \frac cd is \frac dc, and the reciprocal of an integer n is \frac1n. Zero has no reciprocal, so you cannot divide by 0, and dividing by the fraction \frac03 is just as forbidden.

Signs follow the integers: same signs give a positive answer, different signs a negative one, so \left(-\frac12\right)·\left(-\frac23\right)=\frac13. Negative exponents are reciprocals as well, since 2^-3=\frac18, as the section on zero, negative and fractional exponents shows.

Try the calculator above with the other two operations.

How do you convert between mixed numbers and improper fractions?

To turn an improper fraction into a mixed number, divide the top by the bottom: the quotient is the whole part and the remainder is the new numerator over the same denominator; to go back, multiply the whole part by the denominator and add the numerator. The only tool is division with remainder, the same one used in the integers article.

In -3\frac12 the minus applies to the whole number: it means -\left(3+\frac12\right)=-3.5, not -3+\frac12. Mixed numbers are fine for reading a quantity, such as 2\frac12 cups, but improper fractions are easier to calculate with, so convert before you multiply or divide.

How do fractions become decimals and percentages?

To turn a fraction into a decimal, divide the top by the bottom; the decimal terminates exactly when the denominator, in lowest terms, has no prime factor other than 2 and 5, and otherwise it eventually repeats periodically. Ten is 2·5, so only denominators built from 2s and 5s fit into a power of ten.

Two numbers describe the repeat. The digits before it number \max(x,y) when the denominator is 2^x·5^y· q with q coprime to 10 (that is why \frac512 has 2). The block is as long as the number of 10s you need to multiply to get a remainder of 1 modulo q: 1 for \frac13, 6 for \frac17 and for \frac113, 16 for \frac117, 96 for \frac197. It always eventually repeats because long division has fewer than q possible remainders, so one must come back. The proof, and how to turn a repeating decimal back into a fraction, are in the rational numbers section of the real numbers article; the same conversion is a sum of a geometric series.

Percent means "per hundred": p\%=(p)/(100). To convert a fraction, divide to get the decimal and shift the point two places: \frac38=0.375=37.5\%. To convert back, 25\%=(25)/(100)=\frac14.

Percentages apply to a base. A 200 dollar jacket with 15% off costs 85% of 200, which is 0.85·200=170 dollars. A rise of 20% followed by a fall of 20% does not return to the start, because the second percentage is of a larger number: 100→120→96.

Type any fraction below to see its decimal and why it looks that way. The same rule holds in other bases: in base 2 only denominators that are powers of 2 terminate, as the article on binary numbers shows.

How do ratios and proportions use rational numbers?

A ratio a:b compares two quantities and is the fraction \frac ab; a proportion says two ratios are equal, \frac ab=\frac cd, and is solved by cross-multiplying: ad=bc. Problems that scale a recipe, convert a map distance or price a quantity are all proportions.

Two habits prevent most mistakes. Keep the same quantity on the same level of every fraction, flour over cookies on both sides. And check that the answer is sensible: 20 cookies need more flour than 12, so 5 cups is plausible but 2 cups is not. A unit rate is the ratio per one: 20 dollars for 5 kg is 4 dollars per kilogram, and then any quantity is the unit rate times the amount.

How do you work with fractions in Python and JavaScript?

Python has an exact fraction type, fractions.Fraction, that keeps numerator and denominator as integers; JavaScript has none, and its / always produces a binary floating-point number. Floats store only the nearest binary value, so \frac13 and even \frac110 cannot be held exactly: see the floating-point section of the real numbers article.

>>> from fractions import Fraction
>>> Fraction(6, 8)                    # reduced automatically
Fraction(3, 4)
>>> Fraction(1, 3) + Fraction(1, 6)
Fraction(1, 2)
>>> Fraction(3, 4) / Fraction(9, 10)
Fraction(5, 6)
>>> Fraction(-7, 2).numerator, Fraction(-7, 2).denominator
(-7, 2)
>>> float(Fraction(1, 3))
0.3333333333333333
>>> 0.1 + 0.2 == 0.3
False
>>> Fraction(1, 10) + Fraction(2, 10) == Fraction(3, 10)
True
>>> Fraction(0.1)                     # the float 0.1 is not one tenth
Fraction(3602879701896397, 36028797018963968)
>>> Fraction('0.1')                   # the text 0.1 is
Fraction(1, 10)
>>> Fraction(1, 0)
ZeroDivisionError: Fraction(1, 0)

0.1 + 0.2                  // 0.30000000000000004
3 / 4                      // 0.75: JavaScript has no integer division

const gcd = (a, b) => (b === 0n ? (a < 0n ? -a : a) : gcd(b, a % b))
const frac = (n, d) => {
  const g = gcd(n, d)
  return d < 0n ? [-n / g, -d / g] : [n / g, d / g]
}
const add = ([a, b], [c, d]) => frac(a * d + c * b, b * d)

add(frac(1n, 3n), frac(1n, 6n))   // [1n, 2n]

#include <iostream>
#include <cstdio>
using namespace std;

int gcd(int a, int b) { return b == 0 ? (a < 0 ? -a : a) : gcd(b, a % b); }

void show(int n, int d) {                                  // a fraction in lowest terms
  if (d == 0) { cout << "undefined: the denominator is 0" << endl; return; }
  if (d < 0) { n = -n; d = -d; }
  int g = gcd(n, d);
  cout << n / g << "/" << d / g << endl;
}

int main() {
  show(6, 8);                                              // 3/4: reduced
  show(1 * 6 + 1 * 3, 3 * 6);                              // 1/3 + 1/6 = 1/2
  show(3 * 10, 4 * 9);                                     // (3/4) / (9/10) = 5/6
  show(-7, 2);
  show(1, 0);
  printf("%.16f\n", 1.0 / 3);                              // float(1/3)
  cout << (0.1 + 0.2 == 0.3) << endl;                      // 0: the floats are not equal
  cout << (1 * 10 + 2 * 10 == 3 * 10) << endl;             // 1: with whole numbers (tenths) they are
  printf("%.20f\n", 0.1);                                  // the float 0.1 is not one tenth
  return 0;
}

/* output:
3/4
1/2
5/6
-7/2
undefined: the denominator is 0
0.3333333333333333
0
1
0.10000000000000000555
*/

The pitfalls, in order of how often they bite:

Use Fraction when you need an exact answer from exact input, and floats when speed matters and a tiny rounding error is harmless.

Where do fractions come from?

Fractions are among the oldest mathematics: the Egyptians used them around 1550 BCE, and our way of writing them was assembled over about three thousand years.

• Egypt. The Rhind Papyrus writes almost every fraction as a sum of different unit fractions (numerator 1): \frac25=\frac13+\frac115. Only a few, such as \frac23, had a symbol of their own.
• India and the Arab world. Fractions were written with one number above the other. Arab mathematicians added the dividing bar.
• Europe. Fibonacci's Liber Abaci (1202) used the fraction bar and helped spread the Hindu-Arabic numerals in Europe.
• Decimals. Simon Stevin's De Thiende (1585) argued for writing fractions in tenths, hundredths and thousandths, which is the decimal notation we use.

Today the same numbers show up as fractions in a recipe, as decimals on a calculator, as percentages in a shop and as ratios on a map: one idea with four faces.

What are the common mistakes with fractions?

The most common mistakes with fractions are the nine below, each with the correct statement and a number that shows why.

Practice: test your understanding

These questions mix everything above. A wrong answer costs nothing here: read the hint and try again.

Summary: rational numbers at a glance

• Rational: \frac pq with integers p,q and q≠0; every integer is rational; a number has many fractions, one in lowest terms.
• Equal fractions: \frac ab=\frac cd\iff ad=bc; divide top and bottom by the GCD to simplify; cancel factors, never terms.
• Compare: cross-multiply; the larger denominator makes the smaller piece; between two rationals lies their average.
• Add and subtract: common denominator (the LCM), then combine the tops; never add the denominators.
• Multiply and divide: tops times tops, bottoms times bottoms; divide by flipping the second fraction; zero has no reciprocal.
• Mixed numbers: divide with remainder to convert; -3\frac12=-\left(3+\frac12\right).
• Decimals: a lowest-terms fraction terminates exactly when the denominator has only 2s and 5s; otherwise it eventually repeats periodically; p\%=\frac p100.
• Code: use Fraction for exactness in Python; floats cannot hold \frac110.

Glossary: key terms

Rational number

A number that can be written as a fraction of two integers with a non-zero denominator, such as 3 over 4 or minus 7 over 2.

Fraction

A way of writing a number as a numerator over a denominator, which names how many equal parts are taken out of how many.

Numerator

The top number of a fraction, which says how many parts are taken.

Denominator

The bottom number of a fraction, which says how many equal parts the whole is cut into and can never be zero.

Equivalent fractions

Different fractions that name the same number, such as 1 over 2 and 2 over 4, found by multiplying or dividing the top and bottom by the same number.

Lowest terms

A fraction whose numerator and denominator share no factor except 1, found by dividing both by their greatest common divisor.

Proper and improper fraction

A proper fraction is smaller than 1 in size, like 3 over 4; an improper fraction is 1 or more in size, like 17 over 5.

Mixed number

A whole number together with a proper fraction, such as 3 and 2 over 5, which means 3 plus 2 over 5.

Reciprocal

The fraction turned upside down; a number times its reciprocal is 1, and zero has no reciprocal.

Common denominator

A denominator that two fractions are both rewritten over so that they can be added or compared; the least common multiple of the denominators is the smallest choice.

Terminating decimal

A decimal expansion with finitely many digits, such as 0.375; a fraction in lowest terms has one exactly when its denominator has only the prime factors 2 and 5.

Repeating decimal

A decimal expansion that eventually repeats periodically: from some digit onward a block of digits repeats without end, such as 0.333 and so on for 1 over 3.

Percent

A fraction with denominator 100, so that 25 percent means 25 over 100, which is 1 over 4.

Ratio

A comparison of two quantities written a to b or as the fraction a over b.

Frequently asked questions

References

1. James Stewart, Lothar Redlin and Saleem Watson (2016). Precalculus: Mathematics for Calculus (7th ed.). Cengage Learning.
2. Arnold Buffum Chace (1927). The Rhind Mathematical Papyrus. Mathematical Association of America.
3. Laurence Sigler (2002). Fibonacci’s Liber Abaci: A Translation into Modern English of Leonardo Pisano’s Book of Calculation. Springer.
4. Simon Stevin (1585). De Thiende (The Art of Tenths).
5. Florian Cajori (1928). A History of Mathematical Notations. Open Court Publishing Company.
6. Python Software Foundation. The Python Standard Library: fractions, rational numbers. docs.python.org. https://docs.python.org/3/library/fractions.html.

Related articles

• Real Numbers: Definition, Types, Examples and Properties: Real numbers explained: definition, types (rational and irrational), examples, properties and the number line, with interactive practice, a proof and FAQ.
• Irrational Numbers: Definition, Examples, Proofs and Approximations: Irrational numbers explained: why √2 is irrational, which roots are, sums and products, π, e, transcendental numbers and fraction approximations, with practice.
• Integers: Number Line, Operations, Divisibility, GCD and LCM: Integers explained: the number line, adding and multiplying negatives, divisibility, primes, GCD and LCM, and remainders, with worked examples and practice.
• Exponents and Radicals: Laws, Rules, Examples and Practice: Exponents and radicals explained: the five laws, zero, negative and fractional exponents, simplifying roots, rationalizing and scientific notation, with practice.