What makes a number rational
Whole numbers, negatives, fractions, mixed numbers and decimals look like different kinds of number. Is there one test that takes them all in, and is there anything it leaves out?
Where you are 10 in Rational numbers
Why this exists
By now you have met several families of numbers, each learned on its own: the counting numbers, zero, the negatives that make up the Integers, Fractions such as \displaystyle \frac{3}{4}, mixed numbers such as 2\tfrac{1}{3}, and decimals such as 0.75 and -4.07. Each came with its own rules, and it is fair to ask whether -6 is “a fraction”, whether 0.75 is, or whether these are really different things.
They are not different things. Every one of them can be written as one integer divided by another, and that single description is what this lesson is about. It does three jobs. It joins the families into one, so one arithmetic serves them all. It gives a test you can run on any number you meet. And it marks an edge: some numbers, such as the length of the diagonal of a square with sides of length 1, pass no such test, and they lead on to the rest of the number line.
Definition
A number is rational if it can be written as a fraction whose numerator is an integer and whose denominator is a nonzero integer:
with a and b both integers.
The set of all rational numbers is written \mathbb{Q}, for quotient. So \displaystyle \frac{3}{4} \in \mathbb{Q} is read “three quarters is in the set of rational numbers”. The word rational comes from ratio, a comparison of two whole amounts; it has nothing to do with being sensible.
Three parts of the definition do the work.
- “Can be written.” One name of the right form is enough. A number does not have to be shown as a fraction right now; 0.75 is rational because it is also \displaystyle \frac{3}{4}.
- “Integer.” The top and the bottom are whole numbers or their negatives. A fraction bar with a decimal or a square root above or below it does not count by itself, though the number it names may still have another name that does.
- “Nonzero.” A denominator of 0 names no number at all, so it is ruled out.
| The number | One name as integer over nonzero integer |
|---|---|
| 5 | \displaystyle \frac{5}{1} |
| -2 | \displaystyle \frac{-2}{1} |
| 0 | \displaystyle \frac{0}{1} |
| 0.3 | \displaystyle \frac{3}{10} |
| -1.25 | \displaystyle \frac{-5}{4} |
| 2\tfrac{1}{3} | \displaystyle \frac{7}{3} |
Intuition
The fraction \displaystyle \frac{a}{b} with b positive says: cut the distance from 0 to 1 into b equal parts, then count a of those parts from 0, to the right if a is positive and to the left if a is negative. Every rational number is a point you can reach that way. Figure 1 marks four such points.
Each point carries many names, and that is the reason the definition says “can be”. Multiplying the top and the bottom of a fraction by the same nonzero integer gives a new name for the same number, which is what Equivalent fractions are, so every rational number has endlessly many names: \displaystyle \frac{3}{4} = \frac{6}{8} = \frac{30}{40} = 0.75. Among them exactly one is in lowest terms with a positive denominator, and that is the name usually given as the answer.
A minus sign can sit in three places without changing the number:
All three say “3 divided by 2, then made negative”, because dividing numbers with opposite signs gives a negative result. When both top and bottom are negative, the result is positive: \displaystyle \frac{-3}{-5} = \frac{3}{5}.
Rational numbers crowd the number line. Between any two of them there is another, their average, so there is no “next” rational number after \displaystyle \frac{1}{2} (see Finding a fraction between two fractions). Even so, they do not fill the line. There are points on it, such as the one at distance \displaystyle \sqrt{2} from 0, that no fraction of integers names. That is the story of Irrational numbers.
Results
Every integer is rational. For any integer n,
So the whole numbers sit inside the integers, and the integers sit inside the rational numbers, as in Figure 2. The reverse fails at each step: -7 is an integer but not a whole number, and \displaystyle \frac{3}{4} is rational but not an integer.
Every terminating decimal is rational. A decimal with k digits after the point is an integer divided by 10^k. For example \displaystyle 4.07 = \frac{407}{100}.
Every repeating decimal is rational, and every rational number’s decimal either terminates or repeats. So a decimal that goes on forever with no block of digits repeating names a number that is not rational. The bar notation for repeating decimals is in Terminating and repeating decimals, and the way back from a repeating decimal to a fraction is in Turning a repeating decimal into a fraction.
Rational numbers are closed under the four operations. Adding, subtracting, multiplying or dividing two rational numbers gives a rational number, as long as you never divide by 0. With b and d nonzero integers:
Testing whether two names give the same number. For nonzero integers b and d,
Why it is true
Integers. The fraction \displaystyle \frac{n}{1} means n parts, each of size 1, which is n. Its top is the integer n and its bottom is the nonzero integer 1, so it meets the definition.
Terminating decimals. Place value turns each digit after the point into tenths, hundredths and so on, and all of these can be put over the last place used:
In general the digits, read with the point removed, form an integer, and the denominator is 10^k, which is a nonzero integer.
Closure. In each formula above, the new top and the new bottom are made from integers by adding, subtracting and multiplying only, and those operations on integers always give integers. The new bottom is never 0: in bd neither factor is 0, and a product of nonzero numbers is nonzero. Division needs c \neq 0 as well, so that bc \neq 0; that is the one exception. The formulas themselves come from rewriting both fractions over the common denominator bd for adding and subtracting, multiplying tops and bottoms for products, and multiplying by the reciprocal \displaystyle \frac{d}{c} for quotients.
The same-number test. Multiplying both sides of \displaystyle \frac{a}{b} = \frac{c}{d} by bd gives ad = bc, and dividing ad = bc by bd (which is not 0) gives back \displaystyle \frac{a}{b} = \frac{c}{d}. Each equation forces the other.
Why a fraction’s decimal must stop or repeat. Divide a by b by long division. Once the digits of a are used up, each new step brings down a 0 and divides the current remainder, so what happens next depends only on that remainder. A remainder after dividing by b is one of 0, 1, 2, \ldots, b - 1. If it is ever 0, the division stops and the decimal terminates. If not, there are only b - 1 possible remainders, so within b steps one comes round again, and from then on the same steps, and the same digits, repeat forever. For \displaystyle \frac{3}{7} the remainders run 2, 6, 4, 5, 1, 3 and then 2 again, so the digits 428571 repeat: \displaystyle \frac{3}{7} = 0.428571428571\ldots
That repeating decimals are always rational is shown in Turning a repeating decimal into a fraction: the method there turns any repeating decimal into an integer over a nonzero integer.
Worked examples
Integers and mixed numbers
Show that each number is rational by writing it as an integer over a nonzero integer: 12, -5, 0, 3\tfrac{1}{4} and -2\tfrac{2}{3}.
SolutionAn integer goes over 1: \displaystyle 12 = \frac{12}{1} and \displaystyle -5 = \frac{-5}{1}. Zero is \displaystyle \frac{0}{1}; any nonzero denominator would do, since 0 parts of any size is 0.
A mixed number is a whole number plus a fraction. Write the whole part in the same parts as the fraction, then add:
In -2\tfrac{2}{3} the minus sign applies to the whole mixed number, so it means \displaystyle -(2 + \frac{2}{3}):
All five are rational.
Terminating decimals and a percent
Write each as an integer over a nonzero integer in lowest terms: 4.07, -2.35, 0.0008 and 6.5\%.
SolutionRead each decimal by its last place.
4.07 ends in the hundredths place, so \displaystyle 4.07 = \frac{407}{100}. Since 407 = 11 \times 37 and 100 = 2^2 \times 5^2 share no factor, this is already in lowest terms.
-2.35 ends in hundredths: \displaystyle -2.35 = -\frac{235}{100}. Dividing top and bottom by their common factor 5 gives \displaystyle -\frac{47}{20}.
0.0008 ends in the ten-thousandths place: \displaystyle \frac{8}{10000}. Dividing top and bottom by 8 gives \displaystyle \frac{1}{1250}.
A percent means “out of a hundred”, so \displaystyle 6.5\% = \frac{6.5}{100}. The top is not an integer yet, so multiply top and bottom by 10, which gives an equivalent fraction:
Same number, different names
Which of these name the same number as 0.6?
First give 0.6 a fraction name: \displaystyle 0.6 = \frac{6}{10} = \frac{3}{5}. Now compare each with \displaystyle \frac{3}{5}.
\displaystyle \frac{9}{15}: dividing top and bottom by 3 gives \displaystyle \frac{3}{5}. Same number.
0.60: a 0 added at the end changes nothing, since \displaystyle 0.60 = \frac{60}{100} = \frac{3}{5}. Same number.
\displaystyle \frac{-3}{-5}: a negative divided by a negative is positive, so this is \displaystyle \frac{3}{5}. Same number.
\displaystyle \frac{2}{3}: by the same-number test, compare 2 \times 5 = 10 with 3 \times 3 = 9. They differ, so \displaystyle \frac{2}{3} \neq \frac{3}{5}.
\displaystyle \frac{15}{25}: dividing by 5 gives \displaystyle \frac{3}{5}. Same number.
\displaystyle \frac{6}{100}: this is six hundredths, 0.06, not 0.6. Different number.
So \displaystyle \frac{9}{15}, 0.60, \displaystyle \frac{-3}{-5} and \displaystyle \frac{15}{25} all name 0.6.
A fraction bar with other things in it
Decide whether each number is rational. If it is, write it as an integer over a nonzero integer.
None of these is written as integer over integer, so the question is whether some other name is.
(a) Multiplying top and bottom by 100 keeps the value and clears both decimals: \displaystyle \frac{1.2}{0.08} = \frac{120}{8} = 15 = \frac{15}{1}. Rational.
(b) \displaystyle \sqrt{36} = 6, since 6^2 = 36. So \displaystyle \frac{\sqrt{36}}{10} = \frac{6}{10} = \frac{3}{5}. Rational.
(c) This is \displaystyle \frac{2}{3} \div \frac{5}{6}. Dividing by a fraction is multiplying by its reciprocal: \displaystyle \frac{2}{3} \times \frac{6}{5} = \frac{12}{15} = \frac{4}{5}. Rational.
(d) Suppose \displaystyle \frac{\sqrt{2}}{2} were equal to some \displaystyle \frac{p}{q} with integers p and q \neq 0. Multiplying both sides by 2 would give \displaystyle \sqrt{2} = \frac{2p}{q}, an integer over a nonzero integer, which would make \displaystyle \sqrt{2} rational. But \displaystyle \sqrt{2} is not rational, as Irrational numbers proves. So \displaystyle \frac{\sqrt{2}}{2} is not rational either, even though it is written with a fraction bar.
Closure at work
Write each result as an integer over a nonzero integer in lowest terms.
Closure says each answer must be rational before any work is done; the work finds which one.
(a) The common denominator is 21: \displaystyle \frac{1}{3} + \frac{2}{7} = \frac{7}{21} + \frac{6}{21} = \frac{13}{21}.
(b) \displaystyle 0.25 = \frac{1}{4}, so the product is \displaystyle \frac{1}{4} \times \frac{-8}{3} = \frac{-8}{12} = -\frac{2}{3}.
(c) Inside the brackets, over 12: \displaystyle \frac{5}{6} - \frac{1}{4} = \frac{10}{12} - \frac{3}{12} = \frac{7}{12}. Then \displaystyle -0.5 = -\frac{1}{2}, and dividing by it means multiplying by -2:
Reading the decimal
Decide whether each number is rational.
(a) 0.272727\ldots, where the block 27 repeats forever.
(b) 0.1010010001\ldots, where each 1 is followed by one more 0 than the last.
(c) 5.125
(d) 3.14, and the number \pi
(e) the number a calculator shows for \displaystyle \sqrt{2}: 1.414213562
Solution(a) Rational, because the decimal repeats. In fact dividing 3 by 11 gives 0.2727\ldots, so it is \displaystyle \frac{3}{11}.
(b) Not rational. Suppose that from some point on a block of k digits repeated. The runs of zeros keep growing, so somewhere in that repeating part there is a run of at least k zeros, and any k digits in a row there contain every digit of the block. So the block would be all zeros, and every digit from then on would be 0. But a 1 keeps appearing after every run. The decimal neither stops nor repeats, so by the results above it is not rational. It has a pattern, but a pattern is not the same as a repeating block.
(c) Rational: \displaystyle 5.125 = \frac{5125}{1000} = \frac{41}{8}.
(d) 3.14 is rational: \displaystyle 3.14 = \frac{314}{100} = \frac{157}{50}. The number \pi is not rational, a fact that has been proved, though the proof is well beyond this lesson. So 3.14 is a rational number close to \pi, not \pi itself.
(e) The display 1.414213562 is a terminating decimal, so it is rational: it is \displaystyle \frac{1414213562}{10^{9}}. It is not \displaystyle \sqrt{2}. A number ending in the digit 2 has a square ending in 4, so the square of 1.414213562 ends in a 4 in its last decimal place and cannot be exactly 2. The calculator shows a rational number near \displaystyle \sqrt{2} because a screen can only show finitely many digits.
Solving keeps you inside the rational numbers
A number n satisfies
Show that n is rational and find it.
SolutionAdd \displaystyle \frac{2}{5} to both sides; adding the same number to both sides keeps the equation true:
Multiply both sides by \displaystyle \frac{4}{3}, the reciprocal of \displaystyle \frac{3}{4}:
Check. \displaystyle \frac{3}{4} \times \frac{5}{3} = \frac{5}{4} = 1.25, and 1.25 - 0.4 = 0.85.
The same reasoning works for any equation ax + b = c in which a, b and c are rational and a \neq 0. Its solution is \displaystyle x = \frac{c - b}{a}, made from rational numbers by one subtraction and one division by a nonzero number, so by closure it is rational. Equations that need more than the four operations can leave the rational numbers: x^2 = 2 has no rational solution, and solving it is what pushes mathematics out to Irrational and real numbers.
Boundaries
The denominator must not be 0. “Integer over integer” alone is not enough. \displaystyle \frac{5}{0} would have to be a number that gives 5 when multiplied by 0, and there is none, since every number times 0 is 0. \displaystyle \frac{0}{0} fails the other way: every number times 0 is 0, so no single number is named. A 0 on top is fine: \displaystyle \frac{0}{5} = 0.
A fraction bar does not make a number rational. \displaystyle \frac{\pi}{4} and \displaystyle \frac{\sqrt{2}}{2} are written as fractions, but no name of either is an integer over a nonzero integer. The test is about the number, not about how it happens to be written: \displaystyle \frac{1.2}{0.08} looks untidy and is the integer 15.
A screen cannot settle the question. A display shows finitely many digits, so it always shows a terminating decimal, and that decimal is rational whatever number it came from. Whether a number is rational depends on the rule that produces all of its digits, not on the first ten of them.
Closure runs one way only. Two rational numbers always combine to a rational number, but two numbers that are not rational can combine to one that is: \displaystyle \sqrt{2} \times \sqrt{2} = 2, and \displaystyle \sqrt{2} + (-\sqrt{2}) = 0.
Where this goes next
Everything else about rational numbers builds on this test. Their decimals either stop or repeat, and Terminating and repeating decimals gives the bar notation for the repeating ones, while Which fractions give terminating decimals tells from the denominator alone which will stop. Turning a repeating decimal into a fraction runs the trip backwards. Rational numbers can sit left of zero (Negative fractions and decimals on the number line), can be compared and put in order (Comparing and ordering rational numbers), and because they are closed under the four operations, one set of rules covers adding and subtracting them (Operations with rational numbers) and multiplying and dividing them (Multiplying and dividing rational numbers).
The edge of the test matters as much as the test. Numbers such as \displaystyle \sqrt{2} and \pi fail it, and Irrational numbers proves the first case. Together with the rational numbers they make up the real numbers, which fill the The real number line with no gaps, and Sorting a number into its sets sorts any number into whole, integer, rational or irrational.
Exercises
Writing a number as integer over integer
Write each number as an integer over a nonzero integer in lowest terms.
- -14
- 5\tfrac{2}{7}
- -1\tfrac{5}{6}
- 0.45
- -3.008
- 12.5\%
Same number or not
- Do \displaystyle \frac{14}{35} and 0.4 name the same number?
- Do \displaystyle \frac{-4}{6} and \displaystyle \frac{6}{-9} name the same number?
- Do 0.35 and \displaystyle \frac{7}{25} name the same number?
- Give three different names for -0.75, each an integer over a nonzero integer, and say which name is in lowest terms with a positive denominator.
Rational or not
Decide whether each number is rational. If it is, write it as an integer over a nonzero integer.
- \displaystyle \frac{1.5}{0.25}
- \displaystyle \frac{\sqrt{49}}{\sqrt{4}}
- \displaystyle \frac{\pi}{4}, given that \pi is not rational
- 0.363636\ldots, where the block 36 repeats forever
- \displaystyle \frac{0}{-8}
- \displaystyle \frac{7}{0}
Combining rational numbers
Write each result as an integer over a nonzero integer in lowest terms.
- \displaystyle \frac{2}{5} - \frac{3}{4}
- \displaystyle 1.6 \times \frac{5}{12}
- \displaystyle -0.3 \div \frac{9}{10}
- The solution of \displaystyle \frac{2}{3}x + \frac{1}{2} = 0.75
Reasoning
- Explain why every integer is a rational number, and give an example of a rational number that is not an integer.
- Show that the average of two rational numbers is always rational.
- Show that if r is rational and x is not, then r + x is not rational.
- A decimal is built as 0.12112111211112\ldots: a 2 comes after one 1, then after two 1s, then after three, and so on. Explain why this number is not rational.
- A calculator shows 0.333333333. Is the number on the screen rational? Is it equal to \displaystyle \frac{1}{3}?
Answers
- \displaystyle \frac{-14}{1}. Every integer is itself over 1.
- \displaystyle \frac{37}{7}. \displaystyle 5\tfrac{2}{7} = \frac{35}{7} + \frac{2}{7}.
- \displaystyle -\frac{11}{6}. \displaystyle -1\tfrac{5}{6} = -\left(\frac{6}{6} + \frac{5}{6}\right).
- \displaystyle \frac{9}{20}. \displaystyle 0.45 = \frac{45}{100}, then divide top and bottom by 5.
- \displaystyle -\frac{376}{125}. \displaystyle -3.008 = -\frac{3008}{1000}, then divide top and bottom by 8; 125 = 5^3 and 376 = 8 \times 47 share no factor.
- \displaystyle \frac{1}{8}. \displaystyle 12.5\% = \frac{12.5}{100} = \frac{125}{1000}, then divide by 125.
- Yes. \displaystyle \frac{14}{35} = \frac{2}{5} after dividing by 7, and \displaystyle 0.4 = \frac{4}{10} = \frac{2}{5}.
- Yes. Both are \displaystyle -\frac{2}{3}: divide \displaystyle \frac{-4}{6} by 2 and \displaystyle \frac{6}{-9} by 3; each has one negative part, so each is negative.
- No. \displaystyle 0.35 = \frac{35}{100} = \frac{7}{20}, while \displaystyle \frac{7}{25} = \frac{28}{100} = 0.28.
- For example \displaystyle \frac{-75}{100}, \displaystyle \frac{-6}{8} and \displaystyle \frac{3}{-4}, all equal to -0.75. The one in lowest terms with a positive denominator is \displaystyle \frac{-3}{4}, usually written \displaystyle -\frac{3}{4}.
- Rational: \displaystyle \frac{1.5}{0.25} = \frac{150}{25} = 6 = \frac{6}{1}, multiplying top and bottom by 100.
- Rational: \displaystyle \frac{\sqrt{49}}{\sqrt{4}} = \frac{7}{2}, since 7^2 = 49 and 2^2 = 4.
- Not rational. If \displaystyle \frac{\pi}{4} = \frac{p}{q} with integers p and q \neq 0, then \displaystyle \pi = \frac{4p}{q} would be rational, and it is not.
- Rational, because the decimal repeats. It equals \displaystyle \frac{36}{99} = \frac{4}{11}, as Turning a repeating decimal into a fraction shows; dividing 4 by 11 gives 0.3636\ldots
- Rational: \displaystyle \frac{0}{-8} = 0 = \frac{0}{1}.
- Not rational, because it is not a number at all. No number times 0 gives 7.
- \displaystyle -\frac{7}{20}. \displaystyle \frac{8}{20} - \frac{15}{20} = -\frac{7}{20}.
- \displaystyle \frac{2}{3}. \displaystyle 1.6 = \frac{8}{5}, and \displaystyle \frac{8}{5} \times \frac{5}{12} = \frac{40}{60} = \frac{2}{3}.
- \displaystyle -\frac{1}{3}. \displaystyle -\frac{3}{10} \times \frac{10}{9} = -\frac{30}{90} = -\frac{1}{3}.
- \displaystyle x = \frac{3}{8}. Subtract \displaystyle \frac{1}{2} from both sides: \displaystyle \frac{2}{3}x = \frac{3}{4} - \frac{1}{2} = \frac{1}{4}. Multiply by \displaystyle \frac{3}{2}: \displaystyle x = \frac{3}{8}.
- Any integer n equals \displaystyle \frac{n}{1}, an integer over the nonzero integer 1. The number \displaystyle \frac{1}{2} is rational but not an integer: it lies strictly between 0 and 1, and no integer does.
- If r and s are rational, then r + s is rational by closure, and \displaystyle 2 = \frac{2}{1} is a nonzero rational number, so \displaystyle \frac{r + s}{2} is rational too. With \displaystyle r = \frac{a}{b} and \displaystyle s = \frac{c}{d} it is \displaystyle \frac{ad + bc}{2bd}.
- Suppose r + x were some rational number s. Then x = s - r would be a difference of two rational numbers, which is rational by closure. That contradicts x not being rational, so r + x is not rational.
- Suppose a block of k digits repeated from some point on. The runs of 1s keep growing, so the repeating part contains a run of at least k ones, and any k digits in a row there contain every digit of the block. So the block would be all 1s and no 2 could appear after that point, but a 2 appears after every run. The decimal neither stops nor repeats, so the number is not rational.
- The number on the screen is rational: it is the terminating decimal \displaystyle \frac{333333333}{1000000000}. It is not \displaystyle \frac{1}{3}: 3 \times 333333333 = 999999999, so three times the display is 0.999999999, not 1. The display is a rational number close to \displaystyle \frac{1}{3}, and from the screen alone you cannot tell what number it was cut from.