Rational numbers

Numbers · level 5, lesson 2 of 7

A question to start

What is the next whole number after 3? Easy: 4. What is the next integer after \(-3\)? Also easy: \(-2\).

Now try this. What is the next fraction after 0? The smallest fraction bigger than zero. Is it \(\tfrac{1}{10}\)? No, \(\tfrac{1}{100}\) is smaller. Is it \(\tfrac{1}{1\,000\,000}\)? No, \(\tfrac{1}{2\,000\,000}\) is smaller still. However small a fraction you name, someone can name a smaller one. So there is no "next" fraction. Fractions are packed together in a way whole numbers are not. This lesson is about that family of numbers, negative ones included.

The idea

What a rational number is

A rational number is any number you can write as a fraction

\[\frac{p}{q}\]

where \(p\) and \(q\) are integers (whole numbers, positive, negative or zero) and \(q\) is not 0. The word comes from ratio, a comparison of two whole amounts.

Many numbers you already know are rational:

• Every integer. \(5 = \tfrac51\) and \(-4 = \tfrac{-4}{1}\) and \(0 = \tfrac01\).
• Every fraction, such as \(\tfrac23\) or \(\tfrac{17}{5}\).
• Every decimal that stops. \(0.35 = \tfrac{35}{100} = \tfrac{7}{20}\) and \(-1.2 = -\tfrac{12}{10} = -\tfrac65\).
• Negative fractions, such as \(-\tfrac34\).

Mathematicians write \(\mathbb{Q}\) for the set of all rational numbers (\(\mathbb{Q}\) for quotient, the answer to a division).

Where the minus sign goes

\(-\tfrac34\) can be written three ways, and they are all the same number:

\[-\frac34 = \frac{-3}{4} = \frac{3}{-4}\]

Why? Each one means "3 divided by 4, with one negative sign", and from the last lesson, dividing with different signs gives a negative answer. Usually we put the minus sign in front, or on the top, and keep the bottom positive.

What about \(\tfrac{-3}{-4}\)? Two negatives, same signs, so it is positive: \(\tfrac{-3}{-4} = \tfrac34\).

Rational numbers on the number line

Every rational number has its own point on the number line. Negative ones sit to the left of 0, as mirror images of the positive ones.

To place \(-\tfrac52\): it is \(-2\tfrac12\), so go 2 whole steps left of 0 and then half a step more. To place \(-\tfrac34\): cut the gap between 0 and \(-1\) into four equal parts and count three to the left.

Comparing negative fractions

On the number line, the number further left is smaller. For negative numbers that means the one with the bigger size is the smaller number: \(-5\) is less than \(-2\), because 5 below zero is colder than 2 below.

To compare \(-\tfrac34\) and \(-\tfrac23\), give them a common denominator, just as for positive fractions:

\[-\frac34 = -\frac{9}{12} \qquad -\frac23 = -\frac{8}{12}\]

\(-\tfrac{9}{12}\) is further from 0, so it is further left. So \(-\tfrac34 < -\tfrac23\). Notice that this is the opposite order to \(\tfrac34 > \tfrac23\). Flipping to the negative side of 0 reverses the order, just like a mirror reverses left and right.

Adding and subtracting

Nothing new is needed. Use a common denominator (level 4), then add or subtract the tops using the integer rules (level 4 again).

\[-\frac23 + \frac14 = -\frac{8}{12} + \frac{3}{12} = \frac{-8 + 3}{12} = -\frac{5}{12}\]

On the number line: start at \(-\tfrac{8}{12}\) and move \(\tfrac{3}{12}\) to the right. You stop at \(-\tfrac{5}{12}\), still left of 0.

Subtracting a negative is adding: \(\tfrac12 - \left(-\tfrac13\right) = \tfrac12 + \tfrac13 = \tfrac56\).

Multiplying and dividing

Again, nothing new: multiply or divide the sizes as fractions, then use the sign rules from lesson 1.

\[\left(-\frac34\right) \times \frac89 = -\frac{24}{36} = -\frac23 \qquad \text{(different signs: negative)}\]

\[\left(-\frac56\right) \div \left(-\frac{10}{3}\right) = \frac56 \times \frac{3}{10} = \frac{15}{60} = \frac14 \qquad \text{(same signs: positive)}\]

Closed under the four operations

Here is something special about rational numbers. Add, subtract, multiply or divide two of them (but never divide by 0), and the answer is always another rational number. Why? Because each operation on fractions builds a new fraction whose top and bottom are made by adding, subtracting and multiplying integers, and those always give integers. For example

\[\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}\]

and \(ad + bc\) and \(bd\) are integers, with \(bd \ne 0\).

We say the rational numbers are closed under the four operations: you can never calculate your way out of the family. The whole numbers are not closed under subtraction (\(3 - 5\) is not a whole number), and the integers are not closed under division (\(3 \div 5\) is not an integer). The rationals fix both problems at once. (In level 6 you will find an operation that does take you out of the rationals: the square root.)

There is always another rational in between

Take any two rational numbers, say \(a\) and \(b\) with \(a < b\). Their average (mean),

\[\frac{a + b}{2}\]

is halfway between them, so it is bigger than \(a\) and smaller than \(b\). And it is rational, because adding and dividing by 2 keeps you in the family. So between any two rational numbers there is another rational number.

Then do it again: between \(a\) and the average there is another, and another, without end. So between any two rational numbers there are infinitely many rational numbers. This is why there is no "next" fraction after 0: whatever fraction you name, half of it is smaller and still bigger than 0. We say the rationals are dense.

Worked examples

1. Compare \(-\tfrac25\) and \(-\tfrac37\).

1. Common denominator 35: \(-\tfrac25 = -\tfrac{14}{35}\) and \(-\tfrac37 = -\tfrac{15}{35}\).
2. \(-\tfrac{15}{35}\) is further from 0, so it is further left and smaller.
3. So \(-\tfrac37 < -\tfrac25\). (For the positive versions it is the other way: \(\tfrac37 > \tfrac25\).)

2. Add mixed numbers with different signs: \(-2\tfrac12 + 3\tfrac13\).

1. Change to improper fractions. \(-2\tfrac12\) means \(-\left(2 + \tfrac12\right) = -\tfrac52\). And \(3\tfrac13 = \tfrac{10}{3}\).
2. Common denominator 6: \(-\tfrac{15}{6} + \tfrac{20}{6} = \tfrac{5}{6}\).
3. Sense check: a bit more than 3 take away a bit more than 2 should be a bit less than 1. ✓

3. Divide: \(\tfrac79 \div \left(-\tfrac{14}{3}\right)\).

1. Multiply by the reciprocal: \(\tfrac79 \times \left(-\tfrac{3}{14}\right)\).
2. Sizes: \(\tfrac{7 \times 3}{9 \times 14} = \tfrac{21}{126} = \tfrac16\) (divide top and bottom by 21).
3. Different signs, so the answer is \(-\tfrac16\).

4. A rational number between two others. Find a rational number between \(\tfrac13\) and \(\tfrac12\).

1. Average: \(\dfrac{\tfrac13 + \tfrac12}{2} = \dfrac{\tfrac56}{2} = \tfrac{5}{12}\).
2. Check with twelfths: \(\tfrac13 = \tfrac4{12}\) and \(\tfrac12 = \tfrac6{12}\), and \(\tfrac5{12}\) is between them. ✓
3. Another way to find many at once: write both with a big denominator. \(\tfrac13 = \tfrac{20}{60}\) and \(\tfrac12 = \tfrac{30}{60}\), so \(\tfrac{21}{60}, \tfrac{22}{60}, \ldots, \tfrac{29}{60}\) are nine more.

5. Show a decimal is rational. Write \(-0.375\) as a fraction \(\tfrac{p}{q}\).

\(-0.375 = -\tfrac{375}{1000}\). Divide top and bottom by 125: \(-\tfrac{3}{8}\). So it is rational, with \(p = -3\) and \(q = 8\).

Try it

1. Put in order from smallest to largest: \(-\tfrac12\), \(-\tfrac43\), \(0.75\), \(-0.2\), \(\tfrac54\).
2. Which is larger, \(-\tfrac56\) or \(-\tfrac79\)?
3. Work out \(-\tfrac38 + \tfrac56\).
4. Work out \(-\tfrac14 - \tfrac23\).
5. Work out \(\left(-\tfrac45\right) \times \left(-\tfrac{15}{16}\right)\).
6. Work out \(-1\tfrac12 \div \tfrac34\).
7. Find three different rational numbers between \(-\tfrac13\) and \(-\tfrac14\).
8. Is there a smallest positive rational number? Give a reason.

Answers

1. \(-\tfrac43 \approx -1.33\), then \(-\tfrac12 = -0.5\), then \(-0.2\), then \(0.75\), then \(\tfrac54 = 1.25\).
2. Common denominator 18: \(-\tfrac56 = -\tfrac{15}{18}\) and \(-\tfrac79 = -\tfrac{14}{18}\). \(-\tfrac{14}{18}\) is closer to 0, so \(-\tfrac79\) is larger.
3. Common denominator 24: \(-\tfrac{9}{24} + \tfrac{20}{24} = \tfrac{11}{24}\).
4. Common denominator 12: \(-\tfrac{3}{12} - \tfrac{8}{12} = -\tfrac{11}{12}\).
5. Same signs, so positive. \(\tfrac{4 \times 15}{5 \times 16} = \tfrac{60}{80} = \tfrac34\).
6. \(-1\tfrac12 = -\tfrac32\). Then \(-\tfrac32 \times \tfrac43 = -\tfrac{12}{6} = -2\). Check: \(-2 \times \tfrac34 = -\tfrac32\). ✓
7. Write both with denominator 48: \(-\tfrac13 = -\tfrac{16}{48}\) and \(-\tfrac14 = -\tfrac{12}{48}\). Three numbers between them: \(-\tfrac{15}{48} = -\tfrac{5}{16}\), \(-\tfrac{14}{48} = -\tfrac{7}{24}\) and \(-\tfrac{13}{48}\). (Many other answers are right.)
8. No. Suppose \(\tfrac{p}{q}\) were the smallest positive rational. Then \(\tfrac{p}{2q}\) is also positive and rational, and it is half as big, so it is smaller. That contradicts "smallest". So there is no smallest one.

Common mistake

Reading a negative mixed number wrongly. \(-2\tfrac12\) means \(-\left(2 + \tfrac12\right) = -2.5\). It does not mean \(-2 + \tfrac12\), which is \(-1.5\). The minus sign belongs to the whole mixed number. Changing to an improper fraction first (\(-\tfrac52\)) avoids the trap.

Comparing negative fractions by size alone. \(\tfrac34 > \tfrac23\), so it is tempting to say \(-\tfrac34 > -\tfrac23\). But on the negative side the order flips: \(-\tfrac34\) is further from 0, so it is smaller. Picture the number line, or think of temperatures.

Where it leads

• Back: Multiplying and dividing integers · Next: Powers
• Level 5 page · Numbers topic page
• Every fraction can be written in lowest terms in exactly one way, using the greatest common factor: see Number theory.
• "Between any two there is another, so there are infinitely many": the start of thinking about different sizes of infinity in Infinity.
• In level 6 you will meet numbers that are not rational, and see that, dense as they are, the rationals still leave holes in the number line.
• The density of the rationals, proved carefully at university: MAT158 Unit 2.