The real number line
Fractions crowd the number line so tightly that there is always another one between any two. Do they fill the line, and if not, what fills the rest?
Where you are 5 in Irrational and real numbers
Why this exists
A The number line starts from two choices: a point called 0 and a point to its right called 1. Copies of the stretch from 0 to 1, laid end to end in both directions, place the integers, and cutting that stretch into equal parts places every fraction, which is how rational numbers get their points.
Fractions are crowded. Between \displaystyle \frac{1}{3} and \displaystyle \frac{1}{2} sits \displaystyle \frac{5}{12}, between \displaystyle \frac{1}{3} and \displaystyle \frac{5}{12} sits \displaystyle \frac{3}{8}, and the squeezing never stops. It is natural to guess that a line so packed with fractions has no room left for anything else.
The guess is wrong. Draw a square on the stretch from 0 to 1, take its diagonal, and swing the diagonal down onto the line with a compass. Its far end lands on a definite point, and no fraction names that point. Its distance from 0 is \displaystyle \sqrt{2}, and Irrational numbers proves that \displaystyle \sqrt{2} is not a fraction. So the fractions, crowded as they are, leave holes.
The real numbers are what you get by refusing to leave any holes: one number for every point on the line. This lesson is about that line, what fills it, and the order along it.
Definition
Fix a line, a point on it named 0, and a point to its right named 1. The distance from 0 to 1 is the unit of length.
The real number line matches points with numbers. A point at distance d to the right of 0 is the number d, a point at distance d to the left of 0 is the number -d, and the point 0 is the number 0. Every point gets exactly one number this way, and every number gets exactly one point.
A real number is any number that names a point on this line. Written as a decimal, a real number may stop, may repeat a block of digits forever, or may go on forever with no repeating block; every one of these names a point. The set of all real numbers is written \mathbb{R}.
Every real number is one of two kinds, and never both:
- it is rational if it can be written as \displaystyle \frac{a}{b} with a and b integers and b \neq 0;
- it is irrational if it cannot. \displaystyle \sqrt{2} is the first irrational number most people meet.
How the sets sit inside one another is drawn in What makes a number rational, and deciding which sets a given number belongs to is Sorting a number into its sets.
Intuition
Figure 1 is the construction from the opening. Call the diagonal’s length c. A compass centered at 0 draws a circle, every point of which is c from 0, so where the pencil meets the line it marks the number c.
The The Pythagorean theorem, which comes later, gives c. The diagonal cuts the square into two right triangles with shorter sides 1 and 1, and the squares on the two shorter sides add up to the square on the longest side, so
and \displaystyle c = \sqrt{2}. The compass never needs the digits 1.41421\ldots; it puts \displaystyle \sqrt{2} on the line exactly. Swinging the pencil the other way, up and over to the left, marks \displaystyle -\sqrt{2}.
Figure 2 shows how crowded the fractions are. Halfway from \displaystyle \frac{1}{3} to \displaystyle \frac{1}{2} is their average, \displaystyle \frac{5}{12}. Halfway from \displaystyle \frac{1}{3} to that is \displaystyle \frac{3}{8}, then \displaystyle \frac{17}{48}, and the halving never runs out.
Every point in Figure 2 is a fraction. Yet the same stretch also holds irrational points, and so does every other stretch, however short. The rational numbers alone make a line full of pinholes; adding the irrational numbers fills every one of them.
Results
Rational and irrational fill the line. Each point is exactly one real number, each real number is rational or irrational and never both, and the two kinds together fill the whole line.
The real numbers are ordered. For any two real numbers a and b, exactly one of these is true:
a < b means that a lies to the left of b. Order passes along a chain: if a < b and b < c, then a < c.
The average lies between. If a < b, then
When a and b are both rational, so is their average. So between any two different rational numbers there is another rational number, and repeating the step gives infinitely many. No rational number has a “next” rational number, and no real number has a next real number either.
An irrational number lies between too. Between any two different rational numbers there are irrational numbers as well as rational ones.
Why it is true
The average lies between. Start from a < b. Adding a to both sides gives 2a < a + b, and adding b to both sides gives a + b < 2b. Together,
and halving all three parts keeps the order. That is \displaystyle a < \frac{a + b}{2} < b. Nothing in this needs a and b to be rational, so it works for any two real numbers.
The average of two rational numbers is rational. Write \displaystyle a = \frac{p}{q} and \displaystyle b = \frac{r}{s} with p, q, r, s integers and q, s \neq 0. Then
The top is an integer and the bottom is a nonzero integer, so the average is rational.
An irrational number between. Example 4 builds one as a decimal that neither stops nor repeats. There is also a formula. For rational a < b, the number
lies strictly between them: \displaystyle \sqrt{2} is more than 1, so dividing the gap b - a by it gives a positive number smaller than the gap. It is irrational because \displaystyle \frac{b - a}{\sqrt{2}} equals \displaystyle \frac{b - a}{2} \times \sqrt{2}, a nonzero rational number times \displaystyle \sqrt{2}, and a rational number plus a nonzero rational times an irrational number is always irrational. That fact is proved in Irrational numbers.
What is not proved here. That the line has no holes, so that every endless decimal names a point, is taken as given: it is the property that makes a line a line, made precise much later through Limits.
Worked examples
A rational number between two fractions
Find a rational number between \displaystyle \frac{2}{5} and \displaystyle \frac{3}{7}.
SolutionTheir average lies between them and is rational, so compute it. Over the common denominator 35:
Halving gives the average:
Check. Over the denominator 70, \displaystyle \frac{2}{5} = \frac{28}{70} and \displaystyle \frac{3}{7} = \frac{30}{70}, and 28 < 29 < 30. So \displaystyle \frac{29}{70} lies between them.
Over 35 there was no whole number of thirty-fifths between them; doubling the denominator opened room.
Three rational numbers between two negative decimals
Find three rational numbers between -1.2 and -1.1, and write all five numbers in order from least to greatest.
SolutionThe average of the two given numbers is
It lies between them, and it splits the stretch into two halves. Average again inside each half:
On the line, a negative number further from 0 lies further to the left, so it is the smaller one. In order:
Constructing the square root of 17
Construct the point \displaystyle \sqrt{17} on the number line with a compass and straightedge.
SolutionLook for a right triangle whose shorter sides have squares adding to 17. Since 4^2 + 1^2 = 16 + 1 = 17, legs of 4 and 1 will do.
Mark the point 4 on the line. At 4, stand a segment of length 1 straight up, at a right angle to the line (Figure 3). The segment from 0 to the top of it is the longest side of a right triangle with legs 4 and 1, so its length c satisfies
and \displaystyle c = \sqrt{17}. With the compass point at 0 and the pencil at the top of the upright segment, swing down to the line. The pencil stays \displaystyle \sqrt{17} from 0, so it lands on the point \displaystyle \sqrt{17}, a little to the right of 4. Swinging the other way marks \displaystyle -\sqrt{17}.
The same plan works for any whole number that is a sum of two squares, and Exercise 14 shows a chain of steps that reaches the square root of every other whole number too.
An irrational number between two fractions
Find an irrational number between \displaystyle \frac{1}{4} and \displaystyle \frac{1}{3}.
SolutionIn decimals, \displaystyle \frac{1}{4} = 0.25 and \displaystyle \frac{1}{3} = 0.333\ldots. A rational number’s decimal either stops or repeats a block forever (What makes a number rational), so a decimal that does neither is irrational. Build one with a pattern that keeps changing, one more 0 before each 1:
It never stops, and no block can repeat forever, because the runs of zeros keep getting longer. It starts 0.30, so it is at least 0.30 and less than 0.31. Since 0.25 < 0.30 and 0.31 < 0.333\ldots, it lies between \displaystyle \frac{1}{4} and \displaystyle \frac{1}{3}.
Another way. The formula from Why it is true, with the gap \displaystyle \frac{1}{3} - \frac{1}{4} = \frac{1}{12}, gives \displaystyle \frac{1}{4} + \frac{1}{12\sqrt{2}} = \frac{1}{4} + \frac{\sqrt{2}}{24}, which is about 0.309.
No next number
Fumiko says that 0.5000001 is the number that comes right after \displaystyle \frac{1}{2} on the line. Show that it is not, and that no real number comes right after \displaystyle \frac{1}{2}.
SolutionThe average of 0.5 and 0.5000001 is
which is more than 0.5 and less than 0.5000001. So there is a number between them, and 0.5000001 is not next.
The same move works against any candidate. If someone offers a number x with \displaystyle x > \frac{1}{2} as “next”, the average \displaystyle \frac{1}{2}\left(\frac{1}{2} + x\right) lies strictly between \displaystyle \frac{1}{2} and x. Every candidate loses, so no next number exists. Counting numbers are different: after 3 comes 4, with nothing between.
Infinitely many irrational numbers between 0 and 1
Show that there are infinitely many irrational numbers between 0 and 1.
SolutionHalving again and again gives fractions marching toward 0:
These cut the stretch from 0 to 1 into endlessly many pieces that do not overlap, so one irrational number in each piece makes infinitely many, all different. Because \displaystyle 1 < \sqrt{2} < 2 (the squares are 1 < 2 < 4), dividing through by 2, by 4, by 8 and so on gives
and so on. Each of these is a nonzero rational number times \displaystyle \sqrt{2}, such as \displaystyle \frac{1}{8} \times \sqrt{2}, so each is irrational by the fact from Irrational numbers.
Boundaries
Between means strictly between, and needs two different numbers. The average of a number with itself is that number, so there is nothing between a and a. Every result above starts from a < b.
Averaging keeps rational numbers rational, but not irrational numbers irrational. The average of \displaystyle \sqrt{2} and \displaystyle -\sqrt{2} is 0. It is still true that between any two different real numbers there are both rational and irrational numbers, but the proof for two irrational endpoints works with their decimal expansions rather than with averages.
A single denominator does have neighbors. Among the fractions with denominator 7, \displaystyle \frac{3}{7} and \displaystyle \frac{4}{7} have nothing between them. What makes room is allowing any denominator: \displaystyle \frac{1}{2} lies between them.
A compass reaches some real numbers, not all. Right triangles carry the square root of any whole number to the line, but some real numbers, \pi among them, cannot be reached from 0 and 1 by any compass and straightedge construction (a fact proved in the 1800s, far beyond this lesson). Every real number still has its point; it just cannot always be found that way.
Where this goes next
Properties of real numbers names the rules of arithmetic that work on every point of the line. Finding roughly where an irrational number lands is Approximating irrational numbers, and sorting a number into its sets is Sorting a number into its sets. Because the line has no holes, a stretch of it is a solid piece named by its two ends, which is what graphing an inequality draws and Interval and set-builder notation writes down. The no-holes property is also what Limits rely on: numbers that close in on a point always find a real number waiting there.
Exercises
Rational numbers between rational numbers
- Find the average of \displaystyle \frac{1}{6} and \displaystyle \frac{1}{5}, and show that it lies between them.
- Find a rational number between \displaystyle -\frac{3}{4} and \displaystyle -\frac{2}{3}.
- Find a rational number between 2.37 and 2.38.
- Find three rational numbers between 0 and \displaystyle \frac{1}{100}, and write them in order from least to greatest.
- Find a rational number between \displaystyle \frac{7}{9} and 0.78.
Irrational numbers between rational numbers
- Write a decimal that never stops and never repeats, and that lies between 0.4 and 0.5.
- Show that \displaystyle 3 + \sqrt{2} lies between 4 and 5, and explain why it is irrational.
- Use the formula \displaystyle a + \frac{b - a}{\sqrt{2}} to find an irrational number between 1 and 2.
- Use the same formula to find an irrational number between -1 and 0.
- Given that \displaystyle 1.41 < \sqrt{2} < 1.42, find a rational number between \displaystyle \sqrt{2} and 1.5.
Points built with a compass
- Using Figure 1, explain how to mark the point \displaystyle -\sqrt{2}.
- At the point 6, a segment of length 1 stands straight up from the line, and a compass centered at 0 swings the top of it down to the line. Which number does it mark?
- Describe a right triangle, with one leg along the line from 0 and one upright leg, whose longest side the compass would carry down to the point \displaystyle \sqrt{26}.
- Eamonn has marked \displaystyle \sqrt{2}. At \displaystyle \sqrt{2} he stands a segment of length 1 straight up and swings the top of it down to the line with the compass centered at 0. Which number does he mark? He repeats the step from his new point. Which number does he mark next, and what is special about it?
Order on the line
- Given that \displaystyle 1.41 < \sqrt{2} < 1.42, write \displaystyle -\sqrt{2}, -1.5, -1.4, 0 and \displaystyle \sqrt{2} in order from least to greatest.
- Jasper-Rae knows that \displaystyle x < \sqrt{2} and that \displaystyle \sqrt{2} < 1.5. What can she say about x and 1.5? Explain using the line.
Reasoning
- Gideon-Lee says: “There is a fraction between any two fractions, so the fractions fill the whole line.” Explain what is wrong with his reasoning.
- Is the average of two irrational numbers always irrational? Explain.
- Explain why there is no smallest positive rational number.
- Halldora says: “\displaystyle \frac{3}{7} and \displaystyle \frac{4}{7} are next to each other, so \displaystyle \frac{3}{7} does have a next number.” What is right about her statement, and what is wrong?
Answers
- \displaystyle \frac{11}{60}. Over 30, \displaystyle \frac{1}{6} + \frac{1}{5} = \frac{5}{30} + \frac{6}{30} = \frac{11}{30}, and half of that is \displaystyle \frac{11}{60}. Over 60 the two fractions are \displaystyle \frac{10}{60} and \displaystyle \frac{12}{60}, and 10 < 11 < 12.
- \displaystyle -\frac{17}{24}, their average. Over 24 they are \displaystyle -\frac{18}{24} and \displaystyle -\frac{16}{24}; the sum is \displaystyle -\frac{34}{24} and half of it is \displaystyle -\frac{17}{24}, which lies between since -18 < -17 < -16.
- 2.375, their average: \displaystyle \frac{2.37 + 2.38}{2} = \frac{4.75}{2} = 2.375. Other answers such as 2.371 are also correct, here and in Exercises 2, 4 and 5.
- \displaystyle \frac{1}{400} < \frac{1}{200} < \frac{3}{400}. The average of 0 and \displaystyle \frac{1}{100} is \displaystyle \frac{1}{200}; the average of 0 and \displaystyle \frac{1}{200} is \displaystyle \frac{1}{400}; the average of \displaystyle \frac{1}{200} and \displaystyle \frac{1}{100} is \displaystyle \frac{1}{2}\left(\frac{2}{400} + \frac{4}{400}\right) = \frac{3}{400}.
- \displaystyle \frac{701}{900}. Write \displaystyle 0.78 = \frac{39}{50}. Over 900, \displaystyle \frac{7}{9} = \frac{700}{900} and \displaystyle \frac{39}{50} = \frac{702}{900}, so their average is \displaystyle \frac{701}{900}. Note that \displaystyle \frac{7}{9} = 0.777\ldots is the smaller one.
- One answer: 0.4101101110\ldots, with one more 1 in each run of ones. The runs keep getting longer, so no block repeats forever and the decimal is irrational; it starts 0.41, so it lies between 0.4 and 0.5.
- Since \displaystyle 1 < \sqrt{2} < 2, adding 3 throughout gives \displaystyle 4 < 3 + \sqrt{2} < 5. It is a rational number plus an irrational one, so it is irrational: if it were a rational number q, then \displaystyle \sqrt{2} = q - 3 would be rational, and it is not.
- \displaystyle 1 + \frac{\sqrt{2}}{2}, about 1.707. With a = 1 and b = 2: \displaystyle 1 + \frac{1}{\sqrt{2}} = 1 + \frac{\sqrt{2}}{2}.
- \displaystyle -1 + \frac{\sqrt{2}}{2}, about -0.293. With a = -1 and b = 0, b - a = 1, so the number is \displaystyle -1 + \frac{1}{\sqrt{2}} = -1 + \frac{\sqrt{2}}{2}.
- 1.45. Since \displaystyle \sqrt{2} < 1.42 < 1.45 < 1.5, it lies between. Any rational number from 1.42 up to (but not including) 1.5 is safe by the same reasoning, such as 1.43.
- Keep the compass point at 0 and the pencil at the square’s top right corner, and swing up and over to the left. The circle meets the line \displaystyle \sqrt{2} to the left of 0, at \displaystyle -\sqrt{2}.
- \displaystyle \sqrt{37}. The legs are 6 and 1, so c^2 = 6^2 + 1^2 = 37, and the compass lands at \displaystyle \sqrt{37}, a little to the right of 6.
- Legs 5 and 1: stand a segment of length 1 straight up at the point 5. The longest side has c^2 = 5^2 + 1^2 = 26, so \displaystyle c = \sqrt{26}. (A leg of 1 along the line with an upright leg of 5 also works.)
- First \displaystyle \sqrt{3}, since \displaystyle c^2 = (\sqrt{2})^2 + 1^2 = 3. Next 2, since \displaystyle c^2 = (\sqrt{3})^2 + 1^2 = 4. The second point lands exactly on the tick for 2: a construction made of irrational lengths can land on a rational point.
- \displaystyle -1.5 < -\sqrt{2} < -1.4 < 0 < \sqrt{2}. The number \displaystyle -\sqrt{2} is between -1.42 and -1.41, inside the stretch from -1.5 to -1.4.
- x < 1.5. On the line x is to the left of \displaystyle \sqrt{2}, and \displaystyle \sqrt{2} is to the left of 1.5, so x is to the left of 1.5. This is order passing along a chain.
- A fraction between any two fractions means no two fractions are neighbors. It does not mean every point is a fraction. The point \displaystyle \sqrt{2}, marked by the compass in Figure 1, is on the line, and no fraction names it, so the fractions leave holes however crowded they are.
- No. \displaystyle \sqrt{2} and \displaystyle -\sqrt{2} are both irrational, and their average is \displaystyle \frac{\sqrt{2} + (-\sqrt{2})}{2} = 0, which is rational.
- Take any positive rational number r. Then \displaystyle \frac{r}{2}, the average of 0 and r, is rational, positive and smaller than r. So whatever number is offered as smallest, a smaller one exists.
- Right: among fractions with denominator 7, nothing lies between \displaystyle \frac{3}{7} and \displaystyle \frac{4}{7}. Wrong: among all rational numbers, their average \displaystyle \frac{1}{2}\left(\frac{3}{7} + \frac{4}{7}\right) = \frac{1}{2} lies between them, so \displaystyle \frac{4}{7} is not next after \displaystyle \frac{3}{7}, and no number is.