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The idea of the real number system evolved over thousands of years. For
many centuries, the idea that all numbers are natural numbers (1, 2, 3, …)
sufficed for daily life and commerce. It also took centuries for the idea
of nothing to consolidate into the number 0. “How can you have 0 of
something? You don’t have any of it?”
The idea of a fraction as a number took even longer to develop than did the idea of zero as a number. What we now think of as the number $\frac{3}{4}$, for many centuries, was instead thought of as a relationship between two numbers (3 to 4, 6 to 8, …).
When Pythagoras discovered that $\sqrt{2}$ cannot be a ratio of two natural numbers, the Greeks thereafter distinguished between magnitudes (lengths) and numbers (counts). To the Greeks, magnitudes were not numbers and numbers were not magnitudes. Numbers were counts and magnitudes were lengths. It was not until the Middle Ages that Arabian mathematicians, with their use of symbols to represent numbers, came to consider magnitudes and numbers synonymously, and therefore that all lengths and all ratios could be considered as numbers, just like counts. They knew that most lengths cannot be expressed as a ratio of two natural numbers. Hence, the Arabian ways of thinking about numbers opened the door to the idea of irrational numbers.
In the late 1800’s the ideas of natural numbers, integers, rational, irrational, and real numbers were made rigorous. It was proved that each point on a line corresponds to a specific real number once you identify two points on it—one point as 0 and the other point as 1. This was the advent of the number line. Integers are equally spaced on the number line and fractions are placed to the left or right of 0 at a distance that is in proportion to their size relative to 1. For example, $\frac{387}{17}$ is $\frac{387}{17}$ times as large as 1, so the point on the number line that corresponds to $\frac{387}{17}$ is $\frac{387}{17}$ times as far to the right of 0 as 1 is.
The remaining points on the number line correspond to irrational numbers.
It is natural to think that most of the points on a number line correspond to rational numbers. But this is not true. Cantor showed that even though there is an infinite number of fractions between any two integers, there are just as many integers as fractions. Fractions and integers can be put into one-to-one correspondence. See Exercise 3.7.4.
It is also natural to think that there are more real numbers on the entire real number line than there are on any finite interval. But this is not true. See Exercise 3.7.1.
Cantor also demonstrated that even though the number of rational numbers and the number of irrational numbers are both infinite, there are more irrational numbers than rational numbers. Indeed, Cantor showed that if the number of fractions is $X_0$, then the number of irrational numbers is $2^{X_0}$. A visual analogy would be to look at the stars on a dark, clear night. Think of the stars as rational numbers. The empty space between stars would be filled with irrational numbers.
When you hear or read the statement, “Let x be an arbitrary real number”, just think “Let P be an arbitrary point on a number line and let x represent its displacement from 0”. It is this way that we use a number line as a visual aid to thinking about real numbers.
The circle below has a circumference of 1. Explain how the animation suggests that there are just as many real numbers between 0 and 1 as there on the entire real number line. (Play the animation at full screen if you have difficulty seeing its details.)
List 10 rational numbers between $\frac{1}{4}$ and $\frac{1}{3}$.
List 10 rational numbers between $\sqrt{2}$ and $\sqrt{3}$.
A fraction is a symbol of the form “$\frac{a}{b}$” where a and b represent integers, and b ≠ 0. A rational number is the number represented by a fraction. Here is Cantor’s method to show that the counting numbers can be put into one-one correspondence with the set of positive fractions.
How does Cantor’s method of listing the positive fractions guarantee that every positive fraction will appear somewhere in the list?
How does Cantor’s method of counting guarantee that any fraction in the list will eventually be counted? What does this imply about the number of counting numbers and the number of positive fractions?
Consider that $\dfrac{2}{1}$, $\dfrac{4}{2}$, $\dfrac{6}{3}$, ... all represent the same rational number. How many times does the rational number represented by 2/1 appear in Cantor’s list? How many times is every positive rational number represented in the list? What does this imply about the number of counting numbers and the number of positive rational numbers?
Does it strike you as paradoxical that in Exercise 3.7.4 you established that the number of counting numbers is at least as large as the number of positive rational numbers, and yet there is an infinite number of rational numbers between any two counting numbers? Say more about this.
Kaitlin said that it is impossible for a variable’s value to vary even over a tiny interval because even over a tiny interval it would have to vary over an infinite number of numbers that is as large as all the real numbers. What would you say to help Kaitlin understand that while it is true that every interval contains as many numbers as the set of real numbers, it is nevertheless possible to envision a variable varying over all numbers in any interval.