GRADES 9–12
CCSS: The Real Number System:
Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.
The number “π” is an example of a “transcendental number.” Transcendental numbers are the penultimate step in a process that starts with counting numbers and extends them in more and more directions. Counting numbers are “closed” under addition; this means that if we add one counting number to another counting number, the result will always be a counting number. Counting numbers are also closed under multiplication. But counting numbers are not closed under subtraction; if we try to subtract five from three we do not get a counting number.
To correct this, we extend the counting numbers by putting in zero and the negative whole numbers; the result is the set of numbers called the “integers.” Integers are closed under addition, subtraction, and multiplication; if you add, subtract, or multiply two
integers, you will always get another integer. They are not closed under division, though; three divided by five does not give an integer result.
So we extend our numbering system to include fractions; the resulting system is called the “rational numbers.” A rational number can be expressed as the ratio of two integers; it can also be expressed as a decimal, either of finite length or with an infinitely repeating pattern. (A finite decimal can be thought of as having
a repeating pattern of zeroes after the last nonzero digit.)
This almost does the trick, but the system of rational numbers is still not closed under division; dividing something by zero does not give a rational number as the result. There are two ways to deal with this problem. One is simply to live with it. The other is to include a “point at
infinity” as a rational number. Many mathematics teachers maintain strongly that “infinity is not a number” and they are right. Others say that infinity is a perfectly good number; they also are right. One of the beauties of mathematics is that you can make up your own
rules and see what you can derive using them. (For teenagers, you may want to emphasize or omit the “make up your own rules” part.)
The rational numbers are not closed under exponentiation. The ancient Greeks recognized this; the Pythagorean school had a proof that there are no two whole numbers “α” and “b” such that
This is equivalent to
saying that the square root of two cannot be expressed as the ratio of two
integers. More generally, numbers that can
be expressed as the solution to a polynomial with rational coefficients are
called “algebraic numbers.” They can be
expressed as decimals that do not repeat.
After deriving the
existence of algebraic numbers, mathematicians realized that there are other
numbers that can also be expressed as nonrepeating decimals. These are the transcendental numbers; they
are best defined as what they are not.
They are not integers; they are not rational numbers; they are not
algebraic numbers. They are the
remaining thicket of numbers that fill out the number line.
At the beginning of this process I mentioned
that transcendental numbers are the penultimate step in the process. The last step is to include the square root of
negative one, bringing the imaginary and complex numbers into the system. But this is a step beyond the present lesson.
Sixty Years Ago in the
Space Race:
March 1, 1957: The first test launch of a
Jupiter 1C rocket ended in failure 72 seconds after liftoff when the rocket
overheated and exploded.
March 14, 1957: A Jupiter A missile
was launched successfully.
