Download The Pythagorean Theorem and Irrational Numbers

MATH 035
Penn State University
Dr. James Sellers
Handout: Pythagoras and Irrational Numbers
The Set of Real Numbers
As we open this lesson, it is very important that we understand some basic mathematics about
the set of real numbers. The set of real numbers is the set of all numbers which can be written
as a decimal number. These fall into three categories:
1. Those decimal numbers whose decimal representations “terminate”.
2. Those decimal numbers whose decimal representations go on forever as a repeated
infinite decimal.
3. Those decimal numbers whose decimal representations go on forever but do NOT have
a repeated pattern.
Interestingly enough, these sets of numbers fall into two basic categories. The numbers that
fall into categories 1 and 2 are called the rational numbers. This is because those numbers can
be written as ratios of whole numbers (or what we might call “fractions”). The numbers that
fall into category 3 are called the irrational numbers because they CANNOT be written as ratios
of whole numbers.
Remember – in terms of this modern-day language, Pythagoras and his sect believed that all
measurable items in the universe (length, width, height, weight, volume, etc.) were rational. So
in their minds, there were NO irrational numbers. Unfortunately, they were very wrong about
this fact!
Review of Square Roots
Before we get to the “punch line” of this lesson, let’s also take a few moments to remember the
idea of a square root. When we talk about a number like
5 , we are talking about a number
which, when multiplied with itself, gives us 5. In other words,
5 is the unique number x
which satisfies the equation x 2 = 5 . Of course, some square roots can be simplified. For
example, 9 is the same as 3 while 25 equals 5. But in most cases, we cannot simplify square
roots enough to completely get rid of the “square root” symbol.
Crushing Pythagoras’ Main Tenet
Now we want to look directly at how these square roots relate to Pythagoras’ belief system.
Thanks to the Pythagorean Theorem, we know exactly what the length of the hypotenuse is for
a right triangle whose side lengths are both 1. In fact, it must be 2 . So we have a very natural
occurrence of a measurable quantity which equals 2 . Unfortunately for Pythagoras, this
value is NOT a rational number. It is irrational. Let’s look at a classic proof of this result.
Theorem:
2 is irrational.
Proof: Assume, with the goal of contradicting, that 2 is actually rational. That means we are
a
assuming that 2 = for some positive integers a and b in lowest terms (that is, gcd(a,b) = 1).
b
This is actually what we are going to contradict at the end of this proof!
a
, then multiplying both sides of this equation by b yields 2b = a . Squaring both
b
sides gives us 2b 2 = a 2 . This means a 2 is even, so a must also be even. So a = 2t for some
If
2=
integer t. Substituting this back into 2b 2 = a 2 gives us 2b 2 = (2t ) 2 or 2b 2 = 4t 2 or b 2 = 2t 2 . That
means b 2 is even so b is even. But therein lies our contradiction. The numbers a and b cannot
both be even!!! If they are, then their gcd is obviously larger than 1 because a and b have a
common divisor of 2. CONTRADICTION! Therefore, the original assumption, that we could
a
write 2 as with a and b in lowest terms is incorrect. Said another way, 2 cannot be
b
rational. That means
2 is irrational, and our theorem is proved.
Closing Thoughts
1. Note that this same kind of proof can be used to show that numbers like 3 , 5 , and 10
are also irrational. Therefore, it is easy to prove that the set of irrational numbers is infinite.
2. More to the point of this lesson, this result was so destructive to the belief system of
Pythagoras and his followers that legend says the Pythagoreans actually drowned the original
discoverer of this result! Yikes! Of course, that’s probably not the first time, nor the last time,
that someone lost their life over a controversial fact (even when they were correct!).
© 2010, James A. Sellers