Proof: |
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is an Irrational Number |
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Proof: |
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is an Irrational Number |
This proof, known in some form or other for well over 2000 years, is one of the most important proofs mathematics students will see early in their mathematics training. It has the great virtue of being fairly easy to grasp, it has great utility in that it expands the number systems available to students, and it helps students understand how math is built from a few basic axioms (truths that are beyond question) and those axioms are expanded upon with theorems that must be proved. Theorem proving is extremely important in mathematics. Some "proofs" are considered important because they advance math in significant ways or are extremely useful as applications. Some proofs are not very important because they are very limited in how they impact other mathematics. And, finally, some proofs are considered important primarily because they are so "elegent." An elegent proof is one that gets to the heart of the subject quickly using one idea (or a very minimum number of assumptions) that is exceptional insightful. They do not require a long, convoluted trail of if-this, then-that, then-something-else, ... , on and on and on, going around and around in a seemingly senseless pattern, until finally getting the result. The proof shown here, that the square root of 2 is irrational, has been called one of the most elegent proofs that have ever been produced.
The structure of the proof is a "proof by contradiction," where we assume something is true and then show that this must lead to a contradiction, thus the assumption must be false. In this case, we assume square root of 2 is rational, and by a series of logical steps we will get to a point where it will be obvious that this is false. Therefore, if the square root of 2 is cannot be rational, then it must be irrational.
The goal of this proof is to show it is impossible to write any simple fraction that exactly equals the square root of two. The "classic" pattern for this proof is to begin by assuming the square root of two is rational (that you can write it as a fraction) and after a series of logical steps we see that this leads to a contradiction. We then must conclude the assumption was in error and that the square root of two cannot be rational (thus it is irrational). This style of proof is called reductio ab adsurdum or proof by contradiction. [If you are not sure what the terms rational and irrational mean, click here.]
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Assume that the sqaure root of 2 can be written as a fraction of the form a/b and that this fraction is in simplest form, where there is no common factor, other than 1, in the numerator and denominator.
Thus, let
We can perform the following algebraic steps to rewrite this equation:
The last line tells us "two times b squared" equals "a squared." Since "a squared" equals 2 times some number, then a2 must be even (by the definition of even numbers, which states a number is even if it is of the form 2n where n is an integer). Further, if a2 is even, then a must also be even. [Why? This also requires a proof, but the fact that an even number squared also must be even is straight forward and is shown below.]
So, since a is even it can be rewritten as 2n. Let's do that, producing the next line in the proof.
Here we see that b2 must be even and therefore b must be even because it is the square of some even number. This produces a contradiction! We began with an assumption that there existed a fraction a/b in lowest terms (where a and b have no common factors other than 1) that equaled the square root of 2. But we have just got to a point that says a and b are both even, thus they would have a common factor.
Therefore,
… meaning it is impossible to write the square root of 2 as a "rational number" (a simple fraction using integers for the numerator and denominator).
Thus, the square root of 2 is "irrational."
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Early on I said this proof is one that is exceptionally elegent. In the proof you have just seen I have tried to explain steps as they progressed. By taking out some of the "algebra" (steps the quick student could fill in on her or his own), the proof could run as follows:
Theorem: square root of 2 is irrational.
Proof:
Assume square root of 2 is rational, in the form a/b, where the GCF of a and b is 1.
Then 2b2 = a2 and thus a is even. Substitute 2n for a and we see that b2 = 2n2, implying that b is also even. Thus, there is a contradiction: both a and b have been shown to be even (with a common factor of 2), despite the assumption that the fraction, in lowest terms, equaled the square root of 2. Thus, there can be no simple fractional way to represent the square root of 2, and thus the square root of 2 must be irrational.
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Background Information What do the terms rational and irrational mean?
One of the hallmarks of mathematics is its precision. For example, the numbers that all cultures have "discovered" and no one can remember a time when these numbers where not known are called the "Natural Numbers." The Natural Numbers include 1, 2, 3, and so on, without end. The Whole Number set includes all those and the number 0. [Believe it or not, there was a time when cultures did not have a symbol for the number zero!] Next come the numbers mathematicians call the "Integers": 0, 1, 2, 3, ... , and their "negatives" -1, -2, -3, ....
The rational numbers are those numbers that are built by making a ratio of a pair of integers,
.Examples include:
Thus, rational numbers are those that can be written as simple fractions using the integers. An irrational number, then, is one that cannot be written as a fraction. The goal of this proof is to show it is impossible to write any simple fraction that equals the square root of two. The "classic" pattern for this proof is to begin by assuming you can do it and then show that this assumption leads to a contradiction (thus the assumption was in error).
The irrational numbers, then, are those numbers that cannot be expressed as the ratio of a pair of integers, a and b such that a/b.
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Proof: If a perfect square is even then the square root must also be even:
Given a number that is a perfect square, the square root of this number is either even or odd. Thus, there is a number, s, such that
and
where s and r are a whole numbers.
Case 1: Assume the square root, r, is odd. By definition of odd numbers, r can be written in the form
, where k is an integer.
Since.
The square
must be odd because it is in the form of the definition of an odd number. Thus, if r is odd and is also the square root of s, then s must be odd. Similarly …
Case 2: Assume the square root, r, is even. By definition of even numbers, r can be written in the form 2j, where j is an integer.
Since.
The square
must be even because it is in the form of the definition of an even number. Thus, if r is even and is also the square root of s, then s must be even.
Therefore, if a perfect square is odd then its square root is odd and if a perfect square is even then its square root must also be even.
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Some Links on Proofs
- Elementary Number Theory and Methods of Proof
- http://www.ltc.hw.ac.uk/~neilm/maths/topic88.html
- "The Proof" - PBS / Nova documentary on Andrew Wiles' monumental accomplishment
- http://www.pbs.org/wgbh/nova/proof/
- Discussion of the Importance of Proof (and some complete proofs)
- http://www.cut-the-knot.com/proofs/index.html
- False Proofs, Classic Fallacies
- http://mathforum.org/dr.math/faq/faq.false.proof.html
- Animated Proof of the Pythagorean Theorem
- http://www.nadn.navy.mil/MathDept/mdm/pyth.html
- Famous Problems in Mathematics
- http://mathforum.org/isaac/mathhist.html
- The Four Color Theorem
- http://www.math.gatech.edu/~thomas/FC/fourcolor.html
- More to follow ...
