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viren_t2005
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Prove by induction that 2^1/2 is irrational.
AKG said:Induction? Induction on what. There's a standard proof by contradiction for this one.
CRGreathouse said:That's a good question. Usually a proof by induction proves that something is true for n=1 and that if it is true for a given n it is true for n+1. Let's try:
For n=1: Suppose [tex]\sqrt{2}[/tex] is rational. Write [tex]\sqrt{2}=\frac ab[/tex] with (a,b)=1. Then [tex]2b^2=a^2[/tex].
TenaliRaman said:Probably OP meant 2^(1/n) ?
-- AI
shmoe said:They probably want an "infinite decent" type proof where you assume sqrt(2)=p/q then show sqrt(2)=r/t where r<p and t<q
Ray Eston Smith Jr said:(a) The first term in the sequence is 1, which is a rational number.
(b) Assume the nth term in the sequence is rational.
To go from the nth term to the (n+1)th term, you add a rational number (the next digit).
The sum of 2 rational numbers is a rational number,
therefore the (n+1)th term is also rational.
(c) From (a) & (b), by induction, every term in the sequence is rational.
(d) Therefore if 1.413213562373... exists, it is rational.
mathwonk said:the original proof by euclid is by the well ordering principle, i.e. inductiion.
he proves that if 2 B^2 = A^2, then A is even, hence B is even, and then reduces the fraction further. he comments this reduction process cannot go on forever.
indeed the assumption in greathouses proof that it is possible to choose A,B which gcd = 1, is proved by well ordering, since one takes the denominator as small as possible, say.
i.e. induction is so basic to the usual proofs that we have ceased noticing it.
Induction is a mathematical proof technique that involves proving a statement for a specific base case and then showing that if the statement holds for any given case, it also holds for the next case. In this case, we will use induction to prove that 2^1/2 is irrational by showing that if it is irrational for the base case of n=1, then it must also be irrational for all larger values of n.
An irrational number is a real number that cannot be expressed as a ratio of two integers. In other words, it cannot be written in the form of a/b, where a and b are integers and b is not equal to 0. Examples of irrational numbers include pi and the square root of 2.
The base case for proving the irrationality of 2^1/2 is when n=1. This means that we need to show that 2^1/2 is irrational in the case where it is raised to the power of 1. If we can prove this, then we can use induction to show that it is also irrational for all larger values of n.
To use the principle of mathematical induction, we first prove the base case (n=1) by assuming that 2^1/2 is irrational. Then, we show that if 2^1/2 is irrational for some value of n, it must also be irrational for the next value of n. This proves that it is irrational for all larger values of n. By using this principle, we can conclude that 2^1/2 is irrational for all values of n.
Proving the irrationality of 2^1/2 is important because it is a fundamental concept in mathematics and helps to further our understanding of irrational numbers. It also has applications in various fields, such as geometry and number theory. Furthermore, it serves as a demonstration of the power and usefulness of mathematical induction as a proof technique.