Hey there! Sign in to join this conversationNew here? Join for free
x Turn on thread page Beta

Help: Proof that square root of 2 is irrational watch

    • Thread Starter
    Offline

    0
    ReputationRep:
    Hello
    As part of my AS Maths Summer work, i have been asked to 'Search either books or the internet to find a proof ( a clear explanation that would convince a critical reader) that the square root of 2 is an irrational number' - I really cannot find anything, i tried to make one up but they're not really solid enough.......does anyone have any ideas? Thanx in advance, Shane
    Offline

    0
    ReputationRep:
    The classic one goes:

    Any rational number can be expressed in the form p/q where p and q are integers and the fraction can be reduced no more.

    So assume rt2 is rational

    Let rt2 = p/q

    2 = p^2/q^2

    p^2 = 2q^2

    implies that p^2 is even, implies that p is even, since an even x even is even, and odd x odd is odd for all integers.

    so p is even, let p = 2m where m is also an integer (since p is even)

    4m^2 = 2q^2 implies that 2m^2 = q^2, which implies that q^2 is even and so q is even.

    so let q = 2n

    implies that rt2 = 2m/2n, but this cancels to m/n

    our original premise said that the fraction shouldn't cancel any further, but it has done, so our original premise must be wrong and rt2 must be irrational.
    Offline

    0
    ReputationRep:
    proof by contradiction my fellow mathematician!

    Assume that root 2 is rational. If it is rational, it can be represetented as p/q where p and q are integers and have no common factors (Simplest ratio). So 2 = p^2/q^2. So 2q^2 = p^2

    So the square of p is even, so p itself must be even (Because all odd numbers squared are odd, so p is even).

    So lets say p=2r (since p is even)
    then p^2 = 4r^2
    so 2q^2 = 4r^2
    so q^2 = 2r^2

    so q^2 must also be even, so q must be even

    BUT HOLD ON. That means that both p and q are even, and so p/q is not in it's simplest form (since you can divide by 2) so Root 2 can not be rational!
    Offline

    0
    ReputationRep:
    Alas, even easier

    2p^2 = q^2. Consider factors of 2 on both sides, the LHS must have an even factor of two, whilst the right must be even. Contradiction.
    • Thread Starter
    Offline

    0
    ReputationRep:
    Hey thanx, you all helped me loads - i didnt have a clue! Wow, people always reply so quickly for maths - the mathematicians must be the kindest lot on the board. Thanx again, x
    Offline

    11
    ReputationRep:
    The classic proof- and actually conception of the irrational number concept- is the triangle, with sides 1, 1 and hypotenuse square root 2.
    Offline

    0
    ReputationRep:
    (Original post by tomcoolinguk)
    The classic proof- and actually conception of the irrational number concept- is the triangle, with sides 1, 1 and hypotenuse square root 2.
    How does that show that rt2 is irrational?
    Offline

    11
    ReputationRep:
    Right you have your right angled triangle with sides a, b and h.

    a= 1
    b=1

    therefore by pythagoras

    h2= a2+ b2
    h2= 2
    therefore h= sq. root 2
    Offline

    0
    ReputationRep:
    How does that show that h is irrational?
    Offline

    12
    ReputationRep:
    (Original post by fishpaste)
    How does that show that h is irrational?
    Square root 2 = 1.4142...

    When I click the Fraction button on my calculator, the calculator fails to convert the decimal number into a fraction, which it can do if the decimal number is rational.

    e.g.) 4.5 ---> Click Fraction Button -----> Shows 4.5 in Fraction form.

    Hence: Square root 2 is irrational.
    Offline

    12
    ReputationRep:
    (Original post by tomcoolinguk)
    Right you have your right angled triangle with sides a, b and h.

    a= 1
    b=1

    therefore by pythagoras

    h2= a2+ b2
    h2= 2
    therefore h= sq. root 2
    What's that got to do with the question?
    Offline

    0
    ReputationRep:
    (Original post by Invisible)
    Square root 2 = 1.4142...

    When I click the Fraction button on my calculator, the calculator fails to convert the decimal number into a fraction, which it can do if the decimal number is rational.

    e.g.) 4.5 ---> Click Fraction Button -----> Shows 4.5 in Fraction form.

    Hence: Square root 2 is irrational.
    lmao. Nice. The best kind of argument.

    [random observations] I think if you get an ugly enough rational number, it won't convert it, if the period of the digits is longer than the capacity of the calculator. And also your calculator uses a rational approximation to rt2 anyway. Probably from the binomial expansions.
    Offline

    3
    ReputationRep:
    Haha, great, now we can add to Proof by Induction, Proof by Contradiction a brand new proof, "Proof by my calculator says so, nyah!".

    Add to that several proofs my lecturers like such as "Proof because I say so" and "Proof by vigourous hand-waving".
 
 
 
Turn on thread page Beta
Updated: August 2, 2004
Poll
Do you agree with the proposed ban on plastic straws and cotton buds?
Useful resources

Make your revision easier

Maths

Maths Forum posting guidelines

Not sure where to post? Read the updated guidelines here

Equations

How to use LaTex

Writing equations the easy way

Student revising

Study habits of A* students

Top tips from students who have already aced their exams

Study Planner

Create your own Study Planner

Never miss a deadline again

Polling station sign

Thinking about a maths degree?

Chat with other maths applicants

Can you help? Study help unanswered threads

Groups associated with this forum:

View associated groups

The Student Room, Get Revising and Marked by Teachers are trading names of The Student Room Group Ltd.

Register Number: 04666380 (England and Wales), VAT No. 806 8067 22 Registered Office: International House, Queens Road, Brighton, BN1 3XE

Write a reply...
Reply
Hide
Reputation gems: You get these gems as you gain rep from other members for making good contributions and giving helpful advice.