What is the method in constructiong a proof(algebra) square root 2. Not for geometry.?

Question

What is the definition of proof? I would like to prove that square root 2 is rational,reaching to the conclusion that it's actually irrational instead, just to go through the process of proof because I don't understand what the book is telling me. We skipped the page but I believe I should go over it because it is important to go through the problem solving process.
book just gives unorganized steps and its not even details, i would like to put it in 2 colums like geometry. how can i do that?
assume that square root 2 is rational.
then square root 2 can be expressed as p/q where p and q are integers.

assume that p/q is in its lowest terms.

let p/q = square root 2

squaring p^2/q^2 = 2 (canceling out the root)
so p^2=2q^2

this is where i got lost... it said p^2 is even, then p must also be even so one can write
p=2m where m is an integer
4m^2 = 2q^2 (substituting p^2 = 2q^2)
2m^2=q^2
so q^2 s even and therefore q is even. but if both p and q are even then p/q is not low term

Answer 1

You said, [T]his is where [I] got lost ... it said p^2 is even, then p must also be even ..." Let us look more closely at that part of the proof. First, an integer is even if and only if it is a multiple of 2. When it was shown that p^2 = 2q^2, that showed that p^2 is a multiple of 2, hence p^2 is even.

Next, let us examine the next claim, namely that p must be even since p^2 is even. We can use the same indirect method of argument as was used for the whole proof. Suppose p is not even. Then since every number is either even or odd, p must be odd. That is, p is not a multiple of 2; therefore p must have the form p = 2k + 1, for some integer k. (That is, p would leave a remainder of 1 when divided by 2.) Now consider p^2; since p = 2k + 1, p^2 = (2k + 1)^2 =
4k^2 + 4k + 1 = 2(2k^2 + 2k) + 1 = 2K +1 for K = 2k^2 + 2k. But if p^2 = 2K + 1, p^2 is odd, and that is a contradiction. Thus, p^2 is even implies that p is even.

That should get you unlost.

If you would like to put your proofs in two columns, go ahead. Most people don't beyond geometry because it is regarded as mathematically unsophisticated, but there is no absolute prohibition of such a presentation.

Hope this helps. Your query shows an admirable degree of mathematical curiosity.

Answer 2

This illustrates a form of proof which assumes one result, then proves that this leads to a logical contradiction which then means that the original assumption must be false.

This proof shows that sqrt(2) is irrational by (a) assuming that is is rational (ie. can be written as p/q) and (b) then showing that it leads to an internal contradiction.

One of the keys in this proof is assuming that if sqrt(2) can be written as p/q then p & q do not have any common factors. This can certainly be allowed because we can always divide out any such common factors until they're all gone. But assuming that sqrt(2) can be written as p/q shows that p & q must BOTH be even (you need to follow the algebra) and therefore both have 2 as a common factor, which contradicts the assumption above.

Hence, assuming that sqrt(2) is rational leads to a logical contradiction, therefore sqrt(2) must be non-rational (ie. irrational).

Answer 3

alrite here we go....

1) Proof is anything that "states what is is" For eg. 2*2 = 4
but how do we do we prove this?

We take 2 chocolates and give it to the first person....then we take another two chocolates and give it to the second person Hence, in total u have given 4 chocolates

So in 2*2 = 4
The first 2 is what you have
The Second 2 is how many times u distribute that
And the last Result, the 4 is how much u have given

Hence, u proved that if i have 2 things and i have 2 people in waiting then i will be 2 things short and hence i got to get 2 more things which make in total 4!

Irrational no. is any decimal no. that cannot be written as a fraction!

Rational no. is any decimal no that can be written as a fraction in form of p/q where p is the numerator and q is the denominator.

Any decimal no. can be expressed as a fraction if it ends somewhere or if its recurring.

The value of "Square Root 2" can be expressed in decimal but the value in decimal after the decimal point never ends!!

So technically you cant think Square root 2 can be written as p/q. THE REASON IS Mathematics is all about accuracy. WE CANNOT ASSUME!

Hope that explains....

Answer 4

So while we've a sq. root as a denominator, we could desire to rationalize. For a million) we could desire to multiply the numerator and denominator by ?8. so which you get (?8(?3 + 3?5) /(?8 (2?8). this makes (?24 + 3?40)/2?sixty 4 which simplifies to (2?6 + 2?10) / sixteen which eqauls to (?6 + ?10)/8. I divided by 2. for 2) we could desire to multiply by the conjugate of (5+ ?2) that's (5- ?2) to the denominator and numerator. We get (?5)(5- ?2) / (5 + ?2)(5- ?2) which simplifies to (5?5-?10)/(25-(?2)²). This makes (5?5-?10)/(25-2) that's (5?5-?10)/(23). the respond is (5?5-?10)/(23).