Thanks Dr v1n0d! Please be a “Mathematician by day” in this thread so that you can provide some guidance to students if they need help, when I'm not around.

Not too long ago I've came across this intriguing proof that sparked my mind. There are two proofs that show 0=1. This is how it goes:
First proof:
Let x = 1
So x²= (1)²=1
By subtracting x from x², we get:
x²-x=0
x(x-1)=0
x=0,x=1
Ergo, 0=x=1

Second proof:
Let x=y
Multiply x on both sides:
x²=xy
Next, subtract y² from both sides:
x²-y²=xy-y²
By factoring out (x-y) on both sides and dividing them out:
(x+y)(x-y)=y(x-y)
x+y=y
Since x=y:
y+y=y
2y=y
Hence,
2=1
Subtracting 1 from both sides:
1=0

My question is, are both of the 2 proofs above plausible, even if there is something wrong with it, can somebody show me the correct way of dealing with it?

Thank you . I've been placing my hand in different cookie jars , not knowing which to delve deeper into . For now , I have the luxury to do so so I'll jump around different topics , and get some serious work done in January .

The problem which I posed is from the book titled ' Basic Concepts of Mathematics and Logic ' by Michael C . Gemignani , which covers logic , set theory , counting , numbers , functions , ordering , probabilities and Euclidean geometry . It's quite a compact book , not the best designed book but I find it quite entertaining ( although the content comprises mostly of paragraphs of words ) .

That problem comes after this :

In order to disprove the following statement , which statements would we actually prove ?

' Some triangles are isosceles . '

The answer is : All triangles are not isosceles . (the answer doesn't delve deeper into how to prove this statement)

After the topic on Mathematical Disproofs , I ventured into Conjunctions and Disjunctions , which I've seen when learning Mathematical Reasoning in SPM Mathematics , and Boolean Algebra in SPM Physics .

A question : If A , then B is equivalent to ~( A ∧ ~B ) , with the reasoning :

If A , then B , thus if A occurs B must occur as well , or it cannot happen that A occurs but B does not occur , it cannot happen that A  occurs and ~B occurs .

So this statement ~( ~A ∧ B ) is false , right ? Because if A does not occur , B can occur .

At the Big Bad Wolf sale , I bought Calculus ( Pearson International Edition ) textbook for RM 10 , which is a huge college level textbook which covers a lot of concepts in Calculus . I'm still in the preliminaries section , I'm familiar with most of the topics in the preliminaries ( they are covered in SPM Maths / Add Maths ) so I'm learning Logic and Proofs . I've also printed out your tutorials , but not sure where to start with all the resources around me . 

In the Calculus textbook , there are a few problems which I need help with ( college level textbooks only provide solutions for Odd-Number questions    )

1 . We can show that a number divided by zero is meaningless . Suppose a ≠ 0 . If a/0 = b , then a = 0 , which is a contradiction . Now find a reason why 0/0 is also meaningless .

2 . Show that any rational number p/q , for which the prime factorisation of q consists entirely of 2s and 5s , has a terminating decimal expansion .

3 . Show that between two real numbers there is a rational number .

4 . (a) Use the Fundamental Theorem of  Arithmetic to show that the square of any natural number greater than 1 can be written as the product of primes in a unique way , except for the order of the factors , with each prime occurring an even number of times .

(b) Show that √2 is irrational . ( Hint : Try a proof by contradiction . Suppose that √2 = p/q , where p and q are natural ( necessarily different from 1 ) . Then 2 = p^2/q^2 , and 2q^2 = p^2 . Now use (a) to get the contradiction .

Your help is greatly appreciated .  By the way , is this topic under Number Theory ? It's under preliminaries , the author didn't label this sub-topic .

The new stpm system which also known as stpm modular system lacks a lot of detail explanation.... Whereas the old stpm syllabus book contains a lot of detailed explanation....

Conventionally, mathematicians will tell you that there is no number that you can multiply by 0 to get a non-zero number. Hence, there is NO solution, and so any non-zero number divided by 0 is undefined or rather meaningless.

holy smokes...the stuffs you guys wrote are like alien to me. Good work extending the knowledge. Perhaps if my kids need tutorial, i can point to this thread. Anybody tutoring secondary students?

Yes , an equation which combines i , pi , e , 1 and 0 !  Very beautiful equation .

I would appreciate it if you could provide tutorials on Logic .

https://fbcdn-sphotos-h-a.akamaihd.net/hpho...650586348_n.jpg

The link will bring you to a page of 2013 Further Mathematics STPM Repeat Paper . In Q1 , it asks for the validity of propositions , but in my book , it says that a proposition and an argument are two different things . A proposition can be true or false , valid and invalid are incorrect terms to refer to propositions .

I know how to do (a) and (b) , but what about the conclusion ?

Hmmm I kinda fathom the question, but I have no idea on how to show that it's rational or not. A very tricky one I'd say....

Wait , I think I see something .

Suppose that (√2)^(√2) is irrational .

Let (√2)^(√2) = a , where a is an irrational number .

Raising the powers of both sides by √2 , we have (√2)^2 = a^(√2) ;

a^(√2) = 2  ( a is irrational , √2 is also irrational but 2 is rational ) .



Call me a nerd , but this is much more exciting than a lot of things I've seen !

Didn't think of that . 

Is there a general proof ?

∋ means 'such that' ?

Any reasoning behind this , maybe a more detailed explanation ? I'm sorry , I've a lot to go .

Can I say :

For every x∈R, ∃ an inverse y∈R such that x+y = 0 instead of x+y∈R ?

Wait, did I miss something?   If I were to multiply an irrational number(√2) with itself, then I'll get a rational number, which is 2. So, the product of two irrational numbers is rational. Did I left out anything vital?

An irrational to an irrational power may be rational, as what has been shown previously by maximR,(which also involves multiplying the Gelfond–Schneider constant with √2)

But, if in the case of √2*√3, I don't think the result is rational. Correct me if I'm wrong though.

For every x and y such that both are irrational , their product is a rational number .

If the statement is revised as above , can one example prove the statement ?

Oh, thanks for your clarification.
But, again, does this proof is constructive since √2^√3, and that the results is irrational?

I'm sorry, my bad. What I want to point out was that, do we only need to show just one example/proof to imply that the statement is true or false?
Like in the case of a^b, and that both a and b is irrational(suppose that both a and b is √2). And that if we multiply (a^b)^√2, we will get 2, which is rational.
But on the other hand, if a=√2, and that b=√3, the results will surely be irrational. Now, doesn't both examples contradict each other? So, how does one tell or pick one of these examples that will proof that the statement is true?

Done with SPM . Starting on more 'advanced' concepts for now to get used to them before PreU . How are the lecturers at  USM by the way ? How 'rigorous' is the curriculum for Mech Eng ?

I'm not into Mechanical Engineering , I'm more geared towards Physics at the moment . This might change , that's why I'm keeping my options open , planning to do A Levels first ( which means IPTA would be out of the picture for me ) , and I've to really dig deep for scholarships after A Levels .

Do you know how I can be involved with the camp ? I've heard of the IMO one but information about IPO is scarce .

How do you think I should prepare for it ? IPO involves a lot of college level Physics , not sure if I can manage to self-study everything .

I see . Where are you studying right now , if you don't mind me asking .