michel123456 Posted March 3, 2012 Share Posted March 3, 2012 and finally, (personally I don't think you're familiar with this): I like diagrams. Connector will accept that the line segment exists. So all the points that construct the line segment exist. How many points are existing into this segment? An infinity. Because there exist theoretically an infinity of points into this segment, it is difficult to reach exactly a definite point on the segment. [math] \sqrt{2} [/math] is such a point. One between an infinity. [math] \sqrt{2} [/math] exists. And [math] \sqrt{2}+1 [/math] also exists. Link to comment Share on other sites More sharing options...

khaled Posted March 4, 2012 Share Posted March 4, 2012 (edited) Your logic is wrong here; you've already assumed that a exists to make this statement. =Uncool- I'm not good in proof, but I know that [math]a \in \mathbb{R}^+ \;\;\rightarrow\;\; \sqrt{a} \in \mathbb{R}^+[/math] is True Edited March 4, 2012 by khaled Link to comment Share on other sites More sharing options...

uncool Posted March 4, 2012 Share Posted March 4, 2012 I'm not good in proof, but I know that [math]a \in \mathbb{R}^+ \;\;\rightarrow\;\; \sqrt{a} \in \mathbb{R}^+[/math] is True That assumes that [math]\sqrt{a}[/math] exists in the first place. Your statement is true, but the logic doesn't work - you're using circular reasoning. =Uncool- Link to comment Share on other sites More sharing options...

ajb Posted March 4, 2012 Share Posted March 4, 2012 (edited) A quick google will produce lots of notes on how to prove the Square Root Theorem For all real [math]y[/math], if [math] y \geq 0[/math] then there exists one and only one non-negative number [math]x[/math] such that [math]x^{2} = y[/math]. Edited March 4, 2012 by ajb Link to comment Share on other sites More sharing options...

rktpro Posted March 5, 2012 Share Posted March 5, 2012 A quick google will produce lots of notes on how to prove the Square Root Theorem For all real [math]y[/math], if [math] y \geq 0[/math] then there exists one and only one non-negative number [math]x[/math] such that [math]x^{2} = y[/math]. It is taught to all school goers in grade 10. I recall it was root 2 which was first proved irrational and the uniqueness of root 2 was also proved. connector must go through it. Link to comment Share on other sites More sharing options...

The Observer Posted March 5, 2012 Share Posted March 5, 2012 It is taught to all school goers in grade 10. I recall it was root 2 which was first proved irrational and the uniqueness of root 2 was also proved. connector must go through it. I've proved the existence of square roots in an analysis class. It takes the completeness axiom and a certain amount of cleverness. There is no way that it is taught in grade 10. Link to comment Share on other sites More sharing options...

rktpro Posted March 5, 2012 Share Posted March 5, 2012 I've proved the existence of square roots in an analysis class. It takes the completeness axiom and a certain amount of cleverness. There is no way that it is taught in grade 10. What I am saying is that root of an rational number has been proved to be irrational in grade 10. There are over 20 proof for that. Link to comment Share on other sites More sharing options...

The Observer Posted March 5, 2012 Share Posted March 5, 2012 Your grade 10 class proved that the square root of two is irrational? Link to comment Share on other sites More sharing options...

rktpro Posted March 5, 2012 Share Posted March 5, 2012 Your grade 10 class proved that the square root of two is irrational? Yes. Link to comment Share on other sites More sharing options...

ajb Posted March 5, 2012 Share Posted March 5, 2012 It is taught to all school goers in grade 10. That sounds quite advanced for grade 10, but I will take your word for it. Link to comment Share on other sites More sharing options...

imatfaal Posted March 5, 2012 Share Posted March 5, 2012 There is a pretty basic proof using the fact that is you start with assuming you have the rational form of sqrt(2) in its most simplified terms - it is then really simply to show that the numerator must be even, thus the denominator must be odd (otherwise it wasn't simplest form); but then you can show that the denominator must also be even (it is divisible by 4) - you have a proof by contradiction. I was shown it at school - but I am pretty sure it was in ad maths AO (for the older brits) which was age 16 Link to comment Share on other sites More sharing options...

ajb Posted March 5, 2012 Share Posted March 5, 2012 There is a pretty basic proof using the fact that is you start with assuming you have the rational form of sqrt(2) in its most simplified terms - it is then really simply to show that the numerator must be even, thus the denominator must be odd (otherwise it wasn't simplest form); but then you can show that the denominator must also be even (it is divisible by 4) - you have a proof by contradiction. I was shown it at school - but I am pretty sure it was in ad maths AO (for the older brits) which was age 16 A proof that it is irrational I can accept was shown at grade 10. But a proof of the existence and uniqueness? Link to comment Share on other sites More sharing options...

The Observer Posted March 5, 2012 Share Posted March 5, 2012 The proof to which you refer is in the great book A Mathematicians Apology by GH Hardy but I'm quite surprised to hear that you proved it in Grade 10. I suppose an advanced class could have done it, it certainly isn't too hard. However the existence of square roots of all real numbers was quite a bit more advanced than that. Link to comment Share on other sites More sharing options...

MathIsCool Posted March 6, 2012 Share Posted March 6, 2012 I'm assuming that connector doesn't think that ln(2) exists exactly either. =Uncool- [math]ln(2)=\int_{1}^{2}{\frac{1}{t}dt}[/math] Link to comment Share on other sites More sharing options...

imatfaal Posted March 6, 2012 Share Posted March 6, 2012 A proof that it is irrational I can accept was shown at grade 10. But a proof of the existence and uniqueness? No just its irrationality - which I think the initial claim that it was taught in year 10 boiled down to as well Link to comment Share on other sites More sharing options...

Realitycheck Posted March 6, 2012 Share Posted March 6, 2012 Pi is simply the ratio of the circumference to the diameter in a perfect circle. If the circle wasn't perfect, then the equation wouldn't hold true. We're talking about a completely different level, a whole new realm from pesky 2d coordinates. Link to comment Share on other sites More sharing options...

John Cuthber Posted March 6, 2012 Share Posted March 6, 2012 A proof of the uniqueness of root two would be interesting unless someone has redefined "unique". Last time I looked (-x)^2 was equal to x^2 Link to comment Share on other sites More sharing options...

D H Posted March 6, 2012 Share Posted March 6, 2012 (edited) A proof of the uniqueness of root two would be interesting unless someone has redefined "unique". Last time I looked (-x)^2 was equal to x^2 The discussion is with respect to non-negative numbers: For all real [math]y[/math], if [math] y \geq 0[/math] then there exists one and only one non-negative number [math]x[/math] such that [math]x^{2} = y[/math]. Proving that the square root of 2, if it exists, is irrational is something that can be done with high school level mathematics. So is uniqueness. Existence is the tough nut to crack. Even the proofs alluded to having been done in a real analysis class are possibly a bit on the soft side. To truly nail down existence you need to take algebra. Not high school algebra, mind you. I'm talking about the algebra one takes in college after having gone through calculus and analysis. Edited March 6, 2012 by D H Link to comment Share on other sites More sharing options...

ajb Posted March 7, 2012 Share Posted March 7, 2012 Existence is the tough nut to crack. Right, proofs I have seen use language and ideas not really suitable for grade 10. Link to comment Share on other sites More sharing options...

rktpro Posted March 7, 2012 Share Posted March 7, 2012 I recall it was root 2 which was first proved irrational and the uniqueness of root 2 was also proved. Did I say that existence was proved in class 10? Further, debating weather I was taught that is not the main topic here. All schools in different countries have different schooling and syllabus. Link to comment Share on other sites More sharing options...

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