I teach maths, I used Khan Academy myself as a student (especially high level maths such as Linear Algebra and Analysis) and still watch his videos. I know the stuff, he makes it interesting, fun and watchable. Of course his voice is so smooth as well, and I owe a lot to him.
wait, wasn't pi defined by the ratio of a circles circumference to its diameter (pi=c/d)? it has to be rational if so, right? why is this not the case? Also: the sum of rational numbers always is rational, right? 1/1 + 1/4 + 1/9 ... = pi^2/6; therefore pi would also be rational..., can someone explain to me why its not?
Sebastian Hilscher Statement 1: π=c/d, here c itself is irrational hence π is irrational. Statement 2: Similar to what you said 1+1=(√2)² here √2 is irrational. Also e is irrational but its series expansion includes the summation of irrationals so I think "the sum of rationals is a rational quantity" is not valid for infinite series. Someone more knowledgeable than me would be of a better help here.
A rational number in decimal form always terminates or repeats, and pi does not. You say adding a bunch of fractions comes up to pi, which is true for the infinite series you provided. You might also like to note that it is infinite, and it never repeats.
Either c or d is an irrational number in the situation you described, making c/d an irrational number too. A rational number is defined as any number that can be expressed as a ratio of integers, not just expressed as any ratio. (This would immediately contradict the existence of any irrational number; simply take the quotient of that number over 1, and now you would have a so-called rational number.) Also, an infinite sum of rational numbers is not necessarily a rational number. Any such finite sum is, but infinite sums make things funky. (This basically comes down to the fact that the set of rationals does not contain all of its limit points.) If you're more interested, you can actually find videos on RU-vid that define numbers like e (which is irrational) as an infinite sum of rational numbers and go on to prove it is irrational from that fact. Btw, it is fairly difficult to prove that pi is irrational. I believe no such proof exists without the use of calculus or more advanced techniques.
If a,b,c,d are rational and x is irrational than ax+b/cx+d is usually irrational , where do expection accur ? Plz provide this exapmle ...as soon as possible... JazakAllah.
