Why is PI an Irrational Number?!

- Happy belated Pi Day! Math can be fun - or at least intriguing! Use this video as an example. Hope you enjoy!

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MUSIC:
'Avocado Street' by Wes Hutchinson

[leytonelliott]

First video I watched was Minute Science: there's friction between us been watching ever since keep up the good work man.

[MrChinos]

As other people have pointed out, the fact that pi can be calculated as a limit does not imply it is irrational. You could, for example, approximate the area between the parabola y=1-x^2 and the x axis using triangles, and as the number of triangles approaches infinity, you will get a better approximation of the actual area which is 4/3: a RATIONAL number.
Also, the observation that nothing in the world is perfectly curved has nothing to do with pi's irrationality (I'm pretty sure there are no perfect parabolas in nature either). Pi, as any other mathematical object, is defined in terms that are independent from physical reality. Pi, itself, regardless of HOW we calculate it, IS irrational and the proof of this is rather more complicated and does not (and CAN NOT) appeal to our physical reality and our impediment to measure accurately. Math is not an empirical science.
FYI: If the decimal representation of a number does not terminate, that does not mean it is irrational. For example:, 1/7=0.14285714...

[chickenofbristol]

Your definition of an irrational number is wrong. You said it's irrational because it's decimal expansion does not terminate. But 1/3=.3 repeating. You should say that pi cannot be represented by a/b where a and b are integers with no common divisors.

This just bothered me because I'm getting a PhD in math.

[Astfgl]

i: "Be rational!"
π: "Get real!"

[iGamerz]

I'm pretty sure SpongeBob drew a perfect circle, so yeah you should remake this vid.

[NicolasGuerraOficial]

But 1 = 1/2 + 1/4 + 1/8 + 1/16 + ...1/2^n
Infinite operations do not necessarily mean that the number is irrational.

[mrxcs]

Who came for the brazilian meme at thumbnail ?

[asdfrozen]

Just because it's a limit doesn't mean it's irrational ...

[TheHojsimpson]

I saw a Numberphile video about Pi yesterday and then you upload this. Worth of thumbs up.

[jakeconner]

fuck yeah, science. or math in this case. Cool vid, keep up the great work

[alexedelweiss]

I really liked the thumbnail of this video. That woman is a character of a brazilian soap opera and the name of the character is Nazaré. In the soap opera she is a psychopath that used to kill her enemies pushing them form the top of the stairs.

[Tobywan]

I thought i had "learned Pi" when I was studying to become a mechanic (calculating circles/cylinders). But I never really understood it until now.
Thank you for making it so easy to understand.

[shimtest]

if there's no such thing as a perfect circle we are almost definitely living in a simulation

[mdgreathouse]

I feel like I could have gotten all this information from a quick Wikipedia search... but it's much better and more interesting to hear it from you. Thanks! Irrational numbers never really made sense to me until now.

[joeltlahiuz]

I love this "science studio" side of the channel

[BasedWarGoose]

And for Throwback Thursday... a Science Studio video!

[noleamhache]

hahahahahahahahahaha... Nazaré Tedesco in thumbnail!

[Austin]

Salazar Studio knows more about this, but he is holding out on you, oversimplifying. ;P lol

The ending example is a limit problem. Limit problems _can_ have exact solutions. But, the specific limit problem for pi does not have a solution. For example, consider the limit of 1/x as x approaches infinity. As x increases, 1/x decreases. If x is infinitely large, 1/x is infinitely small. Thus, 1/x will approach the limit of 0. In fact, calculus is based on limits, and calculus results in many exact solutions. So, that has nothing to do with why pi is irrational.

Pi is irrational because the limit does not have an exact value. I could use (and have used) calculus and the mathematical definition of a circle to derive an equation for pi. The resulting equation can not be solved for an exact solution. The only way to solve it is by using numerical methods, which would only approximate the value of pi to a certain number of decimal places. And, this is always the same result, no matter how you attempt to determine the value of pi (assuming that you do not make a mistake).

But, what happens if there is a way to calculate pi that _does_ have an exact solution? Well, there isn't. If pi had an exact solution, the numerical methods would find that value. That is the power of mathematics. You can calculate the same thing in extremely different ways and still get the same result. In the case of pi, the decimals never terminate or repeat. It cannot be written as a quotient (as mentioned in the video, you can't really have an exact length of the circumference, so c/d does not count). Therefore, pi is irrational.

The same thing applies to the square root of 3 in the video. There is no exact value for it; it has to be estimated by a numerical method. Perhaps some of you have even estimated its value yourself in high school.

[haris]

Hehe pi is irrational, but many irrational numbers exist in math. You are en engineering major so you probably took 4 semesters of math. Calc 1, 2, linear algebra, and multi variable - those were such fun classes. I think everyone should at least take calc 1.

[kinglumpy]

I really like these different themed videos