Pi number and circumference of a circle - what are they and how to get them

Mark Misin
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INTUITION MATTERS! - Applied Calculus for Engineers-Complete

Calculus + Engineering + PID: Functions, Limits, Derivatives, Vectors, differential equations, integrals: BEST CALCULUS

34:53:37 of on-demand video • Updated October 2020

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English [Auto] Welcome back in this section I'm going to teach you everything that you need to know about trigonometric functions. So let's get started. Let's imagine a circle let's say a ring or whatever circle body. Right. So let's just draw a ring a circle and what do we have with a circle. We have a radius and Let's denote within our and it's length. Let's say that it can be 1 metres it can be five meters. It can be as many metres as you choose. Now the Serko also has an diameter in it and in addition to that we know that the diameter consists of two radii. So if I put another radius here then two times radius would give me a diameter. So this is the diameter. And then this is them after and they are equal. Now if I take this radius and I turn it like this then it will be 300 and 60 degrees right. So what we have in the universe we have a lot of different circles with a lot of different radii with a lot of different diameters. But they are all 360 degrees so that that is that one thing that unites all circles. They all have 360 degrees. But what if we want to compute the circumference of a circle and what is the circumference Well circumference is a unit of length as well and they're in meters kilometers millimeters. And it's just fast going along circle like that and they reach back here. Then I get sick conference and still length of the outer edge of the circle and I would measure it in metres kilometers centimeters and center. Now the fact that all circles have 360 degrees gives us nothing in order to calculate the circumference. So we need another way. Now is there something else that unites all the circles in the world and in the universe for that matter. Well it turns out that there is. Because if I take a random circle and let's say I take it to conference and they denoted with C one and I take the diameter of the circle and I do know with the one and then I take a small smaller circle and then I have C two. And then I have the diameter the two. And if I take a very big circle and then I have see three circumferences three. I mean circumference we do know that it was C-3 and can be whatever length and then 1000 kilometers or maybe the the cross-section of some kind of planet and then it has its own diameter as well which is three. Now it turns out that in our universe all those circles are united by the fact that if you take their conference and divide by their diameter then they that ratio is equal to another circles circumference of its diameter. And again another circles conference divided by its amateur. In fact you can take any circle in the universe and if you calculate its circumference and divide by its own diameter then it would be equal to all other circles. Ratio and that ratio was experimentally found to be 300. Three point one four. And in fact it has an infinite amount of decimal numbers. And of course that number is pi the magical pie. Now how will we calculate the conference of a circle then. Well pi is circumference over diameter. So this is a ratio pi. So in order to get this conference already from the units now or from the notations you can see that all I need to know is the diameter of that circle and then multiply by pi. Then if I imagine pi as a ratio of circumference of diameter I cancel out diameters and I get the circumference. So all I need to know for a circle is its diameter. And then I can calculate its conference. So thats why in-conference equals PI diameter. Now diameter however is two times radius. So I can also write circumference equals two pi are right or two are two are PI. That would be a formula for circumference. And now you know where it's coming from. Now another thing that I want to mention is that pi is an irrational number. Now what is an irrational number. An irrational number is a number that cannot be expressed with a ratio. A and B. Its impossible to suppress an irrational number with a ratio. Now another irrational number would be square root of 2. It means that you are not able. You're not going to be able to to express this number as a ratio. You just can't do it. There is no way to do it. That's a rational number with the number that you can express in a ratio. For example 0.3 3 3 and they would go in into infinity an infinite number of threes. Well you can express it very conveniently with one sir but not square with of do and not pi. But wait a minute. Do we just say that pi is a ratio of circumference over diameter. So how can be an irrational number while it is an irrational number. And the reason for that is very simple you see we can always go more precise with this ratio. We can always take a more precise measurement device and and then calculate a more precise Suq conference and also more precise diameter and then the circumference. This number will change slightly and diameter will also change slightly and then pi will also change slightly. So if I have a circle and then I calculate the circumference and let's say that I use a device that is more precise than the device I used last year. Then I will have let's say more decimal numbers for the conference and for the diameter. So if I think the ratio pi is not something that is constant It's a say these numbers here. They ultimately they are not fixed. You see here they are fixed. One third that's always third it will never change. And if you divide them you will get 0.3 3 3 3 3 3 up into infinity. But with a PI this ratio conference over diameter those two numbers they can always be measured more precisely and therefore they are not fixed and therefore pi is not fixed. It's an irrational number and an irrational number of course has an infinite amount of decimal numbers after its point. And that makes sense because there is no other way an irrational number has to have an infinite amount of small numbers because if you have some kind of number let's say three point 1 4 7 5 and 8 and it's precise and there's nothing after 5 then I can express it as thirty one point four 7 5 over 10. So I can express the ratio. And in that case it's a rational number. So an irrational number by definition is a number that has an infinite amount of numbers after the decimal point and maybe a good way to visualize the entire thing is that if I have some kind of number line and I measure PI on the number line and let's say is zero here and it's not up scale so there is a gap here. So I'll just say that. OK. Here it's three point 1 4. And here it's three point 1 5. And the logic behind it is this. The more precise I become with my devices with my measurement devices the more precisely I will be able to determine my PI in between three point fourteen and three point fifty and right now I think the record is that you have 2.7 billion decimal numbers. After point. She can go very precise already. But of course for real life applications you will never need it. Now does that mean that we are unable to express a circle conference with a 100 percent precision because pi is not the pilot we have. It's not 100 percent precise. Remember pi is approximately three point fourteen. It's approximately well know we can express to conference with 100 percent precision. If we say that radius is five meters Well then circumference would be five times two. I would be 10 PI meters and we just leave it like that. We don't multiply ten times three point fourteen which would be thirty one point four. We didn't do that. We just leave the expression having PI inside and and if we really need to know the entire number we just multiply by three point fourteen. But for the ration of equations and for specific simplifying equations we just leave it as pi.