Gadgetory

science dinner with minute science would any of you like some pie no no not that time this pie all these numbers just kidding that was actually one of the first videos in the minute science playlist one of the first videos on the channel period over a year ago pi day recently passed so I'll be paying tribute to that by repeating that video you just saw a few of the things I'll say in the beginning will sound very redundant but then we'll dive deeper into the topic are you ready would any of you like some pie no no no not that pie this pie all of these numbers in 1706 William Jones was the first person to ever associate the word pie but it's now familiar symbol but what is pie let's start off first the familiar equation you like we saw in geometry circumference equals pi times diameter if we simply isolate pi the right-hand side equals the quotient of circumference and diameter that's it that's the mathematical definition of pi but why then does it never terminate why is pi an irrational number this is where things get dicey though the principle itself is rather rudimentary you ready nothing in nature is perfectly circular got it that's the only thing you need to accept in order for what I'm about to say to make sense everything has ridges grooves microscopic canyons in fact we can't even simulate perfect circles using computers because the math never terminates that and our displays use square pixels to that's a separate issue but even if a perfect circle did exist we could never accurately measure it with man-made tools down to a specific digit granted that's the case for any measurement significant figure is anyone nevertheless straight lines have definite length so that computers can calculate the case however for curved surfaces so here's a way you can attempt to calculate pi good luck good luck let's start with a triangle if we inscribe an equilateral triangle that's the triangle with three equal sides inside of a circle we can use the sine rule to solve for the length of each side of the triangle we assume a circle diameter of one meter each side of the triangle must equal approximately 0.866 eaters now we've already encountered an irrational number root three and this equation never terminates so our calculation from here on out will not be exact but let's keep going anyway with 0.866 meters for one side the perimeter of the triangle is roughly two point five nine eight point eight six six times three as a ratio the triangle has an outer length that is roughly 83% of the circles length accounting for a 17 percent error now let's move on to a square arresting perfectly inside of a circle if the diameter of the circle is 1 meter then with some simple geometry we can calculate the side length of the square one over the square root of 2 or roughly 0.707 again another irrational number let's just ignore it pi times 1 for the circle yields a circumference of 3.14159265 yadda-yadda-yadda it's pi basically and the perimeter of the square is about 2.8 - 8 meters as a ratio the square has an outer length that is roughly 90% of the circle length accounting for in this case only a 10% error can you see where things are headed at this point as we increase the number of equilateral sides to the shape inscribed inside the circle we gradually approach the length of a perfect circle ergo the true value of pi but the limit of the number of sides here approaches infinity no shape no matter how many sides it has will ever become a true circle this inner shape could have 1 billion sides for all we care you'd be very close to PI but not at ty and that's why it's an approximation that's why it never terminates happy late PI day everyone I hope you enjoy the math in this one be sure to give a thumbs up if you thought it was cool thumbs down so the opposite click or subscribe if you haven't already now catch you in the next video this is Salazar studio thanks for learning with this you