Very nice. I have been cataloging proofs that are simple enough to be accessible to the earliest levels of schooling where proofs can be introduced, this is definitely one of them.
Nice and intuitive. But I have a doubt: technically, the length of the sides of the polygon s1 and s2 are functions of the number of sides n. So as n tends to infinity both s1 and s2 tend to zero, but at different rates. Now, I see the point of using the argument of similar triangles, but I'm not sure you can just use this to equalise the two limits. I mean, I'd use the similarity of triangles first (so both c1/2r1 and c2/2r2 are written as functions of, say, s1/r1), and only then apply the limit. I think it would be "cleaner"Another thing that needs to be checked is that the limits actually exist and are finite (although the intuition says it does), because as n goes up the length of the side goes down, so the overall effect on the product is not clear.But great video. I enjoyed and made me think. That's something to be thankful for!
The videos are awesome. Thanks a lot for making the videos. May I request you to make tutorial how to make this kind of video? That will be very helpful.
You have to prove that the triangles are similar. Or in other words, prove that the line segment whose length is equal to S2 is parallel to the the line segment whose length is S1.
pls someone explain: for a rectanguel, to calculate its area we multiply width by lenght , but we dont use the ratio between the longer part between the sorter one. why it is necesary to multiply by pi(a ratio) to find an area of a circle or just the circunference? i dont get why in the rest (square, triangle...) we dont use a ratio? and sorry for my bad english.
+JaimeBermudezTopefius I believe that the true question is what we want know when asking these questions and the properties of a circle. Rectangles can have sides of any length, so they don't actually have a rule of ratio between it's sides, so you just have to formulate how to calculate it's area, side times the different side to find the answer. With circles though, you have a different geometric object, and by the definition of the circle it's center always have the same distance R from all it's sides, and that makes all the circles have the common ratio of Pi between it's circumference and it's radius. Each object have it's proper ratios, like the Pythagoras theorem that states that a² + b² = c² in a rectangle triangle or the law of sines and cosines, we have many ratios.
Why are dividing circumference by 2r1 in lhs and 2r2 in rhs....i mean r1 is greater than r2...so in an eqn, we can multiply by same number on both sides...not different numbers
Sir, I have a doubt, can Pi be expressed as a fraction in the first place, I know Pi is an irrational number, meaning, it cannot be expressed as a rational number of the form (p/q); Please explain
+Vimalesh Muralidharan The circumference is never a rational number (Or if there was an example, that would mean pi is a rational number) therefor it is justifiable
+Joey Sarline the circumference can be a whole integer. For example,set the circumference equal to a whole number n. Then,the diameter of the circle is equal to n/π. However,because of this very fact, the circumference and diameter can never be both integers at the same time. This doesn't prove that Pi is irrational,but it proves that it isn't whole.
pi can be expressed as a fraction. The definition does it. Take any circle and pi is the fraction where the numerator is the circumference and the denominator is the diameter. It can also be easily written as pi/1 or 2pi/2 or ((x+1)(x-1)pi)/(x^2-1). It just can’t be written as p/q where p and q are integers.
The actual proof of the existance and uniqueness of pi, the fact that it has the value we know it has, and the fact that it is the same for all circunferences, all require some heavy real analysis
But if we define pi to be ratio of two lengths then pi becomes expressible as p/q i.e pi becomes rational, Because we can't measure diametre and circumference as irrational.
A quotient of two numbers p and q is rational if and only there exist an m and n in the set of integers such that p/q = m/n. This statement follows from the definition of rational numbers and the closure of the rational numbers under division. Do such numbers exist? Yes, and they are easy to construct. Let p = sqrt(2) and q = 1, p/q = sqrt(2) and sqrt(2) is not rational. A ratio p/q where p and q are is irrational iff at least one of either p or q are irrational (lemma 1). p/q is not necessarily irrational if both p and q are irrational (but it still can be). Using what I've labeled lemma 1 you can demonstrate that at the very minimum either the circumference of a circle or it's radius/diameter must be irrational given the irrationality of pi.
Assuming that these two triangles are similar... In non flat geometry, that's not true. The concept of parallel lines is not true there. When Einstein says space time is curved, how can we make assumption that the circle drawn on paper is flat indeed?
I'm lost, I guess I'll just accept the law of mathematics. >_< , but it is very interesting to see that it is the ratio of circumference/diameter and is always constant whatever the size. Whoever came up with it was a genius.
The details of the proof take time to understand, but as long as you understand the whole point that pi is a constant for any circle. Mainly that Circumference/diameter is constant for any circle. The ancients realized this and decided to use the greek symbol π to represent this value. Therefore π = C/d. Let me know if you have any other questions.
Suspecting it without proof is surprisingly easier than you think. Get a compass, a ruler, a piece of string (Or a ruler, piece of string, and several very circular objects such as cans of different sizes). Construct with compass or gather several circles. Wrap your string taught around the circles, mark the length of string needed to wrap around the circles. Use a ruler to measure the string lengths to get the circumference. Get a piece of graph paper (or use software) and plot the diameter on the x - axis and the circumference on the y. If you do this you will notice that the points form a straight line. If you draw a line through these points the slope will very close to the value of pi! Or put differently as the ancients probably first saw it. When you double the radius the circumference doubles; when you triple the radius the circumference triples. This establishes that there is a direct proportionality between the two, and not knowing the constant of proportionality you can give it any name you want (like pi). Notice this doesn't prove anything, but it gives some intuition about how an ancient mathematician fiddling with a ruler, a compass, and a piece of string could easily have hatched the idea that a number like pi SHOULD exist.
What came first the chicken or the egg ? Similar way what is pi? The pi is circumference of unit circle. The pi is ratio of circumference of a circle and its diameter as you said. By definition radius is half of a diameter. That means 2 times of semi diameter is equal to the diameter. Further proofing is meaningless... ? And I think egg comes first :) Thanks for the nice videos...
There is a flaw in your proof. Beginning around 1:04 you argue that because polygons with a larger number of sides resemble circles more closely, we can conclude that the sequence of the perimeters of the inscribed polygons converges to the circumference of the circle. That is not true. There are several geometrical paradoxes that make use of that bit of false logic. The sequence of perimeters does converge, but not for the reason you stated.
I appreciate your effort to teach math, and the proof is enlightening and presented mostly well, but the video describes your proof poorly. You did not prove that pi is the ratio of the circumference to the diameter of a circle. You proved that said ratio is constant for all circles - that's a different claim entirely. I would like to see this reposted with a correct title, and would upvote you.
The proof is correct. Here Pi is defined as such a constant. You need a definition of pi to start with. The argument is that such a constant exists, and that constant is defined as pi, which is correct. If you mean that you'd like to see a proof that that constant is equal to 3.14.... this same channel has a video about it: ua-cam.com/video/DLZMZ-CT7YU/v-deo.html
you re like a life saver in todays world of mindless memorization of formulae...thanks a lot.
Very nice. I have been cataloging proofs that are simple enough to be accessible to the earliest levels of schooling where proofs can be introduced, this is definitely one of them.
I'm very interested by your list. Could you briefly name the subject of the proofs ?
Class in every second of the video...
Wow!!! even in 2020 I am still amazed!!!
Thanks for the explanation ! It troubled me since this morning ^^
what is the reason at 1:44 ? I never see that kind of argument before
Nice and intuitive. But I have a doubt: technically, the length of the sides of the polygon s1 and s2 are functions of the number of sides n. So as n tends to infinity both s1 and s2 tend to zero, but at different rates. Now, I see the point of using the argument of similar triangles, but I'm not sure you can just use this to equalise the two limits. I mean, I'd use the similarity of triangles first (so both c1/2r1 and c2/2r2 are written as functions of, say, s1/r1), and only then apply the limit. I think it would be "cleaner"Another thing that needs to be checked is that the limits actually exist and are finite (although the intuition says it does), because as n goes up the length of the side goes down, so the overall effect on the product is not clear.But great video. I enjoyed and made me think. That's something to be thankful for!
ua-cam.com/video/3zTpgvgTre0/v-deo.html
AVINOAM ATZABA Bullshit.
My thoughts exactly
Excellent proof! May I ask what software you use to make your animations? They are really professional.
Thanks for watching, I use adobe flash.
+mathematicsonline It must be a hard work. Thank you! Your support means a lot!
Thank you for this. Kudos!
sir,what application are using for drawing diagrams?
Maybe javascript with processing.js or p5.js
The videos are awesome. Thanks a lot for making the videos. May I request you to make tutorial how to make this kind of video? That will be very helpful.
You have to prove that the triangles are similar. Or in other words, prove that the line segment whose length is equal to S2 is parallel to the the line segment whose length is S1.
I wished I learned this in Grade 7 math. Thank you for this explanation.
what kind of psycho teacher teaches kids limits in 7th grade
This is most difficult proof i have ever seen.
@@anjopag31 hahah i laughed so hard, but yes, I am agree with John Charlton
Very nice! Thank you!
wow sir this is out standing
pls someone explain:
for a rectanguel, to calculate its area we multiply width by lenght , but we dont use the ratio between the longer part between the sorter one. why it is necesary to multiply by pi(a ratio) to find an area of a circle or just the circunference? i dont get why in the rest (square, triangle...) we dont use a ratio?
and sorry for my bad english.
+JaimeBermudezTopefius I believe that the true question is what we want know when asking these questions and the properties of a circle.
Rectangles can have sides of any length, so they don't actually have a rule of ratio between it's sides, so you just have to formulate how to calculate it's area, side times the different side to find the answer.
With circles though, you have a different geometric object, and by the definition of the circle it's center always have the same distance R from all it's sides, and that makes all the circles have the common ratio of Pi between it's circumference and it's radius.
Each object have it's proper ratios, like the Pythagoras theorem that states that a² + b² = c² in a rectangle triangle or the law of sines and cosines, we have many ratios.
That's interesting as I can see ratio in area of rectangle. (b/a) * a^2.
This video helped to understand with greater clarity than ever that I just don’t get maths and never will. Nice animations though
I know this video was 7 years ago but how can you find the area and perimeter of a circle that has only 3/4
Thanks! Liked and subscribed!
Ummm.. I have a doubt. You divided the equation by 2r in the law it is divided by r1. Then it should have been S2/2r2 right?
Where can I find referenced 'low of similar triangles'?
Why do we use the radius to find circumference instead of the diameter.
Inductive proof of pi's existence.
Magnificent!
Thank you so much
Why are dividing circumference by 2r1 in lhs and 2r2 in rhs....i mean r1 is greater than r2...so in an eqn, we can multiply by same number on both sides...not different numbers
Brilliant
1:28 when n tends to infinity, triangle will become straight line, so concept of similar triangle is no longer valid.
Where did π come from?
He just explained it, or do you mean the symbol itself?
Thanks
What's the song name for the intro/outro?
1:38 .You didn't explain why are they similar.plz i need a reply!😔
It is because they share an angle and two sides!
great explanation!
Thanks!
+mathematicsonline What are you doing is what I would call REAL mathematics. Clear, precise, and intuitive.
C=ns? what is n?
number of sides of the polygons :D
nice one..
Sir, I have a doubt, can Pi be expressed as a fraction in the first place, I know Pi is an irrational number, meaning, it cannot be expressed as a rational number of the form (p/q); Please explain
+Vimalesh Muralidharan The circumference is never a rational number (Or if there was an example, that would mean pi is a rational number) therefor it is justifiable
+Joey Sarline the circumference can be a whole integer. For example,set the circumference equal to a whole number n. Then,the diameter of the circle is equal to n/π. However,because of this very fact, the circumference and diameter can never be both integers at the same time. This doesn't prove that Pi is irrational,but it proves that it isn't whole.
pi can be expressed as a fraction. The definition does it. Take any circle and pi is the fraction where the numerator is the circumference and the denominator is the diameter. It can also be easily written as pi/1 or 2pi/2 or ((x+1)(x-1)pi)/(x^2-1). It just can’t be written as p/q where p and q are integers.
This particular video only proves why the ratio between circumference and diameter holds same for any circle. But does not prove why it is Pi tho ???
The actual proof of the existance and uniqueness of pi, the fact that it has the value we know it has, and the fact that it is the same for all circunferences, all require some heavy real analysis
But if we define pi to be ratio of two lengths then pi becomes expressible as p/q i.e pi becomes rational, Because we can't measure diametre and circumference as irrational.
A quotient of two numbers p and q is rational if and only there exist an m and n in the set of integers such that p/q = m/n. This statement follows from the definition of rational numbers and the closure of the rational numbers under division. Do such numbers exist? Yes, and they are easy to construct. Let p = sqrt(2) and q = 1, p/q = sqrt(2) and sqrt(2) is not rational. A ratio p/q where p and q are is irrational iff at least one of either p or q are irrational (lemma 1). p/q is not necessarily irrational if both p and q are irrational (but it still can be). Using what I've labeled lemma 1 you can demonstrate that at the very minimum either the circumference of a circle or it's radius/diameter must be irrational given the irrationality of pi.
if you have a rational radius then you wont get a rational circumfrence
First Madhav an indian mathematician siant gave the theory to solve the value π = 3.14... in the form of summation of infinite series
Assuming that these two triangles are similar... In non flat geometry, that's not true. The concept of parallel lines is not true there. When Einstein says space time is curved, how can we make assumption that the circle drawn on paper is flat indeed?
well the 2 triangles shares an angle and each triangle has 2 equal sides so, not assuming anything
1:27 You don't explain why.
Pie does exist. But I eat it every time.
I'm lost, I guess I'll just accept the law of mathematics. >_< , but it is very interesting to see that it is the ratio of circumference/diameter and is always constant whatever the size. Whoever came up with it was a genius.
The details of the proof take time to understand, but as long as you understand the whole point that pi is a constant for any circle. Mainly that Circumference/diameter is constant for any circle. The ancients realized this and decided to use the greek symbol π to represent this value. Therefore π = C/d. Let me know if you have any other questions.
@@mathematicsonline no, and thanks for the explanations and the videos, very helpful.
Suspecting it without proof is surprisingly easier than you think. Get a compass, a ruler, a piece of string (Or a ruler, piece of string, and several very circular objects such as cans of different sizes). Construct with compass or gather several circles. Wrap your string taught around the circles, mark the length of string needed to wrap around the circles. Use a ruler to measure the string lengths to get the circumference. Get a piece of graph paper (or use software) and plot the diameter on the x - axis and the circumference on the y. If you do this you will notice that the points form a straight line. If you draw a line through these points the slope will very close to the value of pi!
Or put differently as the ancients probably first saw it. When you double the radius the circumference doubles; when you triple the radius the circumference triples. This establishes that there is a direct proportionality between the two, and not knowing the constant of proportionality you can give it any name you want (like pi).
Notice this doesn't prove anything, but it gives some intuition about how an ancient mathematician fiddling with a ruler, a compass, and a piece of string could easily have hatched the idea that a number like pi SHOULD exist.
What does this even mean? Is he making up stuff?
I'm assuming you didn't watch the video.
he probably did not know about limits at the time since limits are part of calculus
What came first the chicken or the egg ?
Similar way what is pi? The pi is circumference of unit circle. The pi is ratio of circumference of a circle and its diameter as you said.
By definition radius is half of a diameter. That means 2 times of semi diameter is equal to the diameter.
Further proofing is meaningless... ?
And I think egg comes first :)
Thanks for the nice videos...
What came first the chicken or the egg ? The henhouse came first
There is a flaw in your proof. Beginning around 1:04 you argue that because polygons with a larger number of sides resemble circles more closely, we can conclude that the sequence of the perimeters of the inscribed polygons converges to the circumference of the circle. That is not true. There are several geometrical paradoxes that make use of that bit of false logic. The sequence of perimeters does converge, but not for the reason you stated.
ua-cam.com/video/3zTpgvgTre0/v-deo.html
AVINOAM ATZABA No.
I appreciate your effort to teach math, and the proof is enlightening and presented mostly well, but the video describes your proof poorly. You did not prove that pi is the ratio of the circumference to the diameter of a circle. You proved that said ratio is constant for all circles - that's a different claim entirely. I would like to see this reposted with a correct title, and would upvote you.
The proof is correct. Here Pi is defined as such a constant. You need a definition of pi to start with. The argument is that such a constant exists, and that constant is defined as pi, which is correct. If you mean that you'd like to see a proof that that constant is equal to 3.14.... this same channel has a video about it: ua-cam.com/video/DLZMZ-CT7YU/v-deo.html
This is not simple proof for students who don't know about limits.👎👎