I completely agree! He's so obviously passionate and it's great. If my teachers were like this, I'm pretty sure I would have a lot more fun in my classes.
In case anyone is wondering about the square root thing at 2:15, it's pretty simple. The ratio between the dotted line and 1 has to be the same as the ratio between a and the dotted line, because if you draw lines from the ends of the diameter to the top of the dotted line, the resultant triangles have the same angles. It would be a lot better if I could draw this out, but hopefully you can visualize it. In other words, call x the length of the dotted line, and you have x/1 = x = a/x. Therefore, a = x^2, so x = sqrt(a).
@U.S. Paper Games Maybe your description is unclear but it doesn't sound like you're doing what the video demonstrated. You need a semicircle of *diameter* A+1, with a line segmenting it 1 unit from the perimeter. If our radius is 15, then A (the number we're going to find the sqrt of) is 29. So our dotted line is 14 units from the centre, and forms a right angled triangle with the radius such that its height is the sqrt of (15 squared minus 14 squared), ie root (225-196), ie root 29. Which shows you what's happening in algebraic terms: the length of A (diameter minus 1) is 2r-1, and the Pythagorean formula gives you the sqrt of (r sq minus r-1 sq)... which simplifies to the sqrt of 2r-1. Neat!
If you draw a rectangle and then draw diagonals connecting opposite vertices, the diagonals would bisect each other. So drawing a circle with a center at the intersection point between the diagonals would pass through all four vertices of the rectangle if it passes through one of them. What that means is that any triangle with points on a circle is a right triangle if the hypotenuse is the same length as the circle's diameter. If a line is drawn perpendicular to the hypotenuse passing through the point opposite the hypotenuse, then this will produce two smaller right triangles. Since the sum of a triangle's angles has to be 180 degrees, the smaller triangles will be similar since the original right angle was split into two smaller angles. By that logic, the smaller leg of the smallest triangle would have to be scaled up by a factor of the longer leg to match the size of the other triangle. And that in turn, means the longer leg of the larger triangle has a length equal to the shared leg's length squared.
a) There's no Nobel Prize for Mathematics b) No one's saying you can't solve the problem with Play-Doh. It's only impossible under the rule that you have to do it with nothing but a compass and unmarked straightedge.
I gave it a try: This is for surface area, and I will do volume after. So, let's say 'Sa' = sphere area, and 'Ca' = cube area. Let's give the sphere a radius of five. Therefore, Sa = 4pi(5^2) = 314.16 units^2. Now we have Ca which is an unknown. The formula for the area of a cube is 6a^2, so to get rid of the 6, I divided the area of Sa by six, which gives us (314.16 / 6) = 52.36. Now we're left with a^2 = 52.36, so I took the square root: sqrt(52.36) = 7.236021. So a = 7.236021, now let's plug it into the formula for the surface area of a cube: Ca = 6(7.236021)^2 = ~314.16. Seems like we got surface area, now let's do volume: A sphere with a radius of five (just like the sphere above) = (4/3)pi(5^3) = 523.6 units^3. The formula for the volume of a cube is a^3. We already solved for a when a sphere has a radius of five, so let's plug it in: (7.236021)^3 = 378.88 units^3. The cube appears to have a lesser volume than the sphere. ((523.6 / 378.88) * 100) - 100 = 38.2%. The sphere's volume is about 38.2% larger than the cube. Thanks for taking the time to read, I hope my maths is all correct. (:
Guys I found the solution to this so called 'unsolvable problem' and I will patent it so you have to pay me when you math it out except for my home state Minnesota as a gift to them.
you cannot theoretically square a circle, but realistically you can. in realist terms we are left with approximations determining the effectiveness of theorems in geometry, physics, etc. if you can find me a perfect circle in real life that has exactly an area of x^2xpi, and you can prove it to any digit within pi with no room for error, i'd eat my house.
It is possible to use materials that the Greeks had at their disposal to "square the circle": Draw circle, radius 1 (area=π) Outline circumference with string, straighten out the string, then draw line (this has a length of 2π) Divide length by 2, use triangle scaling method Use the square root finding method thing with the semicircle (to get √π) Side for square has been found Of course, there will be some error due to the elasticity of the string and the human impossibility of perfectly measuring where the string coincides with itself after one rotation among other factors, but theoretically and statistically speaking it is possible
You're allowed to use the compass as a caliper to copy distances, right? So break up an arc length into a series of piecewise line segments, and copy them out to a straight line length. If you solve for the half-width of the square , sqrt(pi/4), you only need to "linearize" 1/8th of the unit circle arc.
Pierre Bierre It wouldn’t be possible to exactly replicate the length of the arc unless you used an infinite number of line segments, which is not allowed, as the construction must be finite.
I love Dr James Grim's enthusiasm when he tries to explain such not so easy math problems. I wish I had such a math teacher. Or all my teachers. Fantastic!!! Thank you.
@@alexeysaranchev6118 It's only correct if you stick to one standard. Either accept contractions or don't. Since contractions are accepted by the vast majority, with the exception of some college teachers, the use of both "y'alls" and "could've" are grammatically correct. Of course, not in the technical sense. However, if half of our country accepts a form of a word, who cares if some college's dictionary accepts it? Language is meant to express meaning, not to be restricted by redundant rules.
Awesome presentation! Thank you! I hated straight edge and compass problems back in junior high (esp. the "is it possible" type, which are way harder than the "construct..." type). I always wondered what the point was. I wish this video had been my introduction to straight edge and compass.
I love this channel and return to it often. Not only fascinating and educational, but the sheer excitement and clarity by Numberphile is a joy to behold!
This isn't how I've heard of "squaring the circle" I'm thinking of something different I guess but I thought it was a (possibly equivalent) problem of dicing up a circle in such a way that you could construct a square from its pieces. And I think this was solved relatively recently, but using some not very satisfying feeling rules.
In case you didn't get it: √2 is an algebraic number because is is the square root of a rational number. Although there is an n where √n = π, there would have to be another number (let's call it 'm') where √m = n, and (let's call the next one 'p') where √p = m, and so on to infinity, That's why π is not an algebraic number.
Thanks a lot for the wonderful videos over the years. Just to highlight that the fact that you can approximate the side of a square that has the same area with a given circle using algebra, doesn't mean that it can actually be done. Since you can only approximate it and not really find it (pi is transcendental), it doesn't exist, no matter the intermediate tools you are using, computers or otherwise. The only tool we have in any case is our mind. Thanks again!
How can one derive the area of a circle? Imagine a circle is superimposed in a square such that the r of the circle is equal to half the side of the square. The area of the square is known as s^2. Suppose one didn't know the formula for the area of a circle. How would he/she derive it from this information?
How does calculating Pi with a calculator work? I did a simple experiment once, I typed in 3.14 instead of using Pi on the calculator, then afterwards I did the same formula again, this time using Pi, and as some of you probably already have guessed, the numbers where quite different. My question is then if the button for Pi on my calculator, is defined with a very long row of numbers, or if there's another method used in the calculator's programming to define Pi?
Its using a lot of numbers (depending on your calculator), but not quite pi. It only comes so close to it, that for us and our practical universe, it doesn't matter anymore. In fact you cant even form a perfect cirle of sphere in real life...
I wish I had math teachers that were excited about math and could rub it off on their students. Well, I did have a few, and their classes were the ones I passed. But other teachers I had, especially my college teachers... Well, I didn't take anything away from them. Now I'm going on a self teaching spree with math.
One thing that I would like to point out is that there are ways of solving this problem as well as the other two famous impossible problems of Euclidean Geometry. The three problems are 1)Squaring the Circle, 2)Doubling the Cube, and 3)Trisecting an Angle. However, it requires us to move away from Euclid's axioms. 1) & 3) can be solved using the Spiral of Archimedes and 2) can be solved using parabolas. Perhaps Numberphile can make a video about those constructions in the future.
This is also new for me so I tried searching for proof but sadly there was'nt any in the net so I made my own proof. Bear with me please. From that semi-circle, make a line from the upper part of the line measuring √(a) and connect it to the center to make a radius. So now we have a right triangle and we can make use of Pythagorean's theorem. The diameter measures (a+1) so we can say that the radius is (a+1)/2, so... HYPOTHENUSE = (a+1)/2 LEG 1 = √(a) Now, leg 2 is just the radius minus 1 right? So that means, LEG 2 = ((a+1)/2) - 1 OR (a-1)/2 Now, using pythagorean's theorem, √(a)^2 + ((a-1)/2)^2 = ((a+1)/2)^2 a + (a^2 - 2a + 1)/4 = (a^2 + 2a + 1)/4 4a + a^2 - 2a + 1 = a^2 + 2a + 1 4a - 2a = 2a 2a = 2a So that's it, hooray or something
If the arc's diameter (a+1) is labeled A_B, put a point C where a and 1 meet then move up perpendicular to A_B until it touches the arc at D. Triangle ABD is a right triangle therefore triangles ACD and BCD are similar. The relationship exists: B_C / C_D = C_D / A_C (1) The lengths are: B_C = 1 (2) A_C = a C_D = b Substitute lengths (2) into (1) to get: b/a = 1/b Therefore: b^2 = a b = √(a)
embustero71 Thank you very much for your proof! :D It works very well, and I understand it! Just one quick question, why is the value of angle ADB a right angle?
Brickfilm Man Draw two intersecting diameters in a circle (they'll cross at the center of course). Take care to notice that the outer hull of the four points where the diameters meet the circle just happen to make a rectangle with the diameter segments being its diagonals.
I wish u were 1 of my teachers in school. I hated math class but seeing someone who not only actually enjoys it but is also passionate about it brings a lot of excitement to the subject.
doggonit numberphile. I'm trying to do math homework; I take a study break, and I decide to watch a silly 4 minute video. Instead of being 5 minutes you string me along for a half hour. errrggg
We are given a square with side length "s." We need to construct a segment with length "r" so that s^2=pi*r^2. Since s is a constructible number, pi*r^2 is constructible. However, we know that pi is transcendental and not constructible so that pi*r^2=s^2 is not constructible, a contradiction. Thus, we cannot construct a circle with an area equal to a given square. Squaring the circle and circling the square are logically equivalent in fact. "Squaring" was a word for what we know call integration. So the problem is really one in just being able to talk about the area of circles in terms of how we normally measure area (i.e. with rectangles). The problem fundamentally is about the nature of pi. And the solution is ehm... really cool.
Algebra is a tool of convenience. Makes sense to me. A lot of what the arabs did was taking greek texts that came from all over the place and just consolidate it into something more interpretable.
Use the last way to construct a number. Draw a line sized pi, add 1, make a circle with pi + 1 and the height will be sqrt(pi). Get this dimension with a compass and draw the square.
Step 1: Make a circle with the radius 1 Step 2: Cut a wire the same size as the circle's circunference Step 3: Wire equals Pi Step 4: Make a line the size of the wire, add the 1 which we used for the radius Step 5: Take the square root of pi Step 6: Cut a wire of that size Step 7: Use wire to draw a square with the sides equal to the square root of pi Done.
but the wire's length would not be exactly equal because of physical limitations (atoms; material decay; acuracy and all that). You'd get, for the length of the square, and approximation of the length "root of pi".
i lovehate this channel so much. its so interesting that i end up clicking video after video in my recommended late into the night and i cant sleep because i need to ABSORB ALL THE KNOWLEDGE IN THE UNIVERSE
Fleegsta no and yes ....actually Area of a circle is exactly pi times r^2, but as u said it can only be approximated because pi can only be approximated
While trying the squaring of the circle, Is it allowed to use a thin string or twine? I mean: If i draw a circle with radius 1, i can messure the lenght of the semi circle with the twine. Now i have the lenght pi and can draw a line of this lenght + 1. Then i can draw the semi circle over this line and can messure the square root of pi like the square root of a in the video. And now i have the length to draw the sides of the square. Or am i making any mistake here?
I was thinking along similar lines in the video about an attempt to legislate that pi = 3.2. Here, the prof emphasises that they were playing by certain rules. You've stepped outside the rules that are considered pure mathematics. But I bet ancient greek engineers didn't rely entirely on the mathematicians. Archimedes invented a simple machine (trammel) which draws ellipses. If it could be made perfectly, they'd be perfect ellipses (proven by mathematicians). But it's less "pure" than just straight-edge and compasses. Who makes the rules?
Can someone explain why the sqrt(a) part of the semi circle is sqrt(a)? or just explain the steps for finding the measurements of the semi circle? thanks!
+Angel Urbina Draw a triangle by connecting the ends of the diameter to where the line sqrt(a) (call this line "h") meets the circumference. This larger triangle is a right triangle. The two smaller triangles are also right triangles. All are Similar (check by adding up angles) in two smaller triangles ratio of a/h is equal to h/1. so h^2 equals a*1 so h equals sqrt (a*1)
You people are wonderful wonderful people. I've never been great at math but it's really fun to watch your videos and enjoy it without worrying about skill
Archimedes merely found one of a long series of approximations. As mentioned in the video, Ramanujan found a very close one too. What happened in 1882 was that it was finally proven that the circle in fact CANNOT be squared using just a straightedge and a compass. When they say the problem was "solved", this is what they mean.
Take a tube with a radius of 0.5. Wrap a sheet of paper around it. Draw a line around the perimeter. Unfold the paper. You now have a line with a length of PI. Done. You just need to use warped space. Next problem?
I've watched this video for years now and I don't understand one thing. Until last week, I couldn't find any other reference of geometric constructions of arithmetic. I don't understand how multiplication/division works. Do I use an arbitrary angle? What about the unlabeled sides to the right? Is it an isosceles right triangle? Thanks to the person who clears this up to me.
Between the diameter and any point on the circle you get a straight triangle. When you add the vertical line he added you get 3 similar triangles. Similar means their ratios are the same. write down the equality between the ratios in the triangles having this vertical line in common. As you will see it shows that the unknown length squared is a.
Search "Radio cube 3".It is a shape mod of another difficult puzzle "Eitan's star".Basically,an icosahedral variant of a Rubik's cube. In my channel you can watch hundreds of videos about that kind of puzzles.Go and do so.
r + (1/9)r approximates the square root of ((L²)/2) Where the circumference of a circle with radius "r" approximates the perimeter of a square with side length "L". This is squaring the circle. Always approximates because of pi.
Here something that has always bugged me, maybe you numberphiles can help. the sum of the product of 9x anything = 9. eg 9x1 =9. 9x2 =18 the sum of the product = 9 (1+8=9) This works for 9 x anything. Why
it's cause it's always missing 1 from 10. u can think of it being +1 instead of +9/-1. so if it counting +1 for each number u got. it's the same as that number . ex 5=+5
Also, things like a := ln(2)/ln(π). Then π^a = 2. But it is certainly correct to say that if c is a nonzero integer, then π^c is irrational. This may have been what Muhammad Abdullah meant.
*pi is constructible* Let you have a wheel with circumference pi Now take that wheel and mark any point A Now put the wheel on the surface and rotate the wheel till point A reaches So, on the surface,point A to point A is equal to PI...........
You know...I randomly clicked on one video and soon watched this. I have to admit. This is far more interesting than what my math classes could teach. Yet...also could be that your British makes it more interesting lol.
+Robert Wilson III Then his comments are not directed at you. I assume you did not whine in school "but why do we have to learn _algebra_, when am I ever gonna neeeed it?"
Thanks for the reason why the squared circle compass excavation could be a 1/4 progression just smaller. When ever a smaller dimension is seen Pi/4 is used-just at those intersections. Is Pi/4 there because of the square or because it facilitated for me. I wrote a mathematical biography on the paper and detailed my name and character. I used non calculus to add the areas simultaneously. I think Pi is an imaginary number-I think its better there. I think the other type of number be put in the transcendental. x^(-2) Pi^-2 SQR(x^2) =i2 2 and -2 Pi squared, like here, SQR(Pi^2) is iPi and Pi. Brett Rauscher 2016 copy this definition of iPi.
There is an increase in accuracy of Pi by 16 times by its most immediate squared Pi in orbit around at radius 1/4 Pi wit radius of Pi and circumference of Pi. Without constructing Pi by that amount (4 Pi) just just draw a radius of Pi-4x will equal Pi. If 4 out the circle must have area 4times the area of the inner circle-the one squared here. Because the geometries connect 4*Pi - 3*Pi = iPi will create a Circle with are Pi and a square that equal the same area.
4:44 Every time I hear "transcendental" think of being a teenager, reading Edgar Allen. Waiting for rides at coffee shops till 2 a.m. Sorry, "not algebraic" will be all else that fits in there. Forever.
Okay! I have a solution! I’m not certain if this counts, but here goes. You take your compass and draw a circle with radius 1. Then you take a piece of paper and roll it up so that it exactly lines up with the circle. Cut of any excess. Unravel the paper. Find the length of the paper. (This is the circumference). Divide circumference by 2(diameter) and there is your pi. We know how to square root so that is not a problem, and to construct a square is just a matter of 4 perpendicular bisectors.
"Then you take a piece of paper and roll it up so that it exactly lines up with the circle. Cut of any excess. Unravel the paper. Find the length of the paper." None of these steps are allowed. You are only allowed to use a compass and unmarked straightedge.
And to be precise, the only allowed operations are to draw a line connecting two existing points, or a circle centered on one existing point and going through another, and then adding all points where the new line/circle intersects the existing lines and circles. (This assumes a "collapsing" compass. It has been shown that a "non-collapsing" compass, which allows taking a distance between two points and drawing a circle with that radius centered on the third point, doesn't allow any more constructions - any point that can be constructed with a non-collapsing compass can also be constructed with a collapsing compass.)
For my undergraduate seminar, I researched and presented on constructable numbers and their role in building polygons, and I myself briefly touched on this ancient problem (as well as another called "Doubling the Cube")
+John Petters If you can construct a circle, you can construct transcendental numbers. But not algebrically. So, Daniel Williams must begin to eat his house.
+Cãtãlin Pomparãu well, of course there would be transcendental numbers on the circle, but how would you know where they are? You can only find them at intersections with other lines and circles.
+John Petters It's a little missunderstanding. The only one transcendental number is the length of circle. Otherwise all points on circle are real. In my view over numbers, you can never draw a whole number, as 1 or any integer. You can only approximate the length of one unit. Integers in mathematics are approximations of a nonlinear phisycal operations, depending on PI. How can you describe in mathematical formalism term of rotation ? You must describe the final result or relations for intermediary steps. That's why the phisycs is so different. Anyway, in mathematics is something missing, and I know what is missing and how to integrate the missing element. I don't know yet why, but is only a problem of hard verifications because there are a lot of ways leading me at the same result, but there must be only 5 ways of transformation. I must verify the similitude of some of them
You might be operating with a different form of construction than the one I used (and what the Greeks used, which is what he refers to in the video). In that method, the numbers you can construct are represented by the distance between points, and you can use existing points to form lines (with a ruler) and circles (with a compass). You are also given a unit length to start with. You can find new points (and thus new numbers) where these lines and circles intersect. Using this, you can build the set of constructible numbers to include all rational numbers and some of the algebraic irrationals. But the transcendental numbers (like pi) remain elusive, hence the impossibility of the problem. Also, judging from your comment, you may have a misunderstanding about transcendental numbers themselves (unless you misspoke). The set of real numbers includes numbers that are transcendental: in fact, "most" of the real numbers are transcendental. The only numbers in the complex system that are not real are the ones that have a nonzero imaginary part.
+John Petters Is a different way to construct geometric figures. The result is obvious the same, but with reasonable accuracy, using simple tools as Euclid did. Coming back to numbers : is a general missunderstang in the way we thought about numbers. Every number is composed of 3 parts : real, imaginary and virtual. That kind of thinking extends the mathematics with a new concept (is a work in progress), for better application in phisycs. We must understand that numbers are only reflexions of reality, abstract objects, that reflects phisycal world. I can't describe here shortly the entire logic of the idea. In that way we slowly slip in number theory, wich is one leg missing. I have not enough time (I'm too old for that) to full grow all aspects, but instead, I create a simple and substantial frame for all further developments needed.
That's the major point of Zeno's paradox: a quantity that is infinitely divisible doesn't mean it is infinite. Numbers are merely symbols to communicate the quantity of something. Something that can be represented with infinite digits (such as 1/3) are not necessarily infinite. The infinite digits of π only means that it can always be divisible in a smaller order of magnitude, but it still has a finite amount (length). The perimeter of the 1-inch-radius circle will be ,and always be, 2π.
> Could a computer square a circle? There is a number of methods to algebraically approximate Pi with any given precision, and any sensible computer capable of working with fractional numbers will have this number, along with other fundamental constants like e and ln2, written in its silicon ready for use.
I could probably do it mechanically. Make a cylinder with diameter 1. Wrap a string around that cylinder. Cut it to the exact circumference. You now have a string of length pi. Add that string to the diameter of your circle. That gives you pi plus 1. Do your half-circle square root trick. You have now drawn a straight line with length of exactly pi^-2.
Yes that would be a solution, but also changes the properties of your operations. See, the interesting thing about ruler and compass constructions is that if you look at the set of points you can construct as complex numbers, the set of points is exactly those you can generate by combinations of addition, substraction, multiplication, division, complex conjugation, and square roots. The string addition, would implicitly add multiplication by pi as an operation.
omg he said pi on 3:14, i can die now.
😮😮😮😮
+Zenytram Searom he said "Pie"
OMG!
All's hidden in numbers ;)
illuminati confirmed
I love how obviously excited you get about math. That more teachers would have such zeal.
I completely agree! He's so obviously passionate and it's great. If my teachers were like this, I'm pretty sure I would have a lot more fun in my classes.
The thing is that I probably learned more from this channel than my math teachers. (Sorry math teacher...)
+Nathan Adam (SchobbishBot3000) don't apologize. These guys do it better.
Aragorn Stellar by v.
Bruh my pure teacher is this excited about maths
In case anyone is wondering about the square root thing at 2:15, it's pretty simple. The ratio between the dotted line and 1 has to be the same as the ratio between a and the dotted line, because if you draw lines from the ends of the diameter to the top of the dotted line, the resultant triangles have the same angles. It would be a lot better if I could draw this out, but hopefully you can visualize it. In other words, call x the length of the dotted line, and you have x/1 = x = a/x. Therefore, a = x^2, so x = sqrt(a).
Neat
@U.S. Paper Games exactly. the ratio couldnt be the same
@U.S. Paper Games Maybe your description is unclear but it doesn't sound like you're doing what the video demonstrated. You need a semicircle of *diameter* A+1, with a line segmenting it 1 unit from the perimeter. If our radius is 15, then A (the number we're going to find the sqrt of) is 29. So our dotted line is 14 units from the centre, and forms a right angled triangle with the radius such that its height is the sqrt of (15 squared minus 14 squared), ie root (225-196), ie root 29.
Which shows you what's happening in algebraic terms: the length of A (diameter minus 1) is 2r-1, and the Pythagorean formula gives you the sqrt of (r sq minus r-1 sq)... which simplifies to the sqrt of 2r-1. Neat!
If you draw a rectangle and then draw diagonals connecting opposite vertices, the diagonals would bisect each other. So drawing a circle with a center at the intersection point between the diagonals would pass through all four vertices of the rectangle if it passes through one of them. What that means is that any triangle with points on a circle is a right triangle if the hypotenuse is the same length as the circle's diameter. If a line is drawn perpendicular to the hypotenuse passing through the point opposite the hypotenuse, then this will produce two smaller right triangles. Since the sum of a triangle's angles has to be 180 degrees, the smaller triangles will be similar since the original right angle was split into two smaller angles. By that logic, the smaller leg of the smallest triangle would have to be scaled up by a factor of the longer leg to match the size of the other triangle. And that in turn, means the longer leg of the larger triangle has a length equal to the shared leg's length squared.
Beautiful!
*Greeks:* Straight edge and compass
*Numberphile:* Straight edge, Ccompass and loads of brown paper.
So much kraft
Sharpies too...
make a circle out of playdoh, then mold it into the shape of a square, wheres my nobel prize?
You would have to keep the playdough perfectly flat and the same height it originally was.
were talking about two dimensions though lol
watch it from above :)
a) There's no Nobel Prize for Mathematics
b) No one's saying you can't solve the problem with Play-Doh. It's only impossible under the rule that you have to do it with nothing but a compass and unmarked straightedge.
That's a compressible material, no nobel prize for you.
But can you cube a sphere?
Yea
No
I gave it a try: This is for surface area, and I will do volume after. So, let's say 'Sa' = sphere area, and 'Ca' = cube area. Let's give the sphere a radius of five. Therefore, Sa = 4pi(5^2) = 314.16 units^2. Now we have Ca which is an unknown. The formula for the area of a cube is 6a^2, so to get rid of the 6, I divided the area of Sa by six, which gives us (314.16 / 6) = 52.36. Now we're left with a^2 = 52.36, so I took the square root: sqrt(52.36) = 7.236021. So a = 7.236021, now let's plug it into the formula for the surface area of a cube: Ca = 6(7.236021)^2 = ~314.16. Seems like we got surface area, now let's do volume: A sphere with a radius of five (just like the sphere above) = (4/3)pi(5^3) = 523.6 units^3. The formula for the volume of a cube is a^3. We already solved for a when a sphere has a radius of five, so let's plug it in: (7.236021)^3 = 378.88 units^3. The cube appears to have a lesser volume than the sphere. ((523.6 / 378.88) * 100) - 100 = 38.2%. The sphere's volume is about 38.2% larger than the cube. Thanks for taking the time to read, I hope my maths is all correct. (:
same system
Figgy Winks Clear NO: you multiply the radius by an infinite number, so that you cant take the 3rd root (or any root in fact) out of it...
soooo.... who's watching this after the 'pi nearly became 3.2' vid
+Mica Santos me
me
+Mica Santos We can also say pi bearly becomes 3.15
me 3
+Mica Santos me
His skin is brighter than my future.
+stripeysoup Making me laugh at 3am... Thank you, sir!
이강민
Vantablack is brighter than mine.
ha ha :)
you almost made me choke laughing loll
and too close.
Guys I found the solution to this so called 'unsolvable problem' and I will patent it so you have to pay me when you math it out except for my home state Minnesota as a gift to them.
thats funny
Tsavorite Prince
Yes, I'll get the Nobel prize for this one
+burnsy96 I think you meant Fields medal.
no i'm pretty sure he meant the nobel prize.
burnsy96 I also live in Minnesota
you cannot theoretically square a circle, but realistically you can. in realist terms we are left with approximations determining the effectiveness of theorems in geometry, physics, etc.
if you can find me a perfect circle in real life that has exactly an area of x^2xpi, and you can prove it to any digit within pi with no room for error, i'd eat my house.
+Daniel Williams (Invents arbitrary unit such that x = 1)
Why is he so shiny?
Battle typhoon truuuuuuuuuuuuuuuuu
Battle typhoon Too much maths. It's coming out his pores.
Battle typhoon He's a robot. His skin is actually plastic.
Battle typhoon He's shiny and chrome to go to valhalla.
Cause he's brilliant. Duh!
It is possible to use materials that the Greeks had at their disposal to "square the circle":
Draw circle, radius 1 (area=π)
Outline circumference with string, straighten out the string, then draw line (this has a length of 2π)
Divide length by 2, use triangle scaling method
Use the square root finding method thing with the semicircle (to get √π)
Side for square has been found
Of course, there will be some error due to the elasticity of the string and the human impossibility of perfectly measuring where the string coincides with itself after one rotation among other factors, but theoretically and statistically speaking it is possible
use a ruler and a compass only, that's the rule.
The Greek problem only permitted compass and straightedge; there is no way to emulate your “straighten out the string” bit under these rules.
Yeah, simple really, it's called string theory !!!
You're allowed to use the compass as a caliper to copy distances, right? So break up an arc length into a series of piecewise line segments, and copy them out to a straight line length. If you solve for the half-width of the square , sqrt(pi/4), you only need to "linearize" 1/8th of the unit circle arc.
Pierre Bierre It wouldn’t be possible to exactly replicate the length of the arc unless you used an infinite number of line segments, which is not allowed, as the construction must be finite.
At time 3:14 he said "Pi"
Aditya Khanna and now your comment likes are 314 XD
i want to like it but i don't want to ruin it XD
comment something else please
+Aditya Khanna You're right....
+Aditya Khanna creepy
+Стилиян Петров I think the Zeno's paradox video doesn't say how you could really "make" a square with side Sqrt(pi).
I love Dr James Grim's enthusiasm when he tries to explain such not so easy math problems. I wish I had such a math teacher. Or all my teachers.
Fantastic!!! Thank you.
10 years later and I still come back to these videos videos 😅
I really wish I had had y'alls videos when I was a kid. I think I would've liked math A LOT more.
What sort of videos could've made you love the English language enough not to use "y'alls"?
@@alexeysaranchev6118 yaull’ses
@@alexeysaranchev6118 It's about as improper as your use of "could've". Sieg grammar I guess.
@@puppergump4117 what's the correct way then?
@@alexeysaranchev6118 It's only correct if you stick to one standard. Either accept contractions or don't. Since contractions are accepted by the vast majority, with the exception of some college teachers, the use of both "y'alls" and "could've" are grammatically correct.
Of course, not in the technical sense. However, if half of our country accepts a form of a word, who cares if some college's dictionary accepts it? Language is meant to express meaning, not to be restricted by redundant rules.
I once ingested an e. It was truly a transcendental experience. #MathJokes
Hopefully you had pi for dessert.
+The Changing Ways Meth Jokes.
+The Changing Ways - Hope it didn't require a transcendentist.
+SpaceGuru5 I can eat a whole pi, but a tau is too much to handle.
levizna Either would be just as irrational.
Algebra rocks. I've been explaining that to people since high school. Algebra is there to make sense of everything. Algebra is like the ABC's of math.
+moonblink Algebra is THE alphabet, words, and sentences of math, yo.
+moonblink Cough, Calculus is more fun, cough
***** But the fundamentals of Calculus differentiate from every other form of Algebra.
***** Really depends on what you're doing with your programs.
Tsavorite Prince
a = c - b
Awesome presentation! Thank you! I hated straight edge and compass problems back in junior high (esp. the "is it possible" type, which are way harder than the "construct..." type). I always wondered what the point was. I wish this video had been my introduction to straight edge and compass.
Dr Grime sure is a bright man, no pun intended!
I really love how passionate he is about mathematics :D it's amazing.
This channel made me like maths
and now I'm an educator sharing problem solvings based on calculations ❤️
lol I love this guy. Great smile and he absolutely enjoys his field.
Thank you for not having distracting background music, like so many others! Like given.
I'm not really into math, but so far I'm enjoying these videos, Good job!
Wish I saw enough videos of numberphile before finishing high school. I would have been more interested in maths, not that I wasn't interested at all.
I love this channel and return to it often. Not only fascinating and educational, but the sheer excitement and clarity by Numberphile is a joy to behold!
This isn't how I've heard of "squaring the circle" I'm thinking of something different I guess but I thought it was a (possibly equivalent) problem of dicing up a circle in such a way that you could construct a square from its pieces.
And I think this was solved relatively recently, but using some not very satisfying feeling rules.
No, the problem your describing is trivial. Just look at the curved parts, you're not gonna get rid of them
@@steffenjensen422 you can actually :)
In case you didn't get it:
√2 is an algebraic number because is is the square root of a rational number.
Although there is an n where √n = π, there would have to be another number (let's call it 'm') where √m = n, and (let's call the next one 'p') where √p = m, and so on to infinity,
That's why π is not an algebraic number.
You left out the crucial point that none of those numbers are rational
Thanks a lot for the wonderful videos over the years. Just to highlight that the fact that you can approximate the side of a square that has the same area with a given circle using algebra, doesn't mean that it can actually be done. Since you can only approximate it and not really find it (pi is transcendental), it doesn't exist, no matter the intermediate tools you are using, computers or otherwise. The only tool we have in any case is our mind. Thanks again!
This is friends talking about cool stuff! Loving it!
Squaring the circle? If you think that's difficult, try Cubing the Sphere! I've been trying to do that for the last 141 years!
The fact that we can't just change our units to solve this also points to something transcendental
How can one derive the area of a circle? Imagine a circle is superimposed in a square such that the r of the circle is equal to half the side of the square. The area of the square is known as s^2. Suppose one didn't know the formula for the area of a circle. How would he/she derive it from this information?
How does calculating Pi with a calculator work? I did a simple experiment once, I typed in 3.14 instead of using Pi on the calculator, then afterwards I did the same formula again, this time using Pi, and as some of you probably already have guessed, the numbers where quite different. My question is then if the button for Pi on my calculator, is defined with a very long row of numbers, or if there's another method used in the calculator's programming to define Pi?
Pi digits can be calculated using taylor series, among other methods, but your calculator is only using a fixed set of digits (10 or 12), most likely.
Slimzie Maygen
Not all of what you said is true, and I fail to see why is it relevant in connection to my reply.
Twinrehz My calculator has a verify mode, and, using this, I found it uses 13 digits of pi.
Its using a lot of numbers (depending on your calculator), but not quite pi. It only comes so close to it, that for us and our practical universe, it doesn't matter anymore. In fact you cant even form a perfect cirle of sphere in real life...
Pi is burned in the prom
He looks so excited at 4:35 talking about transcendental numbers. It's adorable.
I wish I had math teachers that were excited about math and could rub it off on their students. Well, I did have a few, and their classes were the ones I passed. But other teachers I had, especially my college teachers... Well, I didn't take anything away from them. Now I'm going on a self teaching spree with math.
Algebra is brilliant.
I knew it!
+Glenn Beeson (BeesonatotX) You don't say.
+Enlightenment I did say. And you replied.
Glenn Beeson Do you know who I am?
+Enlightenment You know that I don't hence the question. I assume this is going some where correct?
Glenn Beeson I am Ronnie Pickering! Don't you forget! :D
One thing that I would like to point out is that there are ways of solving this problem as well as the other two famous impossible problems of Euclidean Geometry. The three problems are 1)Squaring the Circle, 2)Doubling the Cube, and 3)Trisecting an Angle. However, it requires us to move away from Euclid's axioms. 1) & 3) can be solved using the Spiral of Archimedes and 2) can be solved using parabolas. Perhaps Numberphile can make a video about those constructions in the future.
I wonder what goes through your head when you solve a problem like this
At 2:26, why is the length equal to √(a)?
Thanks for your reply, but I still don't quite understand. What does that have to do with the length?
This is also new for me so I tried searching for proof but sadly there was'nt any in the net so I made my own proof. Bear with me please.
From that semi-circle, make a line from the upper part of the line measuring √(a) and connect it to the center to make a radius. So now we have a right triangle and we can make use of Pythagorean's theorem.
The diameter measures (a+1) so we can say that the radius is (a+1)/2, so...
HYPOTHENUSE = (a+1)/2
LEG 1 = √(a)
Now, leg 2 is just the radius minus 1 right? So that means,
LEG 2 = ((a+1)/2) - 1 OR (a-1)/2
Now, using pythagorean's theorem,
√(a)^2 + ((a-1)/2)^2 = ((a+1)/2)^2
a + (a^2 - 2a + 1)/4 = (a^2 + 2a + 1)/4
4a + a^2 - 2a + 1 = a^2 + 2a + 1
4a - 2a = 2a
2a = 2a
So that's it, hooray or something
If the arc's diameter (a+1) is labeled A_B, put a point C where a and 1 meet then move up perpendicular to A_B until it touches the arc at D. Triangle ABD is a right triangle therefore triangles ACD and BCD are similar.
The relationship exists:
B_C / C_D = C_D / A_C (1)
The lengths are:
B_C = 1 (2)
A_C = a
C_D = b
Substitute lengths (2) into (1) to get:
b/a = 1/b
Therefore:
b^2 = a
b = √(a)
embustero71 Thank you very much for your proof! :D It works very well, and I understand it! Just one quick question, why is the value of angle ADB a right angle?
Brickfilm Man
Draw two intersecting diameters in a circle (they'll cross at the center of course). Take care to notice that the outer hull of the four points where the diameters meet the circle just happen to make a rectangle with the diameter segments being its diagonals.
I wish u were 1 of my teachers in school. I hated math class but seeing someone who not only actually enjoys it but is also passionate about it brings a lot of excitement to the subject.
I have never been so interested in math in my whole life.
04:22 I thought Algebraic numbers are numbers which solve "rational coefficient equations" - not necessarily "constructable numbers". Like ³√2.
doggonit numberphile. I'm trying to do math homework; I take a study break, and I decide to watch a silly 4 minute video. Instead of being 5 minutes you string me along for a half hour. errrggg
What about circling the square?
+McDanny420 If you can find a circle with the area of a square, you have square with the area of a circle, sooooo...?
+McDanny420 Same way
Walking around a square is easy...
McDanny420 you got em there
We are given a square with side length "s." We need to construct a segment with length "r" so that s^2=pi*r^2. Since s is a constructible number, pi*r^2 is constructible. However, we know that pi is transcendental and not constructible so that pi*r^2=s^2 is not constructible, a contradiction. Thus, we cannot construct a circle with an area equal to a given square.
Squaring the circle and circling the square are logically equivalent in fact. "Squaring" was a word for what we know call integration. So the problem is really one in just being able to talk about the area of circles in terms of how we normally measure area (i.e. with rectangles). The problem fundamentally is about the nature of pi. And the solution is ehm... really cool.
Algebra is a tool of convenience. Makes sense to me. A lot of what the arabs did was taking greek texts that came from all over the place and just consolidate it into something more interpretable.
Use the last way to construct a number. Draw a line sized pi, add 1, make a circle with pi + 1 and the height will be sqrt(pi). Get this dimension with a compass and draw the square.
2:32 - mind blown.
Still watching in 2015
watching in 2016
+Desmond Dishwater watching in 2016.02716895
Still watching in 1996.
*****
The video was made in 2013 March. So it's closer to pi years.
4 years later still watching, again. Numberphile
Step 1: Make a circle with the radius 1
Step 2: Cut a wire the same size as the circle's circunference
Step 3: Wire equals Pi
Step 4: Make a line the size of the wire, add the 1 which we used for the radius
Step 5: Take the square root of pi
Step 6: Cut a wire of that size
Step 7: Use wire to draw a square with the sides equal to the square root of pi
Done.
+( ͡° ͜ʖ ͡° )TheNoobyGamer *Looks at comments* Oh, this has been said before? Anyways, can someone figure out
sqrt(π)
?
+( ͡° ͜ʖ ͡° )TheNoobyGamer
1.77245385090551...
Lastrevio
There you go.
but the wire's length would not be exactly equal because of physical limitations (atoms; material decay; acuracy and all that). You'd get, for the length of the square, and approximation of the length "root of pi".
Assuming it would possibly work, the lenght of the wire would equal 2Pi, not Pi.
Numberphile At 0:13 James says that squaring the circle was solved in 1882. Please show us how...
George Sorrell
Thank you for that. :-)
Solved as in proven impossible.
i lovehate this channel so much. its so interesting that i end up clicking video after video in my recommended late into the night and i cant sleep because i need to ABSORB ALL THE KNOWLEDGE IN THE UNIVERSE
Easily! You can make a square with holes in a fractal pattern to get it, that might not count as a square though, so...
what about strings? you cant put a string around a circle of radius 1, and then divide by 2? this would be pi with no doubt
You would not be able to calculate it further after millimeters,microns,atoms,etc.
you can't use strings.
I love James' skill at explanation but can I just say how CUTE he is too?! :D
You can only ever approximate the area of a circle.
As lenght of a segment too
Fleegsta no and yes ....actually Area of a circle is exactly pi times r^2, but as u said it can only be approximated because pi can only be approximated
For Harinadan Nair : But if you put r=Pi the area becomes r^3. Isn't so weird if you use the fact in physics...
Because a polygon of infinite sides can't really exist.
While trying the squaring of the circle, Is it allowed to use a thin string or twine? I mean: If i draw a circle with radius 1, i can messure the lenght of the semi circle with the twine. Now i have the lenght pi and can draw a line of this lenght + 1. Then i can draw the semi circle over this line and can messure the square root of pi like the square root of a in the video. And now i have the length to draw the sides of the square.
Or am i making any mistake here?
I was thinking along similar lines in the video about an attempt to legislate that pi = 3.2. Here, the prof emphasises that they were playing by certain rules. You've stepped outside the rules that are considered pure mathematics. But I bet ancient greek engineers didn't rely entirely on the mathematicians. Archimedes invented a simple machine (trammel) which draws ellipses. If it could be made perfectly, they'd be perfect ellipses (proven by mathematicians). But it's less "pure" than just straight-edge and compasses. Who makes the rules?
i think they would have access to string or twine..
Im so hypnotized by him, thats the stunning thing in these Numberphile clips, these people have a passion with their theme, its so fun to watch.
Can someone explain why the sqrt(a) part of the semi circle is sqrt(a)? or just explain the steps for finding the measurements of the semi circle? thanks!
+Angel Urbina Draw a triangle by connecting the ends of the diameter to where the line sqrt(a) (call this line "h") meets the circumference. This larger triangle is a right triangle. The two smaller triangles are also right triangles. All are Similar (check by adding up angles) in two smaller triangles ratio of a/h is equal to h/1. so h^2 equals a*1 so h equals sqrt (a*1)
+Titurel ohhhhhh... that makes sense. thanks!
You guys need a board or something.
Papyrus has been used too much...
+Oh Kazi but those are recycled paper aren't it? (not papyrus, but the paper used in their videos)
NYEHEHEH...HEH
That's corier new. (I think)
you mean..THE GREAT PAPYRUS
That is Kraft paper
You people are wonderful wonderful people. I've never been great at math but it's really fun to watch your videos and enjoy it without worrying about skill
So, a circle with radius 1 is just a pie with π area
@Fester Blats No, a circle with a diameter of 1 has an CIRCUMFERENCE of pi
Anifco67 No they’re right the area formula is pi times r^2 so if r is 1 then the area would just be pi.
π=3.2
+Clorox Bleach 3.2*
na because 3.14 does not round up
Clorox Bleach r/wooosh
π~3.2
π~~3.2
Archimedes merely found one of a long series of approximations. As mentioned in the video, Ramanujan found a very close one too. What happened in 1882 was that it was finally proven that the circle in fact CANNOT be squared using just a straightedge and a compass. When they say the problem was "solved", this is what they mean.
He is addicted to mathamphetamine
Why is he so shiny xD
Rare Pokemon
why are you so shiny
When a reply gets more likes than the original comment
Numberphile still going strong during the corona-lockdown! Fabulous!
Take a tube with a radius of 0.5. Wrap a sheet of paper around it. Draw a line around the perimeter. Unfold the paper. You now have a line with a length of PI. Done. You just need to use warped space. Next problem?
This problem only works in Euclidean space, you can't use a third dimension.
So the greeks didnt have numbers or algebra but they did have square roots?!
yup
I've watched this video for years now and I don't understand one thing. Until last week, I couldn't find any other reference of geometric constructions of arithmetic.
I don't understand how multiplication/division works. Do I use an arbitrary angle? What about the unlabeled sides to the right? Is it an isosceles right triangle?
Thanks to the person who clears this up to me.
i don't understand how you get the root 'a' part by adding 1?
Between the diameter and any point on the circle you get a straight triangle. When you add the vertical line he added you get 3 similar triangles. Similar means their ratios are the same. write down the equality between the ratios in the triangles having this vertical line in common. As you will see it shows that the unknown length squared is a.
That unsolved rubik's cube was driving me crazy. Anyone else?
Adarsh Singpuri ow yeah
Search "Radio cube 3".It is a shape mod of another difficult puzzle "Eitan's star".Basically,an icosahedral variant of a Rubik's cube.
In my channel you can watch hundreds of videos about that kind of puzzles.Go and do so.
r + (1/9)r approximates the square root of ((L²)/2)
Where the circumference of a circle with radius "r" approximates the perimeter of a square with side length "L". This is squaring the circle. Always approximates because of pi.
If we had no algebra there would be no cities. There probably wouldn't be any computers either, but that's all I'm saying.
Andrew S We wouldn't know the distance of roads with curves.
Dr Scrubbington There is an explanation below a comment about the same question
6:22 you forgot .org
Lol
He said Pie, on 3:14, on March 14, I am complete now, thank you Numberphile for activating the heehoo neurons in my brain.
Just because you don't know all the digits of pi doesn't mean that a square cannot have an area of exactly pi.
Starscream That's not what he claimed. Did you watch the video?
Here something that has always bugged me, maybe you numberphiles can help.
the sum of the product of 9x anything = 9. eg 9x1 =9. 9x2 =18 the sum of the product = 9 (1+8=9)
This works for 9 x anything. Why
it's cause it's always missing 1 from 10. u can think of it being +1 instead of +9/-1. so if it counting +1 for each number u got. it's the same as that number . ex 5=+5
+Paul Dogon
Look up modulo calculation and/or the proof of why a number is divisible with 9 if the digit sum of that number is divisible by 9. :)
+AlsteinLe Can i sue u? U just made me brain wrinkle.
+ʎɯɯıɾ ɔ haha...
+8070alejandro What's your preferred base then?
That you speak about maths with such enthusiasm it makes me so happy.
What if pi^^5 is rational
it isnt. no power of pi is rational
Also, things like a := ln(2)/ln(π). Then π^a = 2.
But it is certainly correct to say that if c is a nonzero integer, then π^c is irrational. This may have been what Muhammad Abdullah meant.
impossible; no power of an irrational number is rational and it's so obvious I would not need to see a proof
No power of an irrational number is rational?
as in, (√2)^2 or 2^(√2)
Because the former is a rational number by definition
Agreed :)
*pi is constructible*
Let you have a wheel with circumference pi
Now take that wheel and mark any point A
Now put the wheel on the surface and rotate the wheel till point A reaches
So, on the surface,point A to point A is equal to PI...........
A wheel is neither a compass nor a straightedge.
Ashish Gupta yep. Simple really.
Goku17yen obviously is an approximation Idiot! Plants, and all of physics is full or pi, e, phi.
You know...I randomly clicked on one video and soon watched this. I have to admit. This is far more interesting than what my math classes could teach. Yet...also could be that your British makes it more interesting lol.
Chinese did this too. Both civilizations found it to be spiritual
I'd prefer learning from a wordy proof based "greek" text book.
+Robert Wilson III Then his comments are not directed at you. I assume you did not whine in school "but why do we have to learn _algebra_, when am I ever gonna neeeed it?"
Okay, I can't watch this full screen. The close-ups kinda freak me out. PERSONAL SPACE, MAN.
"Algebra is an amazing, powerful tool in mathematics."
Thanks Muslims
Tau is better
Pi times 2
After centuries of useless discussions and argumentations, we finally got this:
"Tau is better."
-cit. TheMinecraftBoyz
Quite happy to be strung along by these two!
Thanks for the reason why the squared circle compass excavation could be a 1/4 progression just smaller. When ever a smaller dimension is seen Pi/4 is used-just at those intersections. Is Pi/4 there because of the square or because it facilitated for me. I wrote a mathematical biography on the paper and detailed my name and character. I used non calculus to add the areas simultaneously. I think Pi is an imaginary number-I think its better there. I think the other type of number be put in the transcendental. x^(-2) Pi^-2 SQR(x^2) =i2 2 and -2 Pi squared, like here, SQR(Pi^2) is iPi and Pi. Brett Rauscher 2016 copy this definition of iPi.
If a circle is drawn with circumference C multiplying the diameter produces liner productions of what was plotted of C and new diameters. : )
There is an increase in accuracy of Pi by 16 times by its most immediate squared Pi in orbit around at radius 1/4 Pi wit radius of Pi and circumference of Pi. Without constructing Pi by that amount (4 Pi) just just draw a radius of Pi-4x will equal Pi. If 4 out the circle must have area 4times the area of the inner circle-the one squared here. Because the geometries connect 4*Pi - 3*Pi = iPi will create a Circle with are Pi and a square that equal the same area.
4:44 Every time I hear "transcendental" think of being a teenager, reading Edgar Allen. Waiting for rides at coffee shops till 2 a.m. Sorry, "not algebraic" will be all else that fits in there. Forever.
Okay! I have a solution! I’m not certain if this counts, but here goes.
You take your compass and draw a circle with radius 1. Then you take a piece of paper and roll it up so that it exactly lines up with the circle. Cut of any excess. Unravel the paper. Find the length of the paper. (This is the circumference). Divide circumference by 2(diameter) and there is your pi. We know how to square root so that is not a problem, and to construct a square is just a matter of 4 perpendicular bisectors.
"Then you take a piece of paper and roll it up so that it exactly lines up with the circle. Cut of any excess. Unravel the paper. Find the length of the paper."
None of these steps are allowed. You are only allowed to use a compass and unmarked straightedge.
And to be precise, the only allowed operations are to draw a line connecting two existing points, or a circle centered on one existing point and going through another, and then adding all points where the new line/circle intersects the existing lines and circles. (This assumes a "collapsing" compass. It has been shown that a "non-collapsing" compass, which allows taking a distance between two points and drawing a circle with that radius centered on the third point, doesn't allow any more constructions - any point that can be constructed with a non-collapsing compass can also be constructed with a collapsing compass.)
For my undergraduate seminar, I researched and presented on constructable numbers and their role in building polygons, and I myself briefly touched on this ancient problem (as well as another called "Doubling the Cube")
+John Petters If you can construct a circle, you can construct transcendental numbers. But not algebrically. So, Daniel Williams must begin to eat his house.
+Cãtãlin Pomparãu well, of course there would be transcendental numbers on the circle, but how would you know where they are? You can only find them at intersections with other lines and circles.
+John Petters It's a little missunderstanding. The only one transcendental number is the length of circle. Otherwise all points on circle are real. In my view over numbers, you can never draw a whole number, as 1 or any integer. You can only approximate the length of one unit. Integers in mathematics are approximations of a nonlinear phisycal operations, depending on PI. How can you describe in mathematical formalism term of rotation ? You must describe the final result or relations for intermediary steps. That's why the phisycs is so different. Anyway, in mathematics is something missing, and I know what is missing and how to integrate the missing element. I don't know yet why, but is only a problem of hard verifications because there are a lot of ways leading me at the same result, but there must be only 5 ways of transformation. I must verify the similitude of some of them
You might be operating with a different form of construction than the one I used (and what the Greeks used, which is what he refers to in the video). In that method, the numbers you can construct are represented by the distance between points, and you can use existing points to form lines (with a ruler) and circles (with a compass). You are also given a unit length to start with. You can find new points (and thus new numbers) where these lines and circles intersect.
Using this, you can build the set of constructible numbers to include all rational numbers and some of the algebraic irrationals. But the transcendental numbers (like pi) remain elusive, hence the impossibility of the problem.
Also, judging from your comment, you may have a misunderstanding about transcendental numbers themselves (unless you misspoke). The set of real numbers includes numbers that are transcendental: in fact, "most" of the real numbers are transcendental. The only numbers in the complex system that are not real are the ones that have a nonzero imaginary part.
+John Petters Is a different way to construct geometric figures. The result is obvious the same, but with reasonable accuracy, using simple tools as Euclid did.
Coming back to numbers : is a general missunderstang in the way we thought about numbers. Every number is composed of 3 parts : real, imaginary and virtual. That kind of thinking extends the mathematics with a new concept (is a work in progress), for better application in phisycs. We must understand that numbers are only reflexions of reality, abstract objects, that reflects phisycal world. I can't describe here shortly the entire logic of the idea. In that way we slowly slip in number theory, wich is one leg missing. I have not enough time (I'm too old for that) to full grow all aspects, but instead, I create a simple and substantial frame for all further developments needed.
That's the major point of Zeno's paradox: a quantity that is infinitely divisible doesn't mean it is infinite.
Numbers are merely symbols to communicate the quantity of something. Something that can be represented with infinite digits (such as 1/3) are not necessarily infinite. The infinite digits of π only means that it can always be divisible in a smaller order of magnitude, but it still has a finite amount (length).
The perimeter of the 1-inch-radius circle will be ,and always be, 2π.
> Could a computer square a circle?
There is a number of methods to algebraically approximate Pi with any given precision, and any sensible computer capable of working with fractional numbers will have this number, along with other fundamental constants like e and ln2, written in its silicon ready for use.
This channel is severely underrated
I am terrible at math. Geometry I get, any kind of practical application type math. But, I find these videos fascinating.
I could probably do it mechanically. Make a cylinder with diameter 1. Wrap a string around that cylinder. Cut it to the exact circumference. You now have a string of length pi. Add that string to the diameter of your circle. That gives you pi plus 1. Do your half-circle square root trick. You have now drawn a straight line with length of exactly pi^-2.
Yes that would be a solution, but also changes the properties of your operations.
See, the interesting thing about ruler and compass constructions is that if you look at the set of points you can construct as complex numbers, the set of points is exactly those you can generate by combinations of addition, substraction, multiplication, division, complex conjugation, and square roots.
The string addition, would implicitly add multiplication by pi as an operation.