A Surprising Pi and 5 - Numberphile 

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I'm a bit surprised that after learning about gradians, Brady didn't come up with a new angle measurement known as bradians.

"If there's Pi somewhere, it means that the equation is related to circles." -3B1B

"Our choice is free, we just have to accept the consequences." Truer words.

Engineers be like “I told u guys sinx =x”

The military sometimes use 6400 units (called "Strich" in German, literally "stroke", or "line") to a full circle, which divides very nicely, and is also close enough to 2000 Pi to make distance estimates very simple if you have binoculars with "Strich" grading and know the actual size of objects. A thing of apparent angle of 1 Strich, that is 1 m long, is 1 km away.

There's a fine line between a numerator and a denominator

A more intuitive way would have been to stick with fractions, introduce the unit conversion and refactorize the 1/55555*Pi/180. You end up with: Pi/(55555*180) = Pi/(11111*5*180) = Pi/(11111*900). =Pi/9999900 Each added "5" in the trick adds an other "1" in the refactorization and a "9" in the final fractions. From here it is quite clear why this happens: adding "5" increases the divisor, which gets closer and closer to a full power of 10.

I can't stop wondering where that ladder goes to....

"It's Pi enough to cause a reaction in anybody who has seen Pi before" HAHAHA

These mathematicians all seem so freaking happy

grads are sometimes used to measure latitude on maps of France, because metre was designed so that 1 grad of lat = 100km.

sin(x) = x * laughs in engineering *

If you see π, a circle isn't far.

4:21 I was sure Brady was about to define a new unit and call it "Bradians".

I had a year of college trigonometry and still didn't quite understand radians, and you just explained it to me in about 30 seconds and now it just makes sense. I wish I had teachers that were this clear and concise when teaching.

Gradians are often used here in Sweden in land surveying. They are sometimes refered to as new degrees but best known as gon. From the Greek word “gonia” which means angle. So trigonometry means literally three-angle-measurement. They are used mainly to simplify calculations and to avoid the need for conversions between degrees minutes and seconds which have different bases (multiples).

Thanks for explaining what GRAD means on calculators! I don't think even my maths teachers knew

9:55 In Poland we use brackets for it So it will look like 0.0(000018) and it means 0.0000018000018000018....

Engineers be like: π and 5? I don't see any difference

"Pi/4 radians of pizza please." "What is that, like a slice?!" "More like exactly a slice!"

shame, he clearly should be using the legendary Gaxio

2:28 "we haven't mentioned a circle yet" but... "sin"

One thing that is missing from this explanation is in chapter two: why is the gradient of the sine function (in radians) 1, close to 0? The reason is, that the sine is basically the y coordinate from the point after walking a certain distance along the unit circle starting at (1, 0). And when you walk a tiny bit (e.g. 0.00000001π), you're basically walking straight up (to approx. (1, 0.00000001π)).

I have said this before, Ben Sparks has the best Numberphile videos. Period.

This was fun, and Ben obviously enjoyed leading us through it. Well done, guys!

@2:02 "Yaaaa, pie like"

Interesting to see Russell Crowe teaching mathematics.

Oh wow surprised to see the word vinculum here. Last time i've seen it was at the university, studying law. Vinculum iuris was the legal bound between parts of a contract in ancient Rome [Gaius]. Vinculum is a bind. The numbers here are "bound" together to be repeated. That one will be easy to remember for me !

Every video with Ben Sparks blows my mind!

This was the best recent video that you guys have put out. Props.

There’s a nice generalization of this to numbers other than 5. It turns out that if you use the number n (where n is a positive integer), then the decimal expansion tends to 5pi/n.

I like "mills" to measure angles. 1 mill is the angle of 1 m seen at a distance of 1km. It makes it easy to convert angles into lengths at a certain distance (multiply the angle in mills by the distance in km and you get the length of that angle at that distance). Why would you do ever want to do that? Artillery observers use it to correct the fall of shot. You know the target is 1.5 km away and the first short fell 40 mills to the left of the target. You correct the next shot with a "Right 60"(40 x 1.5) and you should be fine. Of course, that doesn't take into account the errors in each shot, so you actually don't correct until after a number of ranging shots. Then you add (or subtract) all errors and make one average correction. Also, to get the distance of the shot right, you use a binary search algorithm: your first correction is always a 400m "jump", then 200m, then 100m and finally 50 for the "fire for effect". And finally, you compare the fall of each shot with some kind of "standard deviation" (called "F a" or "Fourchette Apparente"). If the shot falls outside of the "F a" measurement, then you don't take it into account as it was not part of the same "statistical family" of shots and shouldn't be used. Being a forward observer, really made you use trigonometrics, mathematics and statistics in a practical way. And no calculators used! You had about 5 seconds to do all these calculations in your head and give the order to correct the next shot. You should make a numberphile about such practical uses of mathematics. (To give some background: that was how it was done up to the 1980s. Now it is all laser range finders and computers, bit I was trained to only use a map, a magnetic compass and binoculars with crosshairs graduated in mills.)

ah, it's happy tim

"Our choice is free, we just have to accept its concequences."

I don’t know if I’m more satisfied with doing it myself,... or the fact he brought out the brown paper screen to explain it right at 3:14.

"2π radians" *angry Tau noises*

7:08, damn, the most accurate handwritten sin wave

Fascinating stuff. And Ben is excellent at explaining.

Always liked Ben Sparks videos, but now he is, to me, a member of the Numberphile Pantheon which includes Cliff Stoll, James Grime, and Holly Krieger, etc. What an excellent maths communicator!

In that case we should expect tau to appear with repeated 25s. Or I suppose 2 followed by 7s and ending in a 5 due to the 2 digit carryover.

Could you do a video on the items in the background? Kind of a Numberphile/Objectivity crossover. Thanks for all the videos during this time!

Fascinating and very enjoyable as usual!

As far as I know, ancient mathematicians used 360 as it had a lot of factors((2^3)*(3^2)*5)making eventual divisions easier. Also, babilonians used a base-60 numerical system, so they might have used 360 as well

In school we did a class trip to do practical trigonometry. Essentially surveying the land around. So we used theodolites. Half of which were in Grad (360° degrees with minutes and seconds) and the other half in Gon or Neugrad(I guess gradians since it was 400 with decimal places). And you had to be careful to use the correct transformations for the final calculations of where to plot the features

I called this the perimeter constant Ω = sin(π/X)*X Where X is no. of sides Therefore perimeter is 2ΩR Where R is length of line between center and vertice Hence lim X→∞ sin(π/X)*X Approaches π

I have to say, even having degrees in both physics and engineering, I have never before encountered before today what the 'grad' on my calculator was. Thank you for enlightening me.

Unexpected pie is my favorite kind of pie

2:25 "We haven't mentioned a circle so far" but isn't referring to sine inherently referring to circles?

Great video! I always teach my 6th grade Pre-Algebra students the words "vinculum" (and its plural, "vincula") and "repetend" and tell them they can impress people by using them at the next cocktail party they attend. (I've taught Latin, too, so I explain the literal meanings whenever I can, which is actually quite frequently in math!) Thanks to this video and the comments I've just read, I can now explain the alternate representations with dots above the beginning and end of the repetend and using parentheses around the repetend. I also never knew that a fraction bar is also called a vinculum in some parts of the world. Very interesting! I think I'll start teaching my wee ones about radians and gradians, too. Or maybe I'll just show them this video and let Ben do it for me... Also, I agree with other viewers that a Bradian (Bradyan?) should be a unit of measure and that Happy Tim needs to make a series of videos about all those cool nerdy things on the shelf behind him!

I absolutely love the way he explains stuff.

Thank you for this video. I now understand this basic concept of trig

I first stumbled across this when I was playing with my calculator in year 8/9 and wondered why x*sin(180/x) approached pi as I put bigger numbers in for x, didn't get the answer from my teacher at the time but this explains it

Theodolites sometimes use gradians, I recently saw one with a scale in gradians. Looks like the gradians are also useful for stepper motors, as they have for example 200 steps per rotation.

Fantastic! The more times the magic "5" is repeated in the denominator, the more zeroes then appear between "18" and next "18" at the fractional part. What a Pi'etic fact!

”How many times would you like me to type the digit: ”5”?” *Me, a Cabtist:* ”3.”

555555 = 5/9 × [10^6 - 1]. If you do your evaluation with that construction, you don't need your repeating decimal with 18 in it and all that hand-waviness at the end.

A surprising “15 seconds ago” video

Excellent explanations, more Ben pls!

Those sound effects make everything way cooler

Pretty sure I worked it out, spoilers below: Basically it's to do with the small angle approximations and the conversion between radians and degrees, since pi/180 is the radians per degree and you're using a number such as 1/5 or 1/55 or 1/555 etc. you end up with something like pi/99900 radians. Since small angle approximations state that for radians sin(dx) ~ dx then you end up with a result of about pi/99900, since we use base 10 this is roughly equal to pi*10^-5. Edit: I showed it a similar way, but I think my way is clearer.

Grad is pretty useful for navigation. Perimeter of earth is 40 000 km. So 1Grad is 1/400 of the 40 000km.=100km. Perfect metric division. If your boat moves 100km on the equator, celestrial objects moves 1grad in the sky. If you measure that an object moved 0.1 grad beween two measures (after correction of earth rotation), your boat has just sailed for 10km. Fairly cool, isn't it ?$

that was seriously so cool! They should teach these sort of cool trick to kids in high school because many thing that math is boring but if they we're to understand the overall trigonometry, then this would blow their minds

5:37 what's to notice though that, as opposed to r, theta is *dimensionless*. For example: r has the dimension of a length, pi*r² has a dimension of an area (i.e the square of a length), 4/3*pi*r³ has a dimesnion of a volume (i.e the cube of a lenght); but 2*pi*r has the same dimension as r (i.e a length), which means the measure of an angle has no dimension. This is not to be confused with units: units of measure are abritrary and useful on a daily basis, whereas the dimensionality of a measure is more fundamental

ill never forget my calculus teacher writing “sin x ≈ x” and asking us to show that it is true in this specific case

Look it's Tim, but smiling!

Great stuff!

I’ve watched a fair amount of these videos but this was the first one I actually tried figuring it out before watching the explanation and got it

2:22 "But we haven't mentioned a circle anywhere..." You literally plugged a number into one of the circle functions.

Do I see a Klein bottle behind him? *Cliff stoll intensifies*

Omg , I study in secondary school , and really you explain it at a very easy level. Great for encouraging people who hate maths!!

Holy moly, it's the first time I learned what this Grad thing is, even if I wondered quite often when staring at my calculator. And that 1 radius turn is also a thing we never explicitly learned, although it's pretty clear if you think about it. Seems I learned about docking a Dragon and angles today. Good day.

If you knew the magnificence of the three, six and nine, you would have a key to the Universe.

In the early days of 3D computer graphics we use what we called "brads" (binary radians I guess) which ran from 0 to 256 - handy because an angle fits into a byte and you don't need such big lookup tables for trig calculations - also when you get large angles that go beyond a full circle, you can just chop off the high order bits and the angle is always between 0 and 255.

This was flipping fantastic.

Nice and very well explained!

The vinculum, never heard of that one!

"Gradians". Brady let slip the opportunity to suggest Bradyans.

I would have thought one of the strongest reasons for using 360 is the number of factors it has, 1,2,3,4,5,6,8,9,10,12,15,16,18,20,24,30,36,40,45,60,72,90,120,180,360 which makes dividing up circles much easier.

Ben! Another BIG Numberphile star!

the more 5's you add, the more accurate it gets - cool

20% off - that's 1/5th off!

Excellent relation between pi and 5...Amazing..

thanks that revived my love for maths :)

This professor absolutely reminds me of Russel Crowe.

It says 55 comments on this page when I loaded it, but I loaded it half an hour or so ago. 5/9 is zero point repeating-5. 9/5 is a term I use for converting between Fahrenheit and Celsius, so I see 1.8 a lot.

It's stuff like this that made me fall in love with math in the first place. I had wanted to study math in college but it was not to be, My dad loved it too.

Had a 400 degree compas back in the 80's, as it was standart in Scouting in Denmark for a few years, but in the end we returned to 360 degree compas. There where a transition periode where I still used the 400 degree compas, and was given a direction in RAD, so had to convert from RAD to GRAD.

My calculator doesn't have the repeating dots :(

I like unexpected pie.

10:08 "they say there's a fine line between a numerator and the denominator" lol

Thanks for teaching me about 'vinculum'!

vinculum he says huh

"Everything to do with a circle is tied up with its radius." Tau: **Shocked Pikachu face**

Nice video. Thanks

Because of the repeating pattern, the error between pi x 10^-k and sin(1/555..55) (with the right k so that the error is minimal), is also very close to pi x 10^-l for some l > k.

I haven't watched past the 2 minutes mark, but here's my attempt at an explanation. Spoiler alert, probably ? Since sin(x) ~ x for small values, I don't think it's in radians, otherwise you'd get something close to 1.8, which, to the best of my knowledge, isn't π. Another thing is that 1/(55555...555) is approaching 1/0.555555... multiplied by 10 to some power. 0.555 = 5/9, so the reciprocal will approach 9/5 (= 1.8) multiplied by 10 to some power, or 180 multiplied by 10 to some power minus 2. If the calculator is not in radians, it's probably in degrees, so you have to multiply by π/180 to get it right, which cancels to get sin(π times 10 to some power), and since it's getting smaller and smaller, the final answer is getting closer and closer to π times some power of 10. Am I right ? I don't know, I'm going to watch the rest of the video now. EDIT : Damn right.

2:25 "We haven't mentioned circles so far." Lies, sine is a circle function.

I have an old compass from the 70's or 80's with 400 degrees. Was something we used for a short time while in the scouts

This video illustrates why I think math and humor can be alike. When someone tells you the kind of joke where you pause, and then it hits you and you start laughing, it's because there's a kind of delight in suddenly understanding an unexpected connection. That happens about halfway through this video. The first part of the "joke" is when he shows a surprising result on a calculator. But the part that won a delighted grin from me was about halfway through when I started to see why it works.