Approximations. The engineering way. 

"for calculation purposes, let asume this cow is perfectly round"

I'm an engineer I see approximation I click

"Approximations" Oh cool "The Engineering way" _oh boi this is gonna be good_

The Forbidden Math

Mathematicians: We need exact solutions! Engineers: Nah, "close enough" is good enough.

I'm only at 0:16 and I'm already having numerical computing class flashbacks (took that class ten years ago). Netwon Raphson, Regula Falsi, Runge-Kutta. It's all coming back.

"Why be right when you can approximate?"

Math never fails to surprise me, I could not even think such a thing could exist

Ah, the fundamental theorem of engineering.

We’re doing this in my calc class rn and I swear to god you explain it better than my professors

Please keep making these so I can make it through college.

That's a really cool formula

first law of engineering: everything is linear

dude I was just expecting to get some stuff like pi = 3 = 3 or g^2 = 10 or something like that, but I actually learned a lot!

Nice Clock and Watch, where can I get one of deeze, Zach? :^D

That square root approximation is elegantly simple. Each guess is just the average of the previous guess, and the number over that previous guess. As you approach the root, it becomes the average of the root and the number over the root (number over root is the root). So beautiful

I did a Bachelors thesis partly on this, when I finally got how it worked when I saw it, it was almost magical.

I had to use the newton raphson method in my engineering career a few years ago to approximate a function (solving a Civil Engineering equation backwards with multiple square roots in weird places) that otherwise converges on a few nonreal/negative answers and one real, positive one I was looking for. I never thought I would actually apply it in my life when I learned it, but it felt so cool to have a real world application for it! Made me realize that weird, theoretical math part of my degree wasn't quite such a waste of time after all!

We ❤ approximations! Honestly, sometimes wanting an exact solution is lazy. People don't realize how much math goes into designing numerical methods and proving their convergence and stability.

Zach : It's possible to get stuck in an infinite loop. Float error : IT'S MY TIME TO SHINE

Interesting topic! This reminds me on programming in BASIC interpreter 40 years ago. At that time the value of PI was not implemented, the solution was : 4*arctan(1), which gave PI with the accuracy of devise's BASIC.

We’re literally on this exact topic in calculus right now

The effort put into these videos is just amazing. And the educational content, truly first class. Keep up the good work Zach!

Really cool to see these real world applications- the way you teach math makes it fun and interesting!

Absolutely beautiful. I learned that stuff year ago at the university, but you described it so so much better.

What a great video 👌 It would have been such a great starting point for me a while back when I was writing GPU algorithms for fast square and cube roots of float 32 and float 64 values. Managed to get them super fast combining Taylor series expansions, the power laws and the good old Newton raphson iteration. If I remember correctly, about 3ns to compute cube root to fp64 precision.

quality content as always

Thanks a lot for the amazing info dude, it's satisfying to get stuff explained by you

Quickly becoming my favorite youtube channel!

Very awesome video Zack.. Keep up the good work..

Thanks for the timely video and inspiration! Just finished related rates in Stewart's calculus and the literal next section is linear approximations. Loved this video and can't wait to be thoroughly confused by that coming numerical analysis video lol

Great video! I was wondering if you would mention the Quake fast inverse square root and then bam! Awesome. Keep up the great work!

As an Engineer I relate to these useful approximations. Thank you so much for theses examples and explanations!

This is so incredibly helpful. I literally had a numerical analysis assignment last week where we had to use Newton Raphson

Applied Numerical Methods. I don't remember the exact name but I remember a technique which converts a definite integral to two (or natural number) terms. Gauss quadrature rule, was it? I honestly was intrigued by this method.

Doing it at engineering school, and very happy to find it on RU-vid ! Thanks

Good stuff. Nice job.

long ago I wrote a integer Square root on a DSP processor. It used the DSP's single cycle multiplier to create the square. then it compared it and set one output bit. after 16 loops I had a 16 bit result.

Great timing. I'm starting my numerical analysis class at uni tomorrow

Looking forward to a video about Numerical Analysis, I'm taking it in the fall!

could have used this video last semester during numerical methods. you explained it better in 14 minutes than my prof did in 3 lectures

Thank you for bringing context to an otherwise "insignificant" topic covered for 15 mins in a first year calculus course! I thought I hated math, but I've just been missing out on how much fun it can be once you wrap your head around the concepts

I heard about it before but was thinking why isn't it too famous thanks for elaborating it. I always wanted to know more about it keep it up😀😀😀👍👍🙏🙏

Numerical analysis is the coolest class of functions that have already been written for you

ooh boi i am going through these in my current semester and already coded the fn for iterattive method and newton raphson, loved to know more on it😊

This video would have been glorious half a year ago... Had a University course in evolutionary game theory and literally all of it was linear approximation because biological/evolutionary models are only estimations and I did not understand what a fixed point was. Seems so easy now... Thanks a lot!

I have a Numerical Analysis midterm in 8 hours so i clicked on this as soon as i saw it in my sub box, thanks ^^

That was one of the coolest videos about a table on my calculus book that I took as magic

Awesome video!

The quake 3 fast inverse square root video got me into watching these kinds of videos. Now that's a meme you'll want to see.

Yas! You posted something on your OG profile! LIT 🔥

great vid as always

Thank you very much for this video.

Damn it's been ages since I did maths "properly", but this was really accessible and a good reminder of how it all slots together. Thank you!

Holy crap thanks for explaining this, the random pdfs that I found on the internet are confusing as hell.

You should definitely talk about the finite element method. Approximating differential equations is a huge deal in engineering (especially civil/mechanical/aerospace)

Beautiful!

Please make videos like this. It was a wonderful video.

Numerical Methods was one of the more rigorous and work-intensive courses in my mechanical engineering workload so far

Dude, I really have to watch all your videos about engineering’s stuff. im in my second year and there is a lot of things i have to be familiar with

I read this under the heading computational methods TODAY!!

10:43 that iteration method is just computing the finite simple continued fractions of the golden ratio, and will converge to its simple continued fraction. A great opportunity to bring up that topic :D

11:53 both solutions of the equation are the golden ratio, but one is the longer side/shorter side and the other one is the reciprical, shorter side/longer side

Very cool video!!

The way your computer calculates square roots (assuming it's a recent computer) is using a related method, Goldschmidt's algorithm. Let Y be an approximation to sqrt(n). Set: x_0 = Y*n h_0 = Y*0.5 And iterate: r_i = 0.5 - x_i * h_i x_{i+1} = x_i + x_i * r_i h_{i+1} = h_i + h_i * r_i Then x_i converges to sqrt(n) and y_i converges to 1/2sqrt(n). As hinted at in the video, some approximations have advantages over others. In this case, the advantage is that the "inner loop" is three copies of the same operation a + b * c, called a "fused multiply-add". This saves on circuitry compared to Newton-Raphson methods.

I took numerical analysis in uni (I think it was called numerical methods) and they recommended to have two scientific calculators to iterate calculations more efficiently (if we're not going to bring our laptops to use excel in class)

Right after watching this video, I listened to Bob Dylan singing "Queen Jane Approximately" from "Blonde On Blonde". Dylan really sucks at rigorous explanation, and Newton-Raphson is also well-presented elsewhere ad nauseam. I understand that going beyond the basics is more difficult, which makes producing lots of videos less likely, and maybe no one will ever even look for the next steps. That is the dilemma of the youtube STEM educator, and is in large part why MIT's OCW series and similar stuff exists and is valuable. That said, it's great that you are reaching out to learners who are just starting out. Well done, man. L'chaim.

Just to contribute an interesting point here. Arguably the most significant piece of evidence we have when it comes the global regularity problem for the Navier Stokes equations is Terence Tao’s work on the subject. His biggest paper on the subject showed that for an approximated form of the Navier Stokes equations (one that has been averaged in an extremely specific and accurate way) blow up results occur. The relevance of this is two fold 1. This may very well be one of if not the most complicated approximations ever thereby showing how approximations are an important part of math and science at every level And 2. It shows that even pure mathematicians can use approximations to create partial progress on the toughest problems ever. That result was huge as it showed both that there is a possible pathway toward a full solution and it also showed that any attempt at proving global regularity in the positive would require methods which delve into the finer nonlinear structures with the full pde that got averaged out in the approximation. In many ways, this paper is why most of the community believes that global regularity for Navier Stokes is going to be solved in the negative whenever it happens.

Some approximations are quite good. If you use 22/7 for the value of Pi then on a 100 ft. diameter circle the circumference error is ~one and a half inches.

Everyone in comments section: it was about time that you decided to finally make a video this

THANK YOU.

Im currently taking a numerical analysis course right now, this 10 minute video made more sense than the whole class has this semester -.-

Another example of numerical approximations of things that are hard to arithmetically calculate is a matrix inverse. Similar to the iteration pf the square root, there is a simple iteration process that leads to a good approximation of the matrix inverse, which takes way longer to compute than the square root, both on a camculator and by hand

0:00 I've been studying and practicing English for the last 22 of my 25 years of age, but only now I found out that, unlike my mother tongue, English has two separate words for clocks and watches, despite I've known and used both words for years now.

pretty cool stuff

i remember this when i took numerical method class. we used loop method to program this

approximations the engineering way: 𝝅=e=3, g=10m/s²=9=𝝅²=e²

Chebyshev Approximations are also very useful.

Just proof lim (xn)n=sqrt(c) but that wouldn’t be engineering style

there are some really cool algorithms. First order methods that use only the derivative and second order methods that need fewer iterations but are damn expensive. @Zach Star Please make a video on gradient descent. Hopefully some of the my students will see the simple version and we can move directly into the more involved variants. There is plain gradient descent, smooth gradient descent, accelerated gradient descent, mirror descent, coordinate descent, BFGS and L-BFGS.

A control theory and applications would be cool to control systems as well as machine learning applications also love the video applications to engineering with algorithms used in matlab and simulink modeling and simulation is such a great field!

¡Gracias!

Video would have helped so much in understanding my numerical methods class if it was a year ago

The clock looks awesome

One that I can remember (I picked it up from one of Clive Sinclair's companies) is that Pi to 6 decimal places is 355/113. Dates back to the early calculators of the 70s, before scientific calculators were available at affordable prices. BTW - 3550001/1130001 does this to 8 decimal places.

would love to see a video about interpolation/extrapolation :)

Where was this video at the start of the semester. Could have saved me so much time trying to get the early chapters in the numerical analysis class I am taking...

Diophantine approximation is a surprisingly interesting area of number theory too.

i also have an amazing aproximation technique it's done like this "hmm root of 17 has to be more then 4 since 4^2 =16 but less then 5 since 5^2=25, it's closer to 4 so for all intents and purposes it's 4"

Legendary

There is a better equation for finding the square root: Xn+1 = A/2 + C/2A, where A is our first estimation and later subsequent numbers as we go for solving X2 and so on... If we get our first estimation near the true value (that shouldn't be too difficult), we can solve the square root of C in 2 - 3 lines only !!

Plugged it into desmos (using the simulations feature) and decided to play around a bit. Set c to 4, got to 2 fairly quickly. Set c to 2, got to sqrt(2) quickly, but slower. Set c to 1, got to 1 reasonably quickly. Set c to 0 and something interesting happened: the value got extremely close to 0, yet it was never exactly 0, fluctuating until I hit the stop button. I have an idea for what to do next :)

I remember doing this in maths. Just not sure if it was GCSE or A-level. Edit: It was A-level

I only learned something similar I think. It was based on Taylor series and you got out of it the approximate value after iteration and how big the error was. I don't remember it very well - it was years ago - but I think if you wanted to know for example sqrt(10) - you just used sqrt(9) and sqrt(16) - so basically bracketed a number, or used (I don't remember if you needed bigger and smaller, or just 1 number that was close and knows) an easy number with the same function (in this case sqrt(x)) to start approximation. Sorry - I learned it, while learning Calculus. It looks a bit like the second method in 12:25 - but I'm not 100% certain.

amazing

Ok, so I just tried the square root formula on excel and it is so damn satisfying.

can't wait for the numerical analysis examples that took way longer than expected :-)

One of the best channels iv seen , i studied business intelligence and math decisions and we had a lot of approximation not arrethmetic ones called #Heuristics it would be good to do a video on someof the most known Also qquestion to people who read this , do we really know how to calculate sqare root ?

How would using for the x_0 the value of the m-th power of c improve the algorithm, where m is 'rootnumber?' ?

Awesome