The Most Useful Curve in Mathematics [Logarithms]

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6 ай бұрын

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References
The History of Mathematical Tables: From Sumer to Spreadsheets - Martin Campbell-Kelly
Navigation - James Pryde
e: the story of a number - Eli Maor
Description of the Wonderful Canon of Logarithms - John Napier
Construction of the Wonderful Canon of Logarithms - John Napier
Arithmetical Logarithmica - Henry Briggs, translated by Ian Bruce www.17centurymaths.com/conten...
The Daring Invention of Logarithm Tables - Klaus Truemper
Henry Briggs MacTutor: mathshistory.st-andrews.ac.uk...
A reconstruction of the tables of Briggs’ Arithmetica logarithmica - Denis Roegel
A reconstruction of the tables of Napier’s descriptio (1614) - Denis Roegel
The HP-35 Design, A Case Study in Innovation - David S. Cochran www.hpmemoryproject.org/wb_pa...
The Polyphase Slide Rule A Self Teaching Manual - William E. Breckenridge
When Slide Rules Ruled - Cliff Stoll

Пікірлер: 449

@adityakulkarni4549

It also describes Welch Labs upload frequency 😢

@AndrewDotsonvideos

Really niche application warning : Logarithms (large ones) permeate so many theoretical nuclear physics calculations, especially ones describing processes where multiple, widely separated scales are relevant (eg. collider events where electron + positron --> 2 jets ). These large logs can ruin so many predictions in perturbative QCD if you're not careful. The expansion parameter (alpha_s) is small, but they multiply these large logs which ruins the convergence of the expansion. People then learned how to "resum" these large logs using things like renormalization group equations and effective field theories to obtain some of the most precise predictions in QCD to date (like extracting the value of alpha_s, the strong coupling constant). Logs almost ruined perturbation theory, but instead they suggested a more powerful way of predicting things perturbatively (N^kLL accuracy: Next-to^k Leading Log accuracy) in a lot of situations.

@JonathanWaltersDrDub

I'm almost convinced we should be giving students slide rules to teach them about logarithms. Sometimes touching the mathematics makes it more real. Thanks for your time in putting this together!

@SinanKaya-cl5ho

For anyone interested, the formula is: -10^7 * ln(x / 10^7)

@richardnineteenfortyone7542

One other comment: taking square roots to 40 decimal places may not be as bad as it sounds. Once you get to a number with 20 zeroes between the leading 1 and the significant digits, you can cheat. Just cut the significant digits in half. For example, working to 10 digits, the square root of 1.0000022222 is 1.0000011111. No square rooting required; just cut something in half. I wonder if Napier cheated in this way... Imagine you are shipwrecked on a remote island with a ton of paper, a crate of pencils, and, for an incredible advantage, a crate of erasers. Something Napier did not have. He had to write in ink with a pen made from a goose feather!

@Googahgee

This is cool! Makes me realize that the term “logbook” is likely directly related to the logarithm, since it came from the “Ship’s Log.” I always used to think that logbooks (and related words) were just coincidentally the same as the word for logarithm, due to “logos” meaning knowledge and stuff, but it’s cool to see the connection between math and language!

@KipIngram

I been suddenly struck by the pervasiveness of KZbin videos on logarithms - most presenting them as this utterly amazing and wonderful thing. And they are, of course. But before I started paying attention to this I just took it for granted that at least 80% of people walking around on the street would KNOW what logarithms were - I've always regarded them as part of basic high school math that everyone learned as they came along. It's a little sad that seems not to be the case.

@philipwatson2407

The good news is that slide rules are not entirely obsolete. They remain the quickest way to calculate the effects on performance and power consumption when changing the driven speed of a centrifugal pump. The flow rate varies proportional to the speed change; the head (pressure) generated varies proportional to the square of the speed change; and the power consumption varies proportional to the cube of the speed change. Thus, if you double the rotational speed of a centrifugal pump, its flow rate will double, its generated head will quadruple, and its power consumption will increase eight times. If you have the manufacturer's performance graph at any given speed, then a single setting of the actual speed against the graphed speed on the C and D scales will allow you to re-graph the entire performance and power consumption characteristics.

@iteerrex8166

I never ran across this amazing piece of history, but I did hear that we used slide rules to do all the science and engineering to go to the moon. Unbelievable! Thanks for a great video 👍

@KipIngram

I think that asking Napier to do that arduous task again was a bit much - I don't blame him for avoiding that.

@klausluger7671

In honor of Henry Briggs I calculated logarithms of 10 from 1 to 10000000 to 16 digits of precision, with following line of python np.log10(np.arange(1,int(1e6))), which instead of 7 years of my life, took around 7 ms of my life

@eskay1891

Before watching : 23 mins is really long

@douglasstrother6584

The "Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables" by Milton Abramowitz & Irene A. Stegun contains a treasure trove of information, and not too expensive. Learning to read function tables is valuable in "sanity checking" hand and computer calculations.

@HiwasseeRiver

I learned all three in school, logs, slide ruler, and cheap calculator - this was back in the 70's. In college we used another marvelous method - Nomographs. Layered onto that was dimensionless groups. You would be shocked how far that will take you in designing the modern world. Did I mention 3D models? We had them, they were physical models, but valuable tools all the same. We also had analog computers for heat transfer. You can use amps, ohms and volts to represent complex geometries. We also had a massive IBM computer and allocation of 1 second of computing time per semester.

@KipIngram

16:34

@felipebarria3204

Fascinating. I had never considered the origin of logarithms, I thought they had been defined simply to complete the triad of operations "Power-Root-Logarithm", but this is much more intuitive. There are certain things I had to pause and write down to get a good understanding, but I feel like I can try to teach this with a more open mind now.

@Samuirai

I never understood these logic tables. Your explanation was so intuitive! Thank you!

@ferenccseh4037

In school, we were taught that "log<base a>(a^c) = c" meaning you can technically export the exponent of a number in a base. I found this explanation adequately useful (and I could remember the formula by saying AssAssin's Creed C [don't ask why that worked for me. Maybe bc it had the right number of letters in the right order?])

@samuelwaller4924

This is an amazing video. I really appreciate that you went into depth on how they were actually calculated. The realization that you can essentially do a binary search with an iterative algorithm to find any value of a function is so, and even cooler when you learn that this is how computers calculate logarithms, trig functions, etc. to this day. Basically any time you can find a relationship where x/2 = f(y) or vice versa, you can do this. It is just so cool that you can do something a crazy as logarithms or trig