Log Tables - Numberphile
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- čas přidán 10. 09. 2024
- Logarithms and log tables - what Professor Bowley used before calculators!
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Professor Roger Bowley is an emeritus professor at the University of Nottingham.
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Numberphile Quite the gem! Thanks.
It certainly is a shortcut. If you want to multiply large numbers, it can take quite a while. For instance, multiplying two five-digit numbers by "long multiplication" requires 25 single-digit multiplications and 20 to 24 single-digit additions. And at each step you can make an error. And for division, the situation is even worse. But using a log table, you just have to look up two numbers in the table, add (or subtract) them, and look up the result in the antilog table.
the antilog table
@@theshuman100 Yeah, that's what it's called.
@@EebstertheGreatis this sarcasm
@@lollol-tt3fx No. This is how people used to multiply. (Well, it's one way some people used to multiply. They also used long multiplication, slide rules, and in many parts of the world, abacuses. And starting around the turn of the century, adding machines became available, so it was possible to multiply mechanically by repeated addition. Rare methods included lattice multiplication, multiplication tables, Napier's bones, and the stepped reckoner.)
Blame the schools for not showing us this i hated math up until i found this channel.
Loving your comments....
But these tables are not a proposal or new idea or some crazy method used by weirdos...
It is a HISTORICAL FACT that people - like engineers and scientists - used these almost universally for many years because it made life easier....
And not every calculation was as simple as 37 x 59... We just used a simple example to show the table in use!
Replies
Mahadev
1256
1256
@@mahadevganujlkhad9799
Ok
A big thanks to Prof. Bowley for explaining it. It's quite embarassing but I missed in school when we first had logs and never understood it therefore. Now I'm nearly done with my chemistry degree and for the first time ever I have an idea what this is all about XD So thank you Mister
I've heard my parents talk about this who both studied maths and I finally understand what they meant. Thank you for this video. Doing maths the old fashioned way instead of using calculators intrigues me.
I cover that sometimes at sixtysymbols (my physics channel)
We really do have it much easier these days with all our electronic tools.
It would have been really cool to see how my great grand father worked as an engineer back before WWII, using log tables and slide rules.
And to think of how the field of engineering were changed while my grand father was working. With the transition from these old techniques, to using computers and pocket calculators.
Engineering slide rules, sometimes lookup results for functions of more than two inputs ;-)
You may also be surprised to hear that 20 years ago undergraduate mechanical engineers were still being taught to use a set of tables referred to as the steam tables. Tabular results of thermodynamic properties for things like water or refrigerants.
You may have been taught things like Boyles law in physics . It's a fair aproximation that is often very useful, but real gases.... they don't follow that to accuracy that can be needed. There are polynormal formulae that have been curve fitted to fit real world thermodynamic properties... but those formulaes themselves need like 20 terms to be accurate. So tables were still much easier.
No calculator I've seen has ever had such a button! Spreadsheets can have functions to do such added in.
I see what you did there! It always makes me smile when 42 is used as an example .
Love the fact that the professor said "42" at 0:42 exactly
I think the editor did some work to get that , seems very deliberate
@@ynmol7830 haha replying to a comment older than most fortnite players
This comment also has exactly 42 likes.
NOBODY MOVE.
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Vv
I love that "whoops" at 0:11. I always do that when writing numbers. I get ahead of myself and write the last digit first. Happens to us all I guess.
10:1
make it a meme lol
I do this all the time in German when I hear 2-digit numbers.
It's "Whoops to the power of two" xD
Not me.
I just missed the log table era, by the time I started O Level Maths in 1980 we were using calculators. Learned how to do all these tricks with logs, but not using tables.
Logs stump me. I once worked out the integral of 1/cabin and got a log cabin.
*a natural log cabin
squ34ky2
He threw it into the sea.
what about the C
The integral of 1/cabin is actually log cabin + C, or houseboat (a log cabin by the sea.)
Actually you would get a natural log of the absolute value of cabin / d/dx(cabin) + C so your joke kinda falls apart there :)
pretty sure if you are already have a calculator and are finding log of numbers, might as well just type in 37x59
With complex numbers it is easier to use "ln", which is logarithm with base of eulers number. It is much harder to multiply two brackets like (25456+321*i)(52516+123*i), when you convert it to an exponencial form: Pol(25456;321)*Pol(52516;123) = [25458.02382*eˇ(0.7224611278*i)][52516.4404*eˇ(0.1341946834)]; then u just multiply eulers number multipliers and add up exponents of eulers number: (25458.02382*52516.4404)*eˇ(0.7224611278*i + 0.1341946834*i) = 1336964791*eˇ(0.8566558112); which we convert back to an algebraic form by Rec(1336964791;0.8566558112) = (1336815356*19988890*i). Show me easier way to do it.
A few years back, my country's 12th graders taking exams at a similar level to a gceo cant use calculator to work out calculus. That book helps tremendously.
but in these situations you have your log tables with you?
Alex Baraban, take it on math test and use it.
Yes, obviously, but there's a relation between the binary representation and the 2log of a number (for the length of the representation n, floor(2log x) = n-1). My question was if higher precision is possible, which might provide interesting insights into the computation of logarithms in computers using exactly the same principle you mentioned.
Kero ganhar
Go ki
No judgement intended, just a common comment. It's great that you actually satisfy my expectations without even asking. I am really thankful for that.
Can't wait for the next part. I want to know how the tables were worked out.
Was there ever a follow up? For functions that can be described/aproximated as polynomials, an algorithm called the method of finite differences can be applied to generate tables row by row
@@stevecummins324 my man casually answering decade old questions here
I got one from my uncle about 15 years back. Thought it was the coolest thing ever. Can still use it a bit. You can keep your pocket calculators.
Just keep in mind log isn't always base 10, somtimes they use e or some other value as the base.
Try it, and you will discover why Napier's Bones, Log Tables and Slide Rules ruled the world until the digital calculator came along. They were all faster than long multiplication (and you made less mistakes). and it was a perfect match for physics because you could easily handle mantissa and exponent separately - and easily check your multiplications.
I'm still waiting for the next part of this...
So am I --Brady filmed it twice, and then forgot about it.
Napier imagined a number (1 -0.1)^(10) and then took powers of it. So the power of 0.2 is ((1 -0.1)^(10))^0.2=(1-0.1)^2=0.81. For 0.3 it is 0.9*0.81=0.729 and so on
To get more accuracy he started with (1-0.001)^1000, and then did the same trick. It took 20 years to work ot the table using 10^7 instead of 1000 or 10.
In the limit of 10 going to infinity the result is1/e where e is Euler's constant.
Roger Bowley I'm obsessed with making everything as intuitive as possible so would you mind explaining why he was imagining lim n -> inf of (1-1/n)^n and how that relates to creating the log table because I don't see where that came from? I've always wondered this since I first saw this definition of e, but nobody ever explained it; they just stated it.
czcams.com/video/vzV50goW_WM/video.html
+Baga Jr It wasn't about approaching e, it was to make the arithmetic easy; I think it would have been easier to use the additive form. The limit of (1-1/n)^n as n rises to infinity is 1/e, so you might as well use (1+1/n)^n. For instance, 1.001^1000 is one that can be used. The point is that there is already an exponent (1000) that can be divided by a power of ten to get fractional powers of the base.
1.001^1000 = b
b^0.001 = 1.001
b^0.002 = 1.002001 (Pascal's triangle :) )
....
b^0.006 = 1.006015020015006001
and so on.
To get your head around rational exponents, consider square roots. You might find it intuitive that a number multiplied by itself "half a time" is its square roots, e.g. 9^(1/2)=root(9)=3.
If you accept this, it's just a matter of writing, say 10^(1.56), as (10^(156))^/(1/100), i.e. the 100th root of 10^(156).
In india during junior college board exams log tables are still being used which are provided by the examiners and calculators are not allowed in exams.
That is a pretty good question. Obviously they are going to cover that in a future video. Using the definition of a Log see how many iterations of a "by hand" binary search it would take to duplicate the accuracy of the table. I'm pretty sure that would be the quickest way to do it if you were stuck on a deserted island.
Logarithms can also be used to calculate square roots by using the formula b^(log_b(n)/2), where base b is real and > 1 and n is a positive number.
It's nice to see Prof Bowley again! Haven't seen him much since he retired. I hope he and his wife are well!
THERE WERE NO CALCULATORS!
(or are you trolling!?)
Y'all get to use calculators? 😂
Surely, there were calculators, but surely they weren't mechanical.
I remember using log tables in Irish high schools in 1980s. Personal calculators in that decade were considerably expensive and they were banned in schools.
Doog
दिसावर में गेम भेजिए सर
I remember doing these in my high school homeschooling. The instructions were minimal and I had no idea what it was used for. So thanks guys.
Hi camera so much I wish you know but I didn't
I think a modern computer has simpler algorithms for doing multiplication. These were used for really big multiplications more complicated than the example shown.
I grew up using the log tables brings back memories thank you next step was slide rule and that made sense because you will be adding logs on a scale. still love my slide rule. still remember interpulating but the was a special scale for that if I remember
1/C
They can be estimated the same way one might estimate pi. Pi was found to a few dozen digits thousands of years ago by segmenting the circle into n-sided polygons. A similar process can be done with logs, testing each possible answer until you get one more digit of accuracy, and repeating until you got tired of it or the rounding error was insignificant.
And slide rules of course, we used them back in the day too.
📏📏📏📏📏📏📏📏📏📏
This is actually quite brilliant. I must get my hands on one of these.
I lost my old log-log when I was a senior in college (about the time I got my first scientific calculator). And just a few years ago in a fit of nostaliga I went on EBay and purchased a replacement (over 30 years after my first one).
I thought the same thing. The only time in school I can recall ever having trouble with math was in 4th grade when we had to do long multiplication and division. It took forever before I finally understood how to do it.
me in
As late as 1995 I was using log, trig, and other assorted tables from not a single tome, but from a voluminous set of books, each page going out to umpteen decimal places.
A-Mazing. loved it. Logs are both a fascination and an issue to me. Thanks numberphile and plz keep bringing them on. I look forward to comments on logs to other bases!
⁸⁹⁵⁰⁵¹⁹⁶⁵⁶..,,ok....ok
There are University U tube channels that post entire lecture series of calculus and other math and science courses. Try something like Stanford University. Maybe MIT. I believe they also direct you to print resources. The lectures are usually free under Creative Commons licencing. Good luck!
In practice, how big do the numbers have to be that doing this extra logarithm work, and flipping through pages beats the time it would take to do it long hand?
At least for me, using logarithms became faster than doing a pair of three digit numbers by hand, especially for division.
doa
Abhi Abhi so kar Uthe khana Banane ja rahe hain
It depends. I find that with two 3 digit numbers, it's already faster to use a table of logs, but that's just me. YMMV
Hi
For multiplication, you are correct. Logarithms do, however, crop up in calculus, and it is often useful to use logs to transform data when doing statistics.
Also, it makes sense to express certain measurements in terms of logarithms because their size varies so much. Earthquakes (Richter Scale) are an example, magnitude 8 earthquake is 10 times stronger than magnitude 7.
Multiplication with logs are a nice way to demonstrate the log rules, however.
Great explanation of log tables. Now do a video on slide rules. (I still have the ones my parents used.)
⁸⁹⁵⁰⁵¹⁹⁶⁵⁶..,,ok.....ok
guess why the first operation the visual system does with the visual signal is logarithm? changes in scene illumination level turn into an additive constant, and brightness ratios between parts of objects turn into subtraction.
I have just got a book of these, and cos and sin and natural log, etc etc...
Sooooo cool cool cool
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Well thanks to this video I now know what logarithms are, after two years in high school where we needed them and one year of university in engeneering...
Logs are used in calculating PH and PoH in chemistry, that why I'm here at least.
That's so cool! In computer science logarithms are one of the many techniques used in complexity analysis to calculate the efficiency of an algorithm
I'm looking forward to the "more logarithms" video. This topic is quite interesting.
I did use these in high school and for the first few weeks of college. Then I acquired my very first mechanical log table ... Looked a bit like a 'RULER' except it was calibrated in a logarithmic scale and had this neat little 'SLIDE' bit so you could multiply and divide numbers just by lining up the bit that would slide then move this neat little 'INDEX' window and read off the answer. It even had 'TRIG' tables built in ... The name of the device was a Log-Log-Deci-Trig-Slide-Rule.
What are log tables. Surface spread. Volume capacity. Etc. Multiplying huge numbers. Significant are the powers. So just add powers and simply multiply the significant digits.
Unless you have a log table at hand, how is it easier to do logarithms than multiplying and dividing
Best explanation actually ..
Might have been a good spot to mention the slide rule...
Chris Gonzales
Slide rules brought me here
Slide rules brought me here too. I usually use one (I have several), but I found my old book of log tables, and thought I would refresh my memory. This explanation is beautifully succinct. My log table book is very old; its first user was in the flyleaf in 1917, and its final user wrote into the page in 1926. Several users; all schoolkids at the same school, presumably.
The :"long hand" method to me is still the better method to use. It may not be as fast, but it is by far easier to double check.
Think about it using this example, with the log table method you have to check three specific numbers within a cluster of other numbers. Not to mention they are written small just so they can fit on a page. It is easy therefore to make a mistake and would effectively take longer to double check.
I remember seeing this kind of tables for the trigonometric functions and the normal distribution as well. Two years ago, my silly teacher at high school made us learn how to use them "for the case you don't have a calculator at hand". XD
It should also be pointed out that log tables have another important use (which is in fact the same).
Calculate the cube root of 5, a = cuberoot(5). We get:
Log a = (1/3) x Log 5
Using the log table we find Log 5 = 0.69897, so Log a = 0.23299. Use the anti-log table to get a = 1.709. The actual answer is 1.709.
You only need to do 1 division to get the root of a number. For the square root it's simple, you divide by 2.
If you're using big numbers and calculators don't exist, then yes it's easier to use the log tables.
^This
Yeah
WTF GRANDAYY ???? What you doin here lol
Iff and only iff
\Q&A
He added the log for 37 and the log for 59 together. He had a table showing him what these logs were for each whole number from 1 to 100 i think.
And what lovely video, I always wondered what the use of logarithms really was!
To make a log table: get the squareroot of 10, the 5'th root of that is x. So log(x) = 0.1. Square x, and double the log, so log(x*x) = 0.1 * 2, and log(x*x*x) = 0.1 * 3, and so on up to 10. That will generate a log table with 1 decimal of accuracy. For more accuracy, the 5'th root of the squareroot of x is y. For 2 decimals of accuracy, each time you multiply by x or y, you add the log of x or y. I made a log table to 5 decimal places, its 3MB, and a log search to 8 decimal places.
Good old days, remember doing this in secondary school.
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Once found two these Tables in the company of seamanship manuals. I decided to look up what they were. Apparently captains who navigated before GPS used these too, all the way up until the late 70's. I infer that they must have been widespread among all sorts of professionals. Then again, I hear the abacus is still sometimes used in the east, and experienced users can beat calculators on time.
I used to think that the secret to life was applying some function to each component, adding them together, and then taking the inverse function of the sum. Works with multiplying with logarithms, the Pythagorean theorem, resistors in parallel, parallel processing, etc. But I could never find the right function to apply.
Slide rules do use a visual representation of logarithms.
Only instead of adding numbers you're adding distances.
No mention of slide rules? I'm surprised
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Now a video about a slide rule would fit perfectly. Slide rule is exactly about multiplying by adding.
Or is there a slide rule video already? I might have missed that.
Although I truly admire Professor Bowley, I think this explanation was a little chaotic. Or is it because of the editing?
They should continue teaching log tables and cursive
Talking about logarithms and things used before calculator, it would be really great if you made an episode about slide rulers. My dad used to do calculations for his diploma thesis with one of them. Awesome tool.
How do they make these tables? I do not understand how you can calculate a number a ^ b.c. For example 10^3.8756. I know how to calculate it if b.c is a whole number..
+Quran Channel Using Napier's base, a would eventually be raised to all sorts of fractional powers, so you end up with log (base a) (something) = something else
How to get the common log? Easy. log (something) = log (base a) (something) / (log (base a) 10).
Cooper Gates how would you find the log using base10 or 2 of 57 without a calculator or table? Any videos for that floating around?
Ryan Rubidoux-Cosman exactly! How they come up with those tables!? I cant find videos of that
Oh, there are lots of ways you can work out the logs of numbers. The simplest method I can think of is to taylor expand log(1+x). This will give you the logs of numbers between 0 and 2 (not including 0 and 2). You can then construct every other log using log(x)+log(y)=log(xy).
AFAIK, using taylor series along with precomputed tables is how modern calculators/computers calculate logs.
This was the sole mean and universally accepted mean of multiplying large numbers for CENTURIES.
Pause at 0:00 - Prof. will be like: :O
Lol 😂
I absolutely enjoyed this video! At 0:51 I think he meant decimal places
Keep up the great work numberphile! (P.S this guy is awesome!)
and thats how slide rules do multiplication and division, its awesome! I love my slide rule!
Ok my andesten
i just learnt from this vid about what logarithms really meant.
9899615673 WhatsApp,
I had to use them less than 10 years ago. Calculators weren't allowed in school (< 12th grade) and this is what we were allowed to use.
Roger is amazing, but then again, all of the professors are
Plus all you need to know is the table for the logs of the numbers between 1 and 10. A big number, like 547 923 = 5.47923 * 10 000, has a log
Log 547 923 = Log 10 000 + Log 5.47923 = 4 + Log 5.47923
And you can do that with any number.
So instead of just doing multiplication, you just need to print out a Log table on a scroll, find the corresponding log, add those numbers together and then look at your scroll to find the answer. How convenient
Max Yakovlev it was convinient before calculators were invented
'This is all we 'ad down t'mine!' Brilliant
Deserves 1/0 likes
@Rebecca Beirne
You just learn the tables by heart. Like you do with the multiplication table. Quite a few people have been known to learn them. You can also learn the natural log tables while you’re at it.
This is the underlying principle on which slide rules are based.
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I had a log book like this in middle school and that was just 10 years ago!
my slide rule just got way more awesome.
I know, but we have to know how to do logs ourselves before being able to program a machine to do it. I would make sense that we did it by hand at first and I'm curious to see how it is done.
Could you expand on Natural Logs too?
I was very lucky 30 years ago that for my university entrance exam for engineer, it was the first year you could use a calculator and didn't need to use log tables. Still, I learned how to do it.
Engineers used this trick on slide rules before calculators came along
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Great to see prof Bowley again!
Thumbs up for selecting 42 as the first number not a multiple of 10!!!!!
Yep
But he did 100 and 1000 before 42.
@@TimothyReeves .
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This made me remember a stick figure comic about AES encryption. You can cover the math on numberphile, and the encryption portion on computerphile.
The Comic is available at the blog moserware, search for AES. Actually he wrote up a number of items that would be good topics.
Note: The comments will not let me reference the site directly.
Thankyou! I recently learnt logarithms and I understood them I just didn't see the point of them. But what is the use of logarithmic graphs? Why not just use an exponential graph?
logs are one of the hardest thing of math for me, never understood em really well.
A 5 minute video just explained logs better then an entire semester of College Algebra.
I wish I had watched this video when I was learning log tables at school. I can't understand why 644 have voted this video down. I can only think they didn't watch all of the video. Brilliant stuff and well worth watching!
Logarithms are really useful for exponentiation as well; a^b = 10^(b*log a)!
Before getting to Napier, I would like to state I have:
a) reformulated definition of logarithms (esp. fractional exponents)
b) used that definition to work out a table of base ten formulated in feet, inches and lines and points (12 points = 1 line, obviously, French subdivision)
c) translated that very scarce table to decimals and found it agrees with usual table fairly well
thereby proving I was right in my reformulation.
if you're going to use a table, you might as well just use a multiplication table and look it up, idk i don't see the benefit.
when you find a multiplication table that has the number 23456 * 98754.67 let me know
You're right. There isn't a multiplication table for those numbers. But while you faff about with your log tables, I'll have the answer by long-hand multiplication. Right off the bat I can see it is going to be around 2.31 billion.
@@expirydate2000 Alrite, now divide those numbers. I'll have the answer in 20 seconds with a log table.
Another video suggestion: the abacus. It's still pretty ubiquitous in many places, and has been, historically, pretty significant.
We still use logs in exams...
Thought for sure it would segue into using a slide rule. Now *that* would be an interesting bit.
Or you could...have a multiplication table?
So easy for him
Not so convenient to find 385 x 12922