Merry Christmas Michael, I really enjoy your videos. I want the history of egyptian fractions. That should be fascinating. I hope 2023 will be the year to exponential growth of your channel. My best wishes for you and your family. Greetings from Venezuela
I think the Egyptian notation is really interesting. I read a book once that had a large section on Egyptian mathematics. Seems to me they had an exception for the fraction 2/3. I lent the book to someone in my last year of university.. all gone. 😊
@@mayanour2394 i wish i could. The last time I saw it was after I lent it to a guy in my last year of university.... that was about 30 years ago. I don't remember the title.
The Rhind Papyrus provides a lot of worked examples illustrating their actual use for practical problems that arose in ancient Egypt like dividing beer or corn among workers. Preferred solutions added an additional requirement that terms beyond the earliest one or two (they permitted more than just two) dominated so tiny residues can be ignored (within the accuracy of their scales). They also permitted 2/3.
I believe you mean "parallel resistors or _series capacitors",_ right? For parallel capacitors, their capacitance values just add, no harmonics involved^^
Interestingly enough I was reading about Egyptian fractions just recently. One simple way of writing a fraction 1/n as a sum of two similar fraction is the elementary formula 1/n=1/(n+1)+1/[n(n+1)]. Now, if we pose the question slightly differently and ask how many ways a simple fraction could be written as a sum of any number of simple fraction (i.e. don't limit it to two terms), then the obvious answer is infinite.
For every d that divides n, the formula 1/n = 1/(n+d) + d/(n(n+d)) yields two unit-fraction summands. That is because we can have a k such that d*k=n, and thus 1/n = 1/(n+d) + 1/(k(n+d)). Interestingly enough, the ancient Egyptians were well aware of at least examples for some pretty large d, even where n wasn't small compared to d. I remember researching it as part of a history of math course, funnily enough, as the only Egyptian in class.
@@MyOneFiftiethOfADollar Apparently, in its invention, it was incentivized by the 'fair split' problem for paying out wages in goods. To this day, such problems are sometimes resolved in this way. In my research, though, some of those proportions (from actual administrative papyri for payment) would've been unreasonable to actually pull off (like, how do you get a portion of 1080 out of roughly a gallon?). Beyond that, sums of non-repeating reciprocals (and 2/3, which was a favourite) were the only way Ancient Egyptians wrote, or even conceptualized (for all we know), non-integer numbers for millenia. They actually developed tables for double odd reciprocals (2/n = sum(1/k_i) for odd n) for the purposes of their doubling-based multiplication algorithms, as well as documented identities to simplify these sums so they are often re-written as short as is mathematically possible.
As for pyramid measurements (let's be honest, that's probably what you're asking about), about the only one noted in my sources is the cotangent of the pyramid's face slope (although some sources dispute this for at least some of the surviving examples). This gives especially small numbers for nearly-vertical obelisks, which probably helped avoid big numbers. Why this would be helpful was lost to time, as far as I'm concerned.
3^3 = 27 And I think at the end the numerator would have +2 to incorporate the 1 over the denominator 2! Love your videos Edit: got it, it was correct at the end!
I realize it's a little early in the morning for me, but no matter how many times and in how many different ways I calculated 3^3 I always came up with 27. I'm glad I'm not losing my mind.
If you allow negative integers, you can take e.g. 100=-4×-25, which gives you 1/10=1/6-1/15. But don't take 100=-10×-10, which will give you 1/10=1/0-1/0, which is an indeterminate form.
This video is *surprisingly relevant* in calculating the *exact expression value* of trigonometric functions for angles that are *rational multiples* of π (pi). e.g. Can you calculate exact expression value of sin or cos(π * p/q) where let's say q = 2^32 - 1 = 4 294 967 295 and p = let's say 16. This requires breaking p/q into it's Egyptian fractions (prime denominators) - e.g. 16 / (2^32 - 1) = 8 (1/65535 - 1/65537) = 4 (1/255 - 1/257) - 8/65537 = 2(1/15 - 1/17) - 4/257 - 8/65537 = 1/3 - 1/5 - 2/17 - 4/257 - 8/65537 (this is the Egyptian fraction decomposition of p/q = 16 / 4,294,967,295). Therefore, sin or cos(16 π / 4,294,967,295) = sin or cos(π/3 - π/5 - 2π/17 - 4π/257- 8π/65537) = expand into sin or cos(A+B+C+D+E) formulas and then substitute with exact expression values of cosA etc. and then simplify the resulting expression. This worked because exact expression values of sin or cos(π * p/q) are known for all q = any Fermat prime 2^(2^n)+1 for 0
Michael Penn - Please make a video on how to calculate exact radical expression value of cos(2π/17). The method can be usually generalized to calculate sin or cos of 2π/FPD where FPD = next Fermat Prime Denominators (after 17, i.e. 257 and 65537).
So if 1/n = 1/a + 1/b then the number of choices for a, b is (1 + divisor sigma(n^2))/2. For n prime there are exactly two solutions: 1/(2 n) + 1/(2 n) and 1/(n + 1) + 1/(n(n +1)).
If you want an excellent resource on Egyptian fractions and how they were used by ancient Egyptian mathematicians, I would highly recommend "Mathematics in the Time of the Pharoahs" by Richard Gillings.
The black board, the concise and well composed lectures, the subject variety, it's like Christmas came 359 days early- assuming I made my difference correctly, Dec 31st 2022 to December 25th 2023, considering 2023 is not a leap year.
It's not simpler for those who are not familiar with the notation. If this were a video about something of that level, then it would make sense to use that notation. Might just be me, but I prefer the presentation that doesn't introduce more concepts and notation than are necessary to understand and solve the problem.
@@atreidesson true, but Michael does make that kind of mistakes, he would say "two" while writing a 1 on the board. I guess it was my mistake to assume that you knew that.
You lot can check out Normal Wildberger's outstanding channel and look up his video on Egyptian fractions for further info on the subject. I think Michael's video and Norman's complement well in fleshing out the subject.
The identity 1/n = 1/(n+1) + 1/n(n+1) can now be called the Egyptian fraction Identity. Your “complete the product” is equivalent to forcing a factoring by grouping. The phrase should replace the lame SFFT or Simons Favorite Factoring Trick which it is commonly called in some math competitions. This was nice combinatorics question and would like to see history video on Egyptian influence in unit fraction development
Merry Xmas to everybody watching! Professor, would you do a video about the result of the goat problem and how the approximation is calculated? Thanks and best wishes from Italy to you and your team!
3^3 isn't 3*3*3, it's 1*3*3*3. when you realize this so much of mathematics that you currently only know by definitions will instantly become intuitive.
Please do not show those “SUBSCRIBE” pop-up banners. Not only are they annoying and distract from your otherwise very engaging content, they are also an insult to viewers who are already subscribed and cannot turn them off. I'm a great fan of your content and it would be unfortunate if I'd have to stop watching because of this.
I was seriously expecting some interesting facts about Egyptian fractions from the title and thumbnail. I was utterly disappointed. This video doesn't talk even a little bit about Egyptian fractions, only about decomposing unit fractions, which, as you might recall from learning the history of math, the Egyptians did not need to do because they already had unit fractions.
Glad I am not the only one confused how 3^3 is 81 when I get 27 :)
Oddly enough my dog whined at that point. Either she's got latent mathematical abilities or she sensed me.
I was confused by that too. I listened 2 times to be sure. Odd that he didn't correct during editing as he often does
Merry Christmas Michael, I really enjoy your videos. I want the history of egyptian fractions. That should be fascinating. I hope 2023 will be the year to exponential growth of your channel. My best wishes for you and your family. Greetings from Venezuela
I think the Egyptian notation is really interesting. I read a book once that had a large section on Egyptian mathematics. Seems to me they had an exception for the fraction 2/3. I lent the book to someone in my last year of university.. all gone. 😊
Hello! Can you tell me the name of the book please? I am working on an assignment and would really appreciate it. Thanks in advance.
@@mayanour2394 i wish i could. The last time I saw it was after I lent it to a guy in my last year of university.... that was about 30 years ago. I don't remember the title.
The Rhind Papyrus provides a lot of worked examples illustrating their actual use for practical problems that arose in ancient Egypt like dividing beer or corn among workers. Preferred solutions added an additional requirement that terms beyond the earliest one or two (they permitted more than just two) dominated so tiny residues can be ignored (within the accuracy of their scales). They also permitted 2/3.
This is pretty useful for systems involving harmonic sums (like parrallel resistors or parrallel capicitors)
I believe you mean "parallel resistors or _series capacitors",_ right? For parallel capacitors, their capacitance values just add, no harmonics involved^^
or any real world situation... that's why they fucking exist. you're so insightful for noticing!
@@sumdumbmick no that's not true a pure mathematician don't care about applications in real world.
3*3*3=27
Another Pennism at 16:00.😄
In respect of the number of likes it got, I will refrain from liking your comment.
Thanks very much for posting. Fascinating and original video. I'm a huge fan of Egyptian fractions so please post more on this topic!
Interestingly enough I was reading about Egyptian fractions just recently. One simple way of writing a fraction 1/n as a sum of two similar fraction is the elementary formula 1/n=1/(n+1)+1/[n(n+1)]. Now, if we pose the question slightly differently and ask how many ways a simple fraction could be written as a sum of any number of simple fraction (i.e. don't limit it to two terms), then the obvious answer is infinite.
For every d that divides n, the formula
1/n = 1/(n+d) + d/(n(n+d)) yields two unit-fraction summands. That is because we can have a k such that d*k=n, and thus
1/n = 1/(n+d) + 1/(k(n+d)).
Interestingly enough, the ancient Egyptians were well aware of at least examples for some pretty large d, even where n wasn't small compared to d. I remember researching it as part of a history of math course, funnily enough, as the only Egyptian in class.
Right, so asking how many ways for a finite number of summands is the only interesting question I can see.
@@minamagdy4126 do you know if unit fractions had any specific “practical value” in, say, Egyptian architecture?
@@MyOneFiftiethOfADollar Apparently, in its invention, it was incentivized by the 'fair split' problem for paying out wages in goods. To this day, such problems are sometimes resolved in this way. In my research, though, some of those proportions (from actual administrative papyri for payment) would've been unreasonable to actually pull off (like, how do you get a portion of 1080 out of roughly a gallon?). Beyond that, sums of non-repeating reciprocals (and 2/3, which was a favourite) were the only way Ancient Egyptians wrote, or even conceptualized (for all we know), non-integer numbers for millenia. They actually developed tables for double odd reciprocals (2/n = sum(1/k_i) for odd n) for the purposes of their doubling-based multiplication algorithms, as well as documented identities to simplify these sums so they are often re-written as short as is mathematically possible.
As for pyramid measurements (let's be honest, that's probably what you're asking about), about the only one noted in my sources is the cotangent of the pyramid's face slope (although some sources dispute this for at least some of the surviving examples). This gives especially small numbers for nearly-vertical obelisks, which probably helped avoid big numbers. Why this would be helpful was lost to time, as far as I'm concerned.
3^3 = 27
And I think at the end the numerator would have +2 to incorporate the 1 over the denominator 2!
Love your videos
Edit: got it, it was correct at the end!
3³ = 27 yes he made that mistake
At the end though it was 1 + (N - 1)/2 so it's 2/2 - 1/2 = 1/2
9:46 27 sir
I realize it's a little early in the morning for me, but no matter how many times and in how many different ways I calculated 3^3 I always came up with 27. I'm glad I'm not losing my mind.
@@vaxjoaberg
Professor Penn deals in super cubes even before he awakes. And mind you, he got the side length right alright.
Yes please. Math history is always good.
Very good today! More nuance than I ever suspected....also...would also love to see a video on egyptian fractions👍
If you allow negative integers, you can take e.g. 100=-4×-25, which gives you 1/10=1/6-1/15. But don't take 100=-10×-10, which will give you 1/10=1/0-1/0, which is an indeterminate form.
This video is *surprisingly relevant* in calculating the *exact expression value* of trigonometric functions for angles that are *rational multiples* of π (pi).
e.g. Can you calculate exact expression value of sin or cos(π * p/q) where let's say q = 2^32 - 1 = 4 294 967 295 and p = let's say 16.
This requires breaking p/q into it's Egyptian fractions (prime denominators) - e.g. 16 / (2^32 - 1) = 8 (1/65535 - 1/65537) = 4 (1/255 - 1/257) - 8/65537
= 2(1/15 - 1/17) - 4/257 - 8/65537 = 1/3 - 1/5 - 2/17 - 4/257 - 8/65537 (this is the Egyptian fraction decomposition of p/q = 16 / 4,294,967,295).
Therefore, sin or cos(16 π / 4,294,967,295) = sin or cos(π/3 - π/5 - 2π/17 - 4π/257- 8π/65537) = expand into sin or cos(A+B+C+D+E) formulas and then substitute with exact expression values of cosA etc. and then simplify the resulting expression.
This worked because exact expression values of sin or cos(π * p/q) are known for all q = any Fermat prime 2^(2^n)+1 for 0
Michael Penn - Please make a video on how to calculate exact radical expression value of cos(2π/17). The method can be usually generalized to calculate sin or cos of 2π/FPD where FPD = next Fermat Prime Denominators (after 17, i.e. 257 and 65537).
Error - It's Fermat Prime Decomposition instead of Egyptian fraction decomposition due to some numerators not equal to +1.
What a fun problem.
Thank you, professor.
I wish you and yours a Merry Xmas.
También te deseo una muy feliz navidad
Iterating this on the denominators can generate some funky nested fractions.
Please make a video about Egyptian fractions! I've recently gotten fascinated with them.
Nice video! Looks like you adjusted the lighting a bit. I think it looks really good!
Merry Christmas, Michael! Thumbs up for the video on Egyptian fractions.
I want to learn more about history of Egyptian math
Now I want the Full video on Egyptian Fraction no just unit fractions.
So if 1/n = 1/a + 1/b then the number of choices for a, b is (1 + divisor sigma(n^2))/2. For n prime there are exactly two solutions: 1/(2 n) + 1/(2 n) and 1/(n + 1) + 1/(n(n +1)).
I think the procedure you adopted for 2 summands extends nicely for 3 summands etc. Not certain, but will investigate later on.
If you want an excellent resource on Egyptian fractions and how they were used by ancient Egyptian mathematicians, I would highly recommend "Mathematics in the Time of the Pharoahs" by Richard Gillings.
MATH HISTORY PLEASE!!!! MATH HISTORY IS SO COOL
Happy holidays math major Michael Penn!
Merry Christmas Michael and everyone here ❤
Math History? 10/10
3*3*3 not = 81
As a sign that 3 to the 3 should not be turned into 3 to the 4, I will refrain from turning the 3 likes on your comment into 4.
Nice. So the number of ways
is the same as the number of proper divisors of n^2 that are less than or equal to n.
The black board, the concise and well composed lectures, the subject variety, it's like Christmas came 359 days early- assuming I made my difference correctly, Dec 31st 2022 to December 25th 2023, considering 2023 is not a leap year.
correct me if im wrong, but can we use the divisor function { σ (n^2) } here for simpler notation?
It's not simpler for those who are not familiar with the notation. If this were a video about something of that level, then it would make sense to use that notation. Might just be me, but I prefer the presentation that doesn't introduce more concepts and notation than are necessary to understand and solve the problem.
Is 3 to the 3 81? Why is there 81 way to make three three-variant choices
Obviously that's a mistake. There's only 3^3=27 possible choices.
@@luisaleman9512 why, would I ask if it was obvious? He corrects mistakes as I remember
@@atreidesson he doesn't correct all the mistakes, his videos are full of comments pointing them out
@@luisaleman9512 well, as I see, he even did say how do we compute this number sooo
@@atreidesson true, but Michael does make that kind of mistakes, he would say "two" while writing a 1 on the board. I guess it was my mistake to assume that you knew that.
Joyeux noël !
Joyeux noël !
3^3 = 27, not 81. Usually you solve it in edition... Not this time...
This video is one of those videos where I have to spend lots of time thinking about because it's very good. Thank you.
Your second example has 14 distinct decompositions, rather than 41, because 3*3*3=27.
Murray Chrustmus from Texas. The video was a nice present!
I've proved to myself why it works but it might be nice for other viewers if you did a video on the Engels Expansion
You lot can check out Normal Wildberger's outstanding channel and look up his video on Egyptian fractions for further info on the subject. I think Michael's video and Norman's complement well in fleshing out the subject.
The identity 1/n = 1/(n+1) + 1/n(n+1) can now be called the Egyptian fraction Identity.
Your “complete the product” is equivalent to forcing a factoring by grouping. The phrase should replace the lame SFFT or Simons Favorite Factoring Trick which it is commonly called in some math competitions.
This was nice combinatorics question and would like to see history video on Egyptian influence in unit fraction development
very nice! but i think he missed about half of the solutions, since the left hand factors may both be negative!
Love 💕💕💕
✅
4:46, a - *10
When a=b, n must be an even number.
Merry Xmas to everybody watching! Professor, would you do a video about the result of the goat problem and how the approximation is calculated?
Thanks and best wishes from Italy to you and your team!
Numberphile did that today.
?
Cool
You see, here natural numbers are considered to be positive integers without zero.
@Michael
You lack consistency.
What an excellent place to stop.
3^3 isn't 3*3*3, it's 1*3*3*3.
when you realize this so much of mathematics that you currently only know by definitions will instantly become intuitive.
i have a challenge : the nth derivate of y is e^(y-n) +n
Please do not show those “SUBSCRIBE” pop-up banners. Not only are they annoying and distract from your otherwise very engaging content, they are also an insult to viewers who are already subscribed and cannot turn them off. I'm a great fan of your content and it would be unfortunate if I'd have to stop watching because of this.
I was seriously expecting some interesting facts about Egyptian fractions from the title and thumbnail. I was utterly disappointed. This video doesn't talk even a little bit about Egyptian fractions, only about decomposing unit fractions, which, as you might recall from learning the history of math, the Egyptians did not need to do because they already had unit fractions.
First !