Numbers

[BigBen]

It just occurred to me that there are several YouTube channels devoted to mathematics and related subjects.  Matt Parker is a British math teacher, whose channel is called, I believe, Stand-up Maths.  Tom Scott, another Brit, covers all sorts of topics, including mathematics.  He has a number of channels, so just search on his name.

The big-time guy is Brady Heron, an Australian who got his start by doing videos for various departments at the University of Nottingham.  Can't remember the name of his maths channel, but search on Periodic Table of Elements (chemistry department, and starring Professor Sir Martyn Poliakoff), and you'll find a listing of all Brady's other channels as well (he does one in conjunction with the Royal Society that is just fascinating). These guys are all friends, so if you find one of them, you are likely to find the others quite easily.

Please forgive a veer off topic, but if you are interested in physics, start with Brady's channel and branch out from there.  Destin Sandler, another friend of Brady's, is an American ballistics engineer with wide-ranging interests in all sorts of topics related to physics and mathematics.  And of course, there is always SciShow!

[Marty]

 

[Drew Espinosa]

Today is 12 February 2021, or 12/02/2021.

[Palantir]

1 hour ago, Drew Espinosa said:

Today is 12 February 2021, or 12/02/2021.

Thank you, Drew. That is so neat!

I thought it must be really rare but I looked it up and found there are 29 occurrences (for dd-mm-yyyy format) in this century. The last one is very unusual as it happens to be a leap day.  29-02-2092

[Timothy M.]

7 hours ago, Drew Espinosa said:

Today is 12 February 2021, or 12/02/2021.

6 hours ago, Palantir said:

Thank you, Drew. That is so neat!

I thought it must be really rare but I looked it up and found there are 29 occurrences (for dd-mm-yyyy format) in this century. The last one is very unusual as it happens to be a leap day.  29-02-2092

I agreee. And they don't use the weird US way of writing the month before the day.  

[Timothy M.]

On 2/7/2021 at 7:46 PM, BigBen said:

I'm a fan of the numbers seventy and above in French.  The names become odd to English ears, as seventy is literally called "sixty-ten", and so on from "sixty-eleven" up to "sixty-nineteen", followed by "four-twenties" for eighty.  They continue on the same way, all the way up to "four-twenty-nineteen" (99).  It took a lot of drill to become automatic when counting, but now it really tickles my fancy.

Try Danish numbers if you want weird. https://satwcomic.com/just-a-number  and this Youtube around 2:00

 

[Marty]

On 2/7/2021 at 6:46 PM, BigBen said:

I'm a fan of the numbers seventy and above in French.  The names become odd to English ears, as seventy is literally called "sixty-ten", and so on from "sixty-eleven" up to "sixty-nineteen", followed by "four-twenties" for eighty.  They continue on the same way, all the way up to "four-twenty-nineteen" (99).  It took a lot of drill to become automatic when counting, but now it really tickles my fancy.

 

5 hours ago, Timothy M. said:

Try Danish numbers if you want weird.

Here's a fascinating article about how the way we actually say numbers may affect our mathematical ability:

https://www.bbc.com/future/article/20191121-why-you-might-be-counting-in-the-wrong-language

Extract:

If I asked you to write down the number “ninety-two”, you wouldn’t have to think twice. By the time we’re adults, the connection between numerals and their names is almost automatic, so we barely give them a second thought. Which is why it might surprise you to hear that the English for 92 isn’t a great way to describe the number, and some languages are even worse.

Other languages do a much better job of describing digits. But it’s not just a matter of semantics – as early as 1798 scientists suggested that that the language we learn to count in could impact our numerical ability. In fact, one Western country actually overhauled its entire counting system within the last century, to make it easier to teach and do mathematics.

[ . . . ]

Numbers in the modern Welsh system are very transparent. Now, 92 is naw deg dau, or “nine ten two”, much like the system used in East Asian languages. In the older, traditional system, (which is still used for dates and ages), 92 is written dau ar ddeg a phedwar ugain, or “two on ten and four twenty”. The new system was actually created by a Welsh Patagonian businessman for accounting purposes, but it was eventually introduced into Welsh schools in the 1940s.

In Wales today, about 80% of pupils are taught maths in English, but 20% are taught in modern Welsh. This provides the perfect opportunity to experiment with children who learn maths in different languages, but study the same curriculum, and who are from a similar cultural background, to see if the East Asian style counting system really is more effective than the ones we use in the West.

Six-year-old children taught in Welsh and English were tested on their ability to estimate the position of two-digit numbers on a blank number line, labelled “0” on one end and “100” on the other. Both groups performed the same on tests of general arithmetic but the Welsh children did better on the estimation task.

“We think that it's because the Welsh medium children had a somewhat more precise representation of two-digit numbers,” says Ann Dowker, lead author on the study and experimental psychologist at the University of Oxford. “They may have had a greater understanding of the relationships between numbers, how large they are relative to other numbers.”

[Timothy M.]

8 hours ago, Marty said:

Here's a fascinating article about how the way we actually say numbers may affect our mathematical ability:

https://www.bbc.com/future/article/20191121-why-you-might-be-counting-in-the-wrong-language

That was very interesting. Maybe this is why Norwegian kids who say ni-ti-to (nine ten two) are better at maths than Danish kids who say to-og-halvfems (two and ninety).

[BigBen]

20 hours ago, Timothy M. said:

Try Danish numbers if you want weird. https://satwcomic.com/just-a-number  and this Youtube around 2:00

I love Tom Scott!  I'd love to marry him, except he is probably not gay.  (Since all the really cute ones are either married or heterosexual . . . [sigh].)

He has a wonderful rant on programming to handle time zones, by the way.

[Marty]

15 hours ago, Timothy M. said:

That was very interesting. Maybe this is why Norwegian kids who say ni-ti-to (nine ten two) are better at maths than Danish kids who say to-og-halvfems (two and ninety).

And I'm reading that halvfems is actually a shortened form of the older word  halvfemsindstyve, which apparently comes from halvfemte (“four and a half”) +‎ sinde (“times”) +‎ tyve (“twenty”).

[Drew Espinosa]

On 2/12/2021 at 4:21 AM, Palantir said:

Thank you, Drew. That is so neat!

I thought it must be really rare but I looked it up and found there are 29 occurrences (for dd-mm-yyyy format) in this century. The last one is very unusual as it happens to be a leap day.  29-02-2092

Also, the last such date to occur in this millennium will be 29/12/2192, with the next one not occurring until 10/03/3001.

[Timothy M.]

3 hours ago, Marty said:

And I'm reading that halvfems is actually a shortened form of the older word  halvfemsindstyve, which apparently comes from halvfemte (“four and a half”) +‎ sinde (“times”) +‎ tyve (“twenty”).

  Danish numbers are beyond weird - they are bloody useless.

[Zombie]

Infinities

I2 is a significant number in human activity throughout history and one example is music - the twelve notes of the musical scale. This derives from the harmonic series and I was planning a post on this but got distracted by the mathematics.

The harmonic series is what’s called divergent, which means there is no finite end. So it is a form of infinity. And there are, so far as I understand it (which is not very well ) an infinite number of infinities.

One puzzle called the “ant (or worm) on the rubber band" supposes an ant crawls along an infinitely-elastic one-metre rubber band at the same time as the rubber band is uniformly stretched. So, if the ant travels 1 centimeter per minute and the band stretches 1 metre per minute, will the ant ever reach the end of the rubber band?

You have a think about it while I go and make myself a nice cup of tea... 

 ...OK, worked it out?

Here’s the “solution” - did you get it right? 

Spoiler

 

Which reminds me of a regular lunchtime walk I used to do with workmates in a previous job along a single track farm road with a straight stretch of about 1 mile along which we would walk to the end, then turn around and come back. 

There was a bridge about half way and then a gate about half way along the second half and one day it occurred to me that if I was walking half the distance (to the bridge) and then half the remaining distance to the gate and then half the remaining distance etc etc etc then, mathematically, this would be an infinite series of “stages”...

...so how did we ever get to the end?  (which obviously we did )

 

Edited May 15, 2021 by Zombie

[Zombie]

The Face Of Evil, 8 January 1977

and the Doctor (Tom Baker) was quite right

Put another way, if you can’t frame the question you probably won’t find the answer - which is why I could never find an answer to my question on lunchtime walks(previous post).

I knew it couldn’t be an original idea because so few of us will ever think up a conundrum no-one else has thought of before. So the answer must be lurking somewhere on the interweb - I just had to find the right search words

And it is

And I did

And it’s called Zeno’s paradox - or one of them. Or possibly two, because two express the same idea just differently (he apparently came up with 9 - but who’s to say no-one else had the same thoughts before him...  )

According to wiki, Zeno of Elea was a pre-Socratic Greek philosopher of southern Italy who was born around 490 BC and died about 430 BC and is best known for his paradoxes which Bertrand Russell called "immeasurably subtle and profound".

Anyway, back to Zeno’s paradoxes (source wiki link below):

The Dichotomy Paradox (my lunchtime walk problem)

Someone wishes to get from point A to point B. First, they must move halfway. Then, they must go half of the remaining way. Continuing in this manner, there will always be some small distance remaining, and the goal would never actually be reached. There will always be another number to add in a series such as 1/2 + 1/4 + 1/8 + 1/16 + .... So, motion from any point A to any different point B seems an impossibility.

The Achilles and the Tortoise Paradox (the ant / worm / rubber band problem)

Achilles is in a race with a tortoise. Achilles allows the tortoise a head start of 100 metres and each moves at a constant speed, one very fast and one very slow  After some finite time, Achilles will have run 100 metres, bringing him to the tortoise's starting point. But during this time, the slower tortoise has moved a much shorter distance and it will then take Achilles some further time to run that distance, by which time the tortoise will have advanced farther etc etc.  So, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go. Therefore, because there are an infinite number of points Achilles must reach where the tortoise has already been, he can never overtake the tortoise...

So what does this mean?

It means that what we experience in the real world v Zeno’s conundrums cannot both be true at the same time. So either there is something wrong with the way we perceive the continuous nature of time or there is no such thing as discrete, or incremental, amounts of time or distance. Or anything else for that matter... Or there is a third picture of reality that “unifies” these two positions (the mathematical one and our common sense everyday experiences) that we do not yet have the philosophical / mathematical tools to fully understand.

Mathematically, when you start adding together the terms in the series 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + etc you’ll see that the sum gets closer and closer to 1, and will never exceed 1. Wiki states that Aristotle noted as the distance (in the dichotomy paradox) decreases, the time to travel each distance gets exceedingly smaller and smaller and he developed a method to get a finite answer for the sum of infinitely many terms which get progressively smaller (such as 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + etc).

Modern calculus achieves the same result by using more rigorous method, and mathematicians such as Carl Boyer think that Zeno's paradoxes are just mathematical problems for which modern calculus provides a mathematical solution.

However, Zeno's questions remain problematic if one approaches an infinite series of steps, one step at a time. This is known as a 'supertask'. Calculus does not actually involve adding numbers one at a time. Instead, it determines the value (called a limit that the addition is approaching.

In other words, mathematics cannot answer all philosophical challenges to real world realities. In the real world travel from A to B is possible and we do it all the time. So the answer to Zeno’s paradoxes lies in physics and not mathematics

https://simple.wikipedia.org/wiki/Zeno's_paradoxes

https://www.forbes.com/sites/startswithabang/2020/05/05/this-is-how-physics-not-math-finally-resolves-zenos-famous-paradox/

 

 

Edited July 3, 2021 by Zombie

[Zombie]

Even professionals don’t always understand the maths

…which is reassuring for slow learners like me

In this case (and there are others ) a physics Prof lost a $10,000 bet that a youtuber’s experiment was wrong because the maths equations “proved” it couldn’t work

This demonstration by the YouTuber…

…looks spookily like the “reveal hidden contents” I used in the earlier post about Infinities (which turned out to be Zeno’s Paradox) :

Now I don’t know if there’s a connection between the two (my head hurts ) but was the physicist’s maths equation simply wrong or is there something else going on here?

P.S. when you’ve watched the video from the point that I bookmarked you might then want to rewind and watch the whole video from the beginning

 

 

Edited July 3, 2021 by Zombie

[Zombie]