The Beauty of Math
This is the English Version of my Post from yesterday. This is not a translation though, I hate doing translations and I really don't want to auto-translate my posts. That said it is hard for me to talk about math in English. For politics that's no problem, I am used to that, but when I think about Math I think completely in German. So I might use some of my Germans thoughts as a blueprint for this post, I will definitely use the same proofs because those are very relevant to me.
Why does Math matter?
Most people know mathematic as the basis for many sciences and while that is all really nice and dandy, I don't care about it too much. There was always that misconception that science should be really my thing, but it is not, I can understand the math behind every science quite easily, but I don't really care too much why my graphic card functions or even worse how my own body functions. It's cool that it works I don't really want to be bothered by the details.
Mhh, I didn't expect to start out with an anti science rant, but being on the topic, if you ever want to have a highly sophisticated discussion with any scientist in any field you at least need to understand the basics of higher Mathematics. I would even go so far and argue that the Math I am about to show you actually enhances critical thinking in fields that you previously thought got nothing to do with mathematics.
Proofs
What I am going to talk about are mathematical proofs. I am not sure how familiar you are with the concept but in math you can proof something against any possible doubt. Take the sentence of Phytagoras for example: we don't have a theory about a²+b²=c², that seems to work as a model and might be disproven with new discoveries in the fields of mathematics. No! We know for sure that a²+b²=c² will be true for all eternity.
Here Math is completely different from any science, there is just one truth, not some dialectic-the-truth-lies -in-the-middle-bs. Mathematicians know everything with 100% certainty. There are some arguments about definitions, like if 0 is a natural number. You can have an opinion on that or just agree THAT 0 IS NOT A NATURAL NUMBER! Sorry, I get heated on this one.
Let's look at our first easy proof 0.999...=1 . I like this one cause you might think that 0,999... is smaller and the number closest to 1 but it is easy to proof it is actually the same as 1:
0.333...=1/3 ---> 0.999...=3/3=1
The Beauty
To me the beauty of a proof can be divided in the following categories: The impressivness or importance of the rule that is proven . The length of the proof, shorter is better. And lastly how understandable your proof is. These criteria are subjective but once you have done some mathematical proofs, it can be really nice to agree sometimes that proofs are either really beautiful or sometimes really ugly.
I present you my favorite: The Proof that there is an unlimited amount of Prim Numbers.
Lets assume we know all Prim Numbers. We can now multiply them all to the Product we shall call P. Now we look at P+1, it can not be divided by any Prim Number because P can divided by all of them. Therefore P+1 has to be a Prim Number itself that is not part of all known Prim Numbers. Therefore our assumption that we can know all Prim Numbers is wrong. Therefore the number of Prim Numbers has to be infinite.
Please comment if you can or can not understand my proof, I would really like know how digestible I made it for the average consumer.
For the Pros
There is one proof I failed in a test. Maybe you can help, I still can't get the solution. If you look at n^4, with n being any natural number, you always get a result that ends on 1,6,5 or 0. In other words, why do you only have two rest classes for n^4 modulo 5?
The first proof is inaccurate. .333 does not exactly equal 1/3, it's an approximation. Likewise with .999.
0.333...is no approximation, it is way to write 1/3, the three dots mark an endlessly repeating series. Same goes for 0.999... or a much more common way to use it 1+2+3+...+10. Not sure if you use the same writing where you live, it is also not the most common one in Germany but it is correct and I really like it :).
My apologies, I didn't realize the dots represented a repeating number, and now that I"m looking at this without being sleep deprived, I should have realized they meant SOMETHING, and looked it up. 😁
Still (and it's been DECADES since pre-calc, so forgive me if this sounds just as dumb), what about .999... =/ 1 ?
Also, aren't decimal numbers more accurately represented as a fraction of some multiple of 10?
I dont understand what you mean to say with that. Can you explain?
Not really. 1/100 or 0.01 has the same accuracy. If you want to avoid rounding numbers you often keep the fractions. I higher math we usually keep all the fractions and rarely use a decimal writing style, it helps you see things.
For the upper one, the odd expression in the middle is an ascii representation for the 'does not equal' sign. So just...
.999... does not equal 1
1/100 DOES express the decimal as a fraction of a multiple of 10. Isn't that more accurate (since decimal numbers can ONLY represent multiples of 10 by their nature) than saying .333... = 1/3?
No, there isn't such a thing as more accurate. In math you are either accurate or you are not and as long as you dont round your decimals, decimals are accurate. 1/3 is exactly 0.333... and 0.999... is exactly 1. 100% accuracy on both.
Well, I decided it was time to do some much deeper reading on this subject, and quite honestly I'm REALLY surprised I haven't stumbled into this conundrum before.
My personal opinion is that it is more of a proof that the number .9... is an imaginary number. I'm also of the opinion that .3... isn't the same as 1/3, it's actually 1.../3..., and likewise .9... would be 9.../10... I didn't read far enough to find out if there's more elegant notation for this.
My argument is this: These proofs are crossing an 'event horizon' between imaginary and real numbers in order to work. I don't know how the math or physics communities feel about calling 'infinite' numbers 'imaginary' numbers, but I don't suppose either community will be swayed much by my opinion about it. 😁
I don't think they would feel great about it since we already have a fixed meaning when we talk about imaginary numbers it is the number i for which i²=-1 is true. Imaginary numbers can also be multiples of i but it is mostly i.
Periodic numbers like 0.3... are actually rational numbers - a really tame beast. You would have to go two steps above to come to the imaginary numbers. In between are irrational numbers like square root 2 and pi.
Rats, I forgot all about irrational numbers!
Infinity is a tricky concept, like zero. I suppose people will always have arguments about ideas that human heads don't comprehend well.
Thank you for the education, and re-education, no need to educate me further on this, I won't remember it anyway! Thanks for the engaging and humbling discussion!
you are welcome. It is always hard to break certain walls in math, once you are through, you dont even know what kept you on the other side.
No, this is the kind of wall that I would never break through, I would just accept that it was there and work around it, or I would get lucky enough to come up with the math to express it more perfectly. Imaginary numbers were easy for me to accept, for some reason. Irrational numbers... let's just say there's a reason I forgot about them!
I would use a the geometrical series to proof the equality to 1
...and now the language is getting outside of my knowledge. This is why I don't post in STEM communities 😂
a good example for numbers that you almost never write as decimals are irrational numbers like the square root of 2
Yes, I get the proof. It is always like watching a magician. And I somehow understand what people find beautiful about math. The same goes for the mathematical trick behind public-key cryptography. But my brain can't accept math, without understanding what it actually is. It's like the exact opposite of what you described. My brain is all about lots and lots of details and complexity. You need to lack curiosity for being good in focussing on math, its kind of a gift.
This math is outside of the domain of statistics so there is no certainty.
I love the magician analogy. There are also the same stages of apprenticeship, the unknowing viewer that thinks it is reall all magic, the apprentice who can understand some magic and knows how the art of Illusion works at its core and then the actual master who can see through all tricks and can even come up with his own original tricks. I feel like I am somewhat of an rogue apprentice :).
I have heard this argument before . Some people dont care about the details of the world that is surrounding them but they want to classify and order it, bring structure to it. Autism comes to my mind, but I never really looked deep into the topic. I am not sure I am really one of those people, even though my social intelligence is rather average, I still feel like I have a good amount of Empathie and I can be charming, but of course there is one mystery that no mathematician can solve and that is the mind of a woman ;) (Maybe it is just easier for me to know what a man thinks cause I am a man myself, but that does not sound cool)
Even though we are not in the field of stochastic the certainty is implicit. Certainty in a non-mathematical context is a achieved when you can proof something to be true beyond any reasonable doubt. In math everything you learn is true beyond any possible doubt. Therefore math is certain.
Wouldn't the graph of y=x^2 look very different for x=1 and x=.9...?
Sorry to keep picking at this, I'll stop if you ask me to.
its ok, keep asking if you are still in doubt. If you want to look at the graph for x-->x² then we only focus on one point x=1=0.99... . At this point the result or the y-coordinate is 1, because 1²=1=(0.99...)².
This is where things get hard for me, and I feel there needs to be a better way to define what's actually happening here.
By accepting the answer above, we accept 1 to represent any expression of .99...^n. While the result of all those expressions still approach infinitely close to 1, they approach at different rates, while the solution offered by substiting 1 doesn't approach 1 at all.
oj, I thought I answered, sry.
yes we do, that is 'consens' (it is more than consens since it is proofable).
1=0.99...^5=0.99...^12=1
These numbers are not approaching 1 they are 1.
I understand 'consens' (I think, consensus in English?) and the usefulness of it.
I also understand the need to constantly question it.
Once upon a time, there was a 'consensus' that Earth was the flat center of the universe.
I'll try to explain what I mean about having 'better language', in (hopefully) better language.
For the recurring decimals we get from whole divisions of 3, (.3..., .6...), by saying these decimals repeat infinitely, it SEEMS like we're saying we can make this an infinitely smaller fraction (3/10, 33/100,333/1000 ...), which makes it seem to define a moving point.
It's a little more accurate to say these decimals exist in an eternal state that represents a precise division of 1/3.
The idea that we need infinitely repeating numbers to represent a precise division just seems 'messy' to me, and in the case of these particular fractions, represents the problem of dividing by 0. Decimal numbers simply can't show fractions of 3 without them. If we used a number system based on units of 12 instead of 10, we could show these fractions without a repeating (do?)decimal.
I'm not saying we should switch to a base 12 number system, by any means. I'm saying maybe there should be a broader discussion about finding a system that doesn't allow us to show the number one both as 1, and something that (seems to) represent an undefinable quantity smaller than one.
I would say we have the same issue with pi... showing it in decimal numbers is impossible, but the relationship between the diameter and circumference of a circle isn't an impossible, eternally moving relationship, we just can't show it with those numbers.
You are basically explaining why the teachers where always so keen on using fractions instead of decimals, it is in away always the cleaner method. We would never write 0.125 or even 0.25 in college but always 1/4 or 1/8.
It made my head hurt thinking about this, you should be right, but I never looked at fractions in things like hexadecimal or even binary. 0.4 should be exactly 1/3 when your basis is 12.
LOL, I'm glad to hear the my head isn't the only line hurting over this! 😂
Let me try this here for the first time, this thread REALLY deserves it 😄
!ENGAGE
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Thank you for this inspiring post. I've done some more reading on infinitesimals and other types of numbers that I hadn't heard of in my calculus class. This led me down a very interesting rabbit hole, and I really can't thank you enough.
happy to have inspired you to look some more into math, there is some really fascinating stuff. I never heard of infinitesimals to be honest, but did you know there are different kinds of infinities?
Different kinds of infinites? I know now 😁. The infinitesimals led me to surreal and hyperreal numbers, then nature reminded me that she doesn't care how well I grasp math, by sending a downpour to ruin some of my spring work... looks like my 'light' reading on mathematical theories will have to wait until this winter!
unlucky, all you need is pen & paper, but it is quite essential :D
You have just inspired me to love maths....@thatgermandude