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The Math Myth

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Re: The Math Myth

#121

Post by chestervonwinch »

sp527 wrote: post #27
NhanH wrote: post #20

Earlier quoted context omitted.

To be fair, 0.9999... = 1 is not quite basic. You need to know things like infinitesimals, the distinction between value and representation of numbers etc.

It's also not strictly, theoretically true. More an 'for all intents and purposes' kind of thing.

The sum, 0.9 * Σ (0.1)^n from n=0 to n=N-1, is 0.999... as N goes to ∞ (write out the terms).

On the other hand, for any finite value of N, the sum [1] is equal to 0.9 * (1-(0.1)^N) / (1-0.1) = 1 - (0.1)^N. This value goes to 1 as N goes to ∞.

More rigorously, for any positive value, ε, there is a value N, such that the value of the finite sum is within ε of 1.

[1]: https://en.wikipedia.org/wiki/Geometric_series#Formula

Re: The Math Myth

#122

Post by kozikow »

yodsanklai wrote: post #105
kozikow wrote: post #97

Earlier quoted context omitted.

Only few people in finance really "make it" - and it mostly consists of portfolio managers (quantitative or else). "Superstar economy" analogy discussed in this thread have very strong effect in finance. Luck is also a huge factor. I know cases of International Olympiad gold Medalists, who didn't make it as portfolio managers. Do you really think you are smarter? If you are mathematically inclined software engineer,…

> To even get considered for the good quant position you need a good phd in applied mathematics or statistics. Are you sure about that? again, it's only anecdotal but I know at least 3 quants that didn't do a PhD (just a MS from reputable schools), including one that worked at GS as a new graduate. But there may very well be the exceptions.

I was editing the post while you replied. I realised that my post wasn't conveying any useful point.

In general, when many people hear Quant they think about quantitative portfolio manager - this is what I was talking about and where the "money" is at. It seems that IBs tend to use "quant" nowadays for a role that is very similar to Silicon Valley "data scientist".

Re: The Math Myth

#123

Post by douche »

visarga wrote: post #104
mseebach wrote: post #33

Earlier quoted context omitted.

> I'm sure a significant number of engineers and STEM professionals feel (as I do) that they're deliberately eschewing those subjects not for a lack of interest, but rather as a response to market demand That's the point -- the "math myth" is specifically in relation to math (and science) skills as being of particular importance as a comparative advantage over other countries (Russia, Germany, Japan, India, China) wh…

> not validate hard math, but "why is that number so low when that is so high, and is revenue in X really only a third of Y" style things Isn't that math pre-level 8? It's just simple algebra or arithmetic. Interesting how little use trigonometry and calculus have.

I've always thought trigonometry was one of the more useful things I did learn in school. Comes in handy a lot with carpentry and building projects. It's kind of cool when you realize that most of the shortcuts and rules-of-thumb that the old timers and carpenters use all the time can be derived from trigonometry, even if they never formally learned it that way.

Re: The Math Myth

#124

Post by kazinator »

jhanschoo wrote: post #111
kazinator wrote: post #62

Earlier quoted context omitted.

0.999... is equal to 1 only if we assign a particular semantics to the "..." notation. Namely if "..." means "the limit of the decimal number to the left, as the repetitions of the last digit grow ever larger", then 0.999... is an alternative notation for 1 since that limit is 1. The actual number formed by repeating 9's an infinite number of times is not constructable. Whereas 1 is constructable. So they cannot be t…

> So they cannot be the same thing. That's because, philosophically, two objects must be identical in every property to be the same object, and constructability is a property. Mathematicians have proposed several different constructions of the reals. If you were familiar with, say, constructing the rationals from the integers or constructing the reals from the rationals, you would know that this is not a problem. One…

I completely agree with you, and understand all that.

> L is the smallest number in E such that L is greater than or equal to all numbers in S.

And note that this doesn't imply that L is in S!

So here we have S = { 0, 0.x, 0.xx, 0.xxx, ... }. E is R, the set of reals. The smallest number in R greater than or equal to anything in S is 1. That doesn't mean 1 is in S. If 1 is not in S, then we're taking a notation based on condensing the names of the elements in S, like "0.x...", or 0.x with a bar over the x, and making that notation denote something not in S, namely 1.

This doesn't appear incompatible with the limit-based construction.

Are there instances in which this supremum approach toward continued decimals disagrees with the limit approach, in establishing the value?

Re: The Math Myth

#125

Post by Asooka »

Nursie wrote: post #57
analog31 wrote: post #25

At my workplace, we have about 60 scientists and engineers. The author's observation is accurate, that most people never use math beyond Excel and 8th grade math. They also never use most of the theory that they learned in their science (including CS) and engineering educations. The typical career arc is to get through college, then sit down at a CAD workstation, or programming terminal, and forget all of your math a…

I do complex mathematics for fun sometimes, it always surprises me when I start talking about the fractal images I generate and a large percentage of other engineers are wowed at how, well, complex it must all be. Then start to glaze over a bit... I loved studying all that stuff at school, pretty pictures or no. But as a coder the opportunities to use much of it in anger are really quite restricted.

Maybe the best argument for teaching people math is to show them how complexity can be tamed via patterns. As in, if you sit down and think logically, you can find solutions or come up with approaches to even the gnarliest-looking problems. And things that sound very simple (e.g. the Fermat conjecture, which can be understood by 5-th graders) can have really complicated solutions. It should teach people how to use their slow thinking rather than their fast thinking and that they absolutely have the capacity to do it. I've similarly never met anyone who had to do 20 accurately timed hops in a row daily, but I don't hear anyone want to remove jumprope and P.E. Still, there should absolutely be a class on "real-world applied mathematics" where you're taught the mathematical principles behind loans, mortgages, taxes, etc.

Re: The Math Myth

#126

Post by dhd415 »

The value of studying more advanced mathematics is not tied strictly to what will be used on a day-to-day basis in one's job. I studied math well beyond what I use in my day-to-day work as a software engineer, but I've found it valuable for at least two different reasons. First, it exposed me to ideas and concepts beyond what is right in front of me every day. If I happen upon the occasional question about computational theory or cryptography or whatever, I am at least aware that there's a field of study around it and I know where to look for solutions to known problems. Second, I don't think I'm entirely unique in that my mastery of lesser math was improved by studying higher math. In other words, I'm pretty rusty on things such as partial differential equations, but because I studied them, I know algebra, trig, basic calculus, etc., cold and that is beneficial both in my day-to-day work and normal life.

Re: The Math Myth

#127

Post by soVeryTired »

kazinator wrote: post #71
soVeryTired wrote: post #65

Earlier quoted context omitted.

They're coming out of the woodwork... Edit: sorry, didn't mean to ad hominem, but I'm going to leave my original comment there all the same. To make my post a bit more constructive, OP, what exactly is your definition of "constructible"? Because it doesn't relate to any mathematical concept I'm familiar with. Other than maybe "finitary".

https://en.wikipedia.org/wiki/Constructivism_(mathematics) Do you have remarks not regarding the semantics of "constructable"?

But the "example from real analysis" section in your link uses exactly the same limiting technique to construct e. That section shows that 0.999... is constructible.

Re: The Math Myth

#128

Post by iopq »

kazinator wrote: post #62
iopq wrote: post #18

I think society would be a lot better if BASIC math and statistics would be better understood. How many times do you see a study posted here with N=23 and people say "the sample size is too small" when it's clearly not? How many people ask for a card deck change to change their luck? How many times do people read a poll like 49% +/- 3% vs. 43% +/- 3% and conclude the two candidates are statistically tied? I could pro…

0.999... is equal to 1 only if we assign a particular semantics to the "..." notation. Namely if "..." means "the limit of the decimal number to the left, as the repetitions of the last digit grow ever larger", then 0.999... is an alternative notation for 1 since that limit is 1. The actual number formed by repeating 9's an infinite number of times is not constructable. Whereas 1 is constructable. So they cannot be t…

The problem with this argument is it assumes you can get to infinity 9s by adding them one-by-one.

It's like adding sides to a regular polygon to get to a circle. No matter how many sides you add, you still have a polygon. You can only say that a circle is a polygon with infinity sides, but you'll never get there adding them one by one.

Re: The Math Myth

#129

Post by NhanH »

nostrademons wrote: post #88

This largely matches my experience - as a software engineer, I spend probably In a market economy, basically all returns come from marginal gains. The vast majority of your lifetime income will come from a dozen or fewer opportunities that you happen to be in a position to take advantage of, whether it's a new job offer or a high-profile project you volunteer for or a startup that takes off. You will qualify for thos…

Wah.

>The vast majority of your lifetime income will come from a dozen or fewer opportunities that you happen to be in a position to take advantage of, whether it's a new job offer or a high-profile project you volunteer for or a startup that takes off

Is this really true? I'm still way too early in my career to know. But I'd have thought if you stick with the more "traditional" route, your income would be fairly consistent (as opposed to something similar to the startup lottery, for example).

Re: The Math Myth

#130

Post by douche »

merpnderp wrote: post #24

How would this same conjecture apply to History, Literature, Biology, Physics, etc etc? How much of any advanced learning do most people use in their day to day lives? All of it in the periphery would be my counter-conjecture. I was always taught trade schools were for learning a particular skill. College was to equip you with the knowledge and ability to think logically required to have a better life.

With history, at least, if you want to understand how the world is in the state it is in today, you need to understand how it got here.
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