Where transcendental numbers hide in everyday math

https://www.quantamagazine.org/where-transcendental-numbers-hide-in-everyday-math-20211027/

by theafh • 5 years ago

82 37 5 years ago

nine_k• 5 years ago

Not arbitrary transcendental numbers, but the e specifically.

Yeah, this was disappointed. Information about e is widely available, but I don't recall ever really reading much about transcendental numbers in general, other than a brief introduction to e.

Definitely a missed opportunity.

lisper3• 5 years ago

OK, here's a fun fact: most numbers are transcendental, which is to say, if you choose a real number uniformly at random in any finite interval the probability of choosing a transcendental number is 1. (The proof is easy and left as an exercise.) But actually describing a transcendental number is very hard. The first transcendental number other than pi and e was not described until the mid 19th century, and even today only a few dozen classes of transcendental numbers are known.

bsedlm• 5 years ago

[deleted by me, the poster]

mywittyname2• 5 years ago

I read in Hacker's Delight a proof that e is the theoretically optimal base for computation. But since we lack the technology to make a base-e computer system, we have to settle for either binary, or base-3.

So you're probably onto something.

adrian_b2• 5 years ago

To be more clear, e is the optimal base for representing approximate numbers, when it is desired to minimize the relative errors.

Old IBM mainframes used base 16 for floating-point numbers, which (like also the base 10) is more distant from e than 2, and that caused larger computation errors than in modern computers with binary floating-point numbers.

For exact numbers (e.g. integers), the base does not matter, except on how it influences the cost of the hardware needed to implement the arithmetic operations. Base 2 normally results in minimal cost.

vishnugupta• 5 years ago

Until very recently I had no clue what "e" really stood for, though I was good at maths in college. I took it as the result of some complicated infinite series.

Until that is I came across this[1] wonderful article. That site is a treasure trove of very good insightful articles. Can't recommend enough.

[1] https://betterexplained.com/articles/an-intuitive-guide-to-e...

gnatman• 5 years ago

I like this explanation from Numberphile as well[1]

[1]https://www.youtube.com/watch?v=AuA2EAgAegE

jstx12• 5 years ago

I mean it is an infinite polynomial. Sometimes it’s a lot more useful to think of it that way and not in terms of compounding.

Or you can think of the function exp() represented by the Taylor series and e just happens to be exp(1). But the particular number itself isn’t as important as the function.

dxbydt• 5 years ago

<<< this is important <<< this is not important

To be fair, neither is important. You don't have to reach for a transcendental if you want infinite polynomials, you can manufacture one for integers, rationals, irrationals...the familiar example -

f(x) = 1 + 1/x + 1/x^2 + 1/x^3 + 1/x^4 + ... f(1) = 2

boulos• 5 years ago

Hmm. Did you end up studying differential equations?

I feel like the most clear statement from that article is "it's the base of exponential growth", but that (for a math inclined audience) the best way to show that is via equations like dy/dt = \alpha y.

huachimingo• 5 years ago

Another good way to see the limit definition is by solving the discrete case of that diff. equation (y' = y, y(0)=1).

Replace dy/dx with ∆y/∆x, so

∆y = y(x+∆x) - y(x)

Once you solve that (use some induction and numerical methods), see what happens when ∆x appraches to 0 and x=1.

For example try with ∆x=1 and then generalize that case to ∆x=n.

hota_mazi• 5 years ago

You didn't justify why you picked y' = y, though, and I'm having a hard time connecting that differential equation with the one that follows "so" in your message.

Could you break it down further?

Gunax• 5 years ago

My favourite use of e is determining how many (independent) trials it would take to encounter an event of some likelihood.

Eg. if an event has a 1/10 chance of occuring, how likely am I to encounter it at least once after 10 trials? What about a 1/1000 event after 1000 trials?

Well, if it has probability p, then there is 1-p of it NOT happening. So the chance of seeing a 1/p event after p trials is:

= 1 - (1-p)^p (looks a lot like e) = 1 - 1 / (1-p)^-p = 1 - 1/e (63.2%)

This comes up a lot in real-life, since it feels like if we do something with a 1% chance 100 times, it should occur, but there is really a more than 1/3 chance against it!

kevin_thibedeau• 5 years ago

I can recommend "e: The Story of a Number". It has a good explanation of transcendentals along with an informative dive into the history and significance of e.

rrauenza• 5 years ago

I was going to post this Mathologer video (https://www.youtube.com/watch?v=CaasbfdJdJg) based on e being "the most irrational number" -- but I misremembered. It's the golden ratio, phi.

But its still a really interesting video!

Smithalicious• 5 years ago

There's something ironic about calling the "most irrational number" the "golden ratio"

User23• 5 years ago

The defining characteristic of the exponential function is that d/dx e^x = e^x. The series construction makes this really obvious, because each element of the series is the first derivative of the next element, ad infinitum. And it wasn't as if this series was somehow luckily discovered, it was constructed from the defining characteristic.

Visual Complex Analysis, by Tristan Needham, covers this lucidly.

chobytes• 5 years ago

I think I find viewing at the result of some series to be more satisfying. After seeing series solutions to differential equations, the number e kind of faded into the background of the numerical process that gives us e.

foxes• 5 years ago

Where did you see e as a series before seeing it as the limit of some growth? I remember seeing the intuitive explanations in school, before anything else.

zarzavat• 5 years ago

IMO the most elegant definition of pi and e comes from differential equations.

e is the base of the exponential function, the solution of

    y = y'

pi arises from sine and cosine, which are the solutions to

    y'' = -y

alejohausner• 5 years ago

An k-ary search tree has k children per node, and k-1 keys per node. To find a value in a k-ary tree with n nodes, the maximum number of comparisons will be (k-1)log_k(n).

This is minimized when k = e.

Where transcendental numbers hide in everyday math

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