Nothing is ever "exponential"

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Please, maths people, confirm my bugbear.

Very few natural processes or even unnatural processes are increasing exponentially. When people say exponentially, they usually mean 1,2,4,8,16 which is geometric I think not exponential (which is something else - what is it again? It's not 1,4,9,16,25 or is it?). Am I right? Am I right to get annoyed by it?

― Sam (chirombo), Thursday, 29 May 2003 11:17 (twenty-two years ago)

erm...well...sam...erm...

― Pinkpanther (Pinkpanther), Thursday, 29 May 2003 11:22 (twenty-two years ago)

1 2 4 8 16 == 2^0 2^1 2^2 2^3 2^4 making it exponential??

But I think "exponential"s more "folk" definition is something where the rate of increase is always increasing. As a physicist i get annoyed when people say weight when they mean mass. (Not really)

― Alan (Alan), Thursday, 29 May 2003 11:22 (twenty-two years ago)

Yes, 1 2 4 8 16 is an exponential sequence.

― caitlin (caitlin), Thursday, 29 May 2003 11:24 (twenty-two years ago)

And lots of natural processes do follow either an exponential curve or an inverse exponential one - radioactive decay, for example.

― caitlin (caitlin), Thursday, 29 May 2003 11:27 (twenty-two years ago)

Bollocks and fuck. I hate it when my bugbears turn out to be chickenbears.

Well then use this thread to list your sciencey pedantries which cause you to feel smug/annoyed.

― Sam (chirombo), Thursday, 29 May 2003 11:27 (twenty-two years ago)

Exponents don't have to be whole numbers, by the way. Not sure why I say this, but...

― Michael Daddino (epicharmus), Thursday, 29 May 2003 11:28 (twenty-two years ago)

See, I thought that your natural processes (e^(-t) etc) were not considered "exponential" which was a third, bigger sort of increase apart from arithmetic (1,2,3,4) and geometric (1,2,4,8 and also the e ones).

― Sam (chirombo), Thursday, 29 May 2003 11:29 (twenty-two years ago)

Personal bugbear: a quantum shift is a _really really_ small one.

― Andrew Farrell (afarrell), Thursday, 29 May 2003 11:33 (twenty-two years ago)

The smallest shift possible, surely?

― caitlin (caitlin), Thursday, 29 May 2003 11:35 (twenty-two years ago)

in non-[hysics englis an uantum is just a fixed ammount of something, which is still not what is generally defined by a quantum shift.

― Ed (dali), Thursday, 29 May 2003 11:38 (twenty-two years ago)

a quantum shift is a sudden change from one fixed amount to another with no gradations in between, hence the folk expression still works in some way -- it just places more weight on the discontinuity than the size

― Alan (Alan), Thursday, 29 May 2003 11:42 (twenty-two years ago)

haha the series 1^0 => 1^1 => 1^2 => 1^3 => 1^4 is also exponential => it is exactly the same always = exponential!!

― mark s (mark s), Thursday, 29 May 2003 11:48 (twenty-two years ago)

it just places more weight on the discontinuity than the size

did you mean "mass"?

― ken c, Thursday, 29 May 2003 11:59 (twenty-two years ago)

what does the e stand for? Surely it should be t to stand for "two point one eight two eight one eight two blah blah".

― joshua, Thursday, 29 May 2003 12:07 (twenty-two years ago)

insert a seven in there somewhere.

― joshua, Thursday, 29 May 2003 12:08 (twenty-two years ago)

"Dark side of the moon." The moon is not a regular polyhedron! It doesn't have sides! Unless you mean that the INside of the moon is dark. And do you? DO YOU??!

― Michael Jones (MichaelJ), Thursday, 29 May 2003 12:10 (twenty-two years ago)

Okay actually yeah in computer science you have O(n) O(c^n) and then O(n^c) where c is a constant as yr. basic rates of increase.

O(n^c) by the way = 1, 2, 9, 16, 25, 36, 49 = 1^2, 2^2, 3^2, 4^2, 5^2, 6^2, 7^2.

But as I recall O(n^c) is termed geometric which makes sense since by geometry 1^2 is the area of a 1" long square, 2^2 the area of a 2" long square etc.

So yes certain things can increase at an increasing rate without being exponential. But for an exponential if c > e then the increasing rate of increase itself increases (for = e it stays the same).

― Sterling Clover (s_clover), Thursday, 29 May 2003 13:49 (twenty-two years ago)

I remember that the graph of y=e^x has the unusual property that at any point the rate of increase is the same as the value of y - in other words, at any point on the graph the rate of increase is the same as the rate that the rate of increase is increasing at.

― caitlin (caitlin), Thursday, 29 May 2003 14:05 (twenty-two years ago)

The key feature of exponentials is that the rate of doubling (or halving in the case of decay) always remains the same.

― Ed (dali), Thursday, 29 May 2003 14:08 (twenty-two years ago)

Beer foam decays exponentialy. Much thanks to the Annals of Improbable Research.

― Mr Noodles (Mr Noodles), Thursday, 29 May 2003 14:12 (twenty-two years ago)

what does the e stand for?

It stands for Euler's Constant.

― Mr Noodles (Mr Noodles), Thursday, 29 May 2003 14:14 (twenty-two years ago)

How about 'Round The Back Of The Moon'?

― the pinefox, Thursday, 29 May 2003 14:16 (twenty-two years ago)

This thread is scaring me, so it might be a good place to post this, which one of my friends saw fit to niggle me with yesterday:

A census taker goes to a house for some information. The lady says she has 3 kids, and the product of their ages is 36. The sum of their ages is equal to the address next door.

So he goes next door, but he soon finds he has to come back to request
more information. The lady tells him the oldest child is asleep
upstairs. What are the three children's ages?

(Or, carry on as you were and I'll go and lie down with a cold flannel on my head.)

― Archel (Archel), Thursday, 29 May 2003 14:19 (twenty-two years ago)

"Dark side of the moon." The moon is not a regular polyhedron! It doesn't have sides! Unless you mean that the INside of the moon is dark. And do you? DO YOU??!

Oh come now -- "side" isn't always used so literally. I sleep on my left side, and I'm on some side of some complex issue. The problem with "dark side of the moon" is that the side we don't see isn't actually "dark" all the time...

― Chris P (Chris P), Thursday, 29 May 2003 14:21 (twenty-two years ago)

2, 3 and 6

― Ed (dali), Thursday, 29 May 2003 14:22 (twenty-two years ago)

or 3 3 4 provided they are on the other side of the street.

― Mr Noodles (Mr Noodles), Thursday, 29 May 2003 14:23 (twenty-two years ago)

Unless two of the children are twins and I reckon she would have mentioned that.

― Ed (dali), Thursday, 29 May 2003 14:23 (twenty-two years ago)

2*3*6 = 32 for those of us not using the new math.

Physist Vs Eng FITE!

― Mr Noodles (Mr Noodles), Thursday, 29 May 2003 14:24 (twenty-two years ago)

Using the same regular conversational rules which led to her mentioning the product of the ages of her children, Ed?

― Tim (Tim), Thursday, 29 May 2003 14:25 (twenty-two years ago)

There's no easy way to do that one, just work out all the 8 ways that the kids could multiply to 36, then see which two of those add to the same number (cause he couldn't tell just by looking at the sum), then pick between on the basis that only one of them has an oldest child (the other has oldest twins).

― Andrew Farrell (afarrell), Thursday, 29 May 2003 14:26 (twenty-two years ago)

This house next door thing is throwing me off. It could be either 2,2 and 9 or 2, 3 and 6 couldn't it? I mean, twins aren't ruled out.

― Archel (Archel), Thursday, 29 May 2003 14:27 (twenty-two years ago)

The oldest child is 4 and the twins are 3.

― Chris P (Chris P), Thursday, 29 May 2003 14:27 (twenty-two years ago)

oh, for christs sake, people: the number next door is 13. So the two ways are 1,6,6 and 2,2,9.

― Andrew Farrell (afarrell), Thursday, 29 May 2003 14:28 (twenty-two years ago)

Or the oldest child is 9 and the twins are 2. But 2+3+6=11, and what's the other set of children that could equal 11?

― Chris P (Chris P), Thursday, 29 May 2003 14:29 (twenty-two years ago)

Oh christ. It's too early for this. 9 and 2 and 2 is right.

― Chris P (Chris P), Thursday, 29 May 2003 14:29 (twenty-two years ago)

Haha this is worst than I thought.

― Archel (Archel), Thursday, 29 May 2003 14:29 (twenty-two years ago)

I don't care if I got it wrong, still can't belive my eagerness to mock engs blinded me to the true value of 6*6.

― Mr Noodles (Mr Noodles), Thursday, 29 May 2003 14:32 (twenty-two years ago)

The address next door is 38 obv obv.

― Tim (Tim), Thursday, 29 May 2003 14:33 (twenty-two years ago)

I don't understand any of this.

― Nick Southall (Nick Southall), Thursday, 29 May 2003 14:36 (twenty-two years ago)

Don't worry Noodles, for a few minutes last night, I was convinced that 6+6+6 was 24.

― Archel (Archel), Thursday, 29 May 2003 14:36 (twenty-two years ago)

"Yes, this is my 36 year old son and 1 year old twins. What do you mean you're from the Sun?"

― Andrew Farrell (afarrell), Thursday, 29 May 2003 14:40 (twenty-two years ago)

Haha yes but what's the other way that it could add up to 38?

― Chris P (Chris P), Thursday, 29 May 2003 14:43 (twenty-two years ago)

She's taken the Sun upstairs to see her layabout 36 year old son?

― Tim (Tim), Thursday, 29 May 2003 14:45 (twenty-two years ago)

Why does it have to add up to 38?

― caitlin (caitlin), Thursday, 29 May 2003 14:48 (twenty-two years ago)

Did someone leave out the address next door? You people start assuming its 11 or 12 or whatever, but her kids could be 1, 2, and 18... Those teenagers sleep all the time.

1+1+36 = 38
1+2+18 = 21
1+3+12 = 16
1+4+9 = 14
1+6+6 = 13
2+2+9 = 13
2+3+6 = 11
3+3+4 = 10

Until you know the address, you might be able to cross through 1,6,6 because there's not an "oldest" but the rest are possibilities until you know more.

― Stuart (Stuart), Thursday, 29 May 2003 14:49 (twenty-two years ago)

Er... maybe not 12 :)

― Stuart (Stuart), Thursday, 29 May 2003 14:50 (twenty-two years ago)

You can have an oldest if they're both 6. Eleven months difference? Couple of hours gap between twins?

― James Ball (James Ball), Thursday, 29 May 2003 14:52 (twenty-two years ago)

No, the point is the guy HAD to come back for more info. If the house number wasn't 13 there was enough information to determine the kids ages uniquely.

― RickyT (RickyT), Thursday, 29 May 2003 14:53 (twenty-two years ago)

True, and I agree with you, but we're dealing with brain teaser "logic". That's the kind of thing I would've gotten into an argument with the teacher about.

― Stuart (Stuart), Thursday, 29 May 2003 14:56 (twenty-two years ago)

RickyT: if she lives at number 15 the extra information the census taker needs could be "next door which way?"

― Tim (Tim), Thursday, 29 May 2003 15:01 (twenty-two years ago)

People in my class would argue he still didn't know cause child birth is generally a sequintial thing and 6-6-1.

The point is this so called lady is being a righteous pain in the ass.

― Mr Noodles (Mr Noodles), Thursday, 29 May 2003 15:04 (twenty-two years ago)

Good point Mr H.

― RickyT (RickyT), Thursday, 29 May 2003 15:08 (twenty-two years ago)

OK, chew on THIS (which I haven't even begun to attempt):

Archel chooses two integers, m and n, each between 2 and 100 inclusive. She tells Andrew the product, mn. The sum, m + n, she tells to Caitlin. Their conversation is as follows:

Andrew: I don't have the foggiest idea what your sum is, Caitlin.
Caitlin: That's no news to me, Andrew. I already knew that you
didn't know.
Andrew: Ah ha, NOW I know what your sum must be, Caitlin!
Caitlin: And likewise Andrew, I have surmised your product!!

What two integers did Archel choose?

― Archel (Archel), Thursday, 29 May 2003 15:09 (twenty-two years ago)

Yeah, it's 9, 2, and 2. And yes, Stuart, you do have to assume that the guy is completely rational and is honestly coming back to say "not enough information" in the math sense, as opposed to "stop dicking me around with your timewasting mindgames, you Sphinxy whore."

I think it's safe to assume with a problem like this that all characters are rational agents and that the answer is never "but ah-hah, what if one of the twins was born first? Thus I deem this problem unsolveable." Similarly you don't fill out standardized test analogies by putting little notes in the margin that say "but actually this second equivalency does not match the first, as in the first both words are Latinate while the proposed solution to the latter actually has a Germanic etymology, rendering the whole thing moot."

― nabisco (nabisco), Thursday, 29 May 2003 15:09 (twenty-two years ago)

What manner of rationality has someone answer a census question "I have 3 kids, and the product of their ages is 36. The sum of their ages is equal to the address next door."

Why would we assume she was rational when the timy amount of evidence we have suggests she's barking?

― Tim (Tim), Thursday, 29 May 2003 15:11 (twenty-two years ago)

Proof that maths != real world and is therefore useless.

― Archel (Archel), Thursday, 29 May 2003 15:14 (twenty-two years ago)

Also: 'rendering the whole thing moot' = the point of ilx, surely?

― Tim (Tim), Thursday, 29 May 2003 15:14 (twenty-two years ago)

And what kinda census is this that doesn't ask for names?

― Mr Noodles (Mr Noodles), Thursday, 29 May 2003 15:15 (twenty-two years ago)

math is merely language or a tool. One that happens to describe the reality.

― Mr Noodles (Mr Noodles), Thursday, 29 May 2003 15:18 (twenty-two years ago)

I've worked out *how* to solve Archel's new problem, but I'm too lazy to actually do it.

― caitlin (caitlin), Thursday, 29 May 2003 15:19 (twenty-two years ago)

Reality schmality. Life is all about illusion.

― Archel (Archel), Thursday, 29 May 2003 15:19 (twenty-two years ago)

No! Life is all about CAKE and CHOCOLATE :-)

― caitlin (caitlin), Thursday, 29 May 2003 15:22 (twenty-two years ago)

If Archel has one cake and has to share it between x people, with x being... nah, fuck it I'll eat it all myself.

― Archel (Archel), Thursday, 29 May 2003 15:33 (twenty-two years ago)

OK, I think I've solved Archel's second problem, but I can't be bothered to check really. I think it's probably 2 and 4.

― caitlin (caitlin), Thursday, 29 May 2003 15:35 (twenty-two years ago)

I think it's 3 and 4.

― Tim (Tim), Thursday, 29 May 2003 15:40 (twenty-two years ago)

But Caitlin doesn't know that the numbers aren't 3&3 / 2&5 respectively, and in both cases Andrew would know.

― Andrew Farrell (afarrell), Thursday, 29 May 2003 15:55 (twenty-two years ago)

No, it's definitely not 3 and 4.

I was assuming that Archel's two numbers were different, in which case as you said it can't be 2 and 4.

― caitlin (caitlin), Thursday, 29 May 2003 16:09 (twenty-two years ago)

Ack, I think I'm wrong.

― caitlin (caitlin), Thursday, 29 May 2003 16:25 (twenty-two years ago)

The numbers are 4 and 13, and you owe me an hour of my life back. All of you.

― Andrew Farrell (afarrell), Thursday, 29 May 2003 16:26 (twenty-two years ago)

Math describes reality = we all get to say things like "I have surmised your product!"

― nabisco (nabisco), Thursday, 29 May 2003 16:28 (twenty-two years ago)

andrew: 'splain please.

― Sterling Clover (s_clover), Thursday, 29 May 2003 17:12 (twenty-two years ago)

Erk, good point. so line by line:

1) If the numbers are two primes, then Andrew can figure them out immediately.

2) So Caitlin can only be certain that Andrew doesn't know if her sum can't be the sum of two primes. So it can be 11, 17, 23, 27, 29....

3) Andrew understands that Caitlin's number is one of these, so if he can figure it out from that, then his number can only be the product of numbers that sum to one of the numbers in 2) in one way.

So for example, it can't be 30, because you can get 30 from 11->(5,6)->30, and 17->(2,15)->30

and so on. The first few contenders after this are

11->(2,9)->18
11->(3,8)->24
11->(4,7)->28
27->(2,25)->50
17->(4,13)->52
29->(2,27)->54
23->(4,19)->76
27->(4,23)->92
35->(3,32)->96
51->(2,49)->98

4) And then if Caitlin can figure it out from _this_ list, then her number must be the sum of numbers that can only multiply to one of the numbers in 3) in one way. So we can see that it isn't 11, or 27. And in fact when you use the rest of the list, it can only be 17->(4,13)->52

Er, any questions. Technology was used in the construction of this album.

― Andrew Farrell (afarrell), Thursday, 29 May 2003 17:50 (twenty-two years ago)

But was it post- or pre-1963 technology?

― nickn (nickn), Thursday, 29 May 2003 19:30 (twenty-two years ago)

So is the problem solvable or not? I got 9,2, and 2, but that means squat-all if you know neither the woman's address nor her neighbors. And that's not what the census guy came back to ask, either -- one must assume he's able to see everyone's house number. So what's the point of this exercise?

― Kenan Hebert (kenan), Thursday, 29 May 2003 19:54 (twenty-two years ago)

No, he looked at the neighbor's house, and that wasn't enough info to solve the problem. But when he found out that one of the kids was the "oldest" one, then he was able to figure out what the answer was (assuming you don't think of twins as having an "older" and "younger" one). Therefore the house next door is #13 and the kids are 9, 2, and 2, instead of being 6, 6, and 1. (Both of which add up to 13 and multiply up to 36.)

― Chris P (Chris P), Thursday, 29 May 2003 20:01 (twenty-two years ago)

Wait a second!

A series that increases arithmetically goes up by adding n to the original: x, x+n, x+2n, x+3n, etc.

An exponential series goes by increasing the exponent by n: x, x^n, x^2n, x^3n, etc. (More or less.)

Doesn't a geometric series go by increasing adding n to a number and then squaring (or cubing or whatevering) it? So: x^y, (x+n)^y, (x+2n)^y, etc.

Which I think means that Sam was backwards at the beginning.

Arithmetic: 1, 2, 3, 4, 5, 6, 7...
Geometric: 1, 4, 9, 16, 25, 36, 49...
Exponential: 1, 2, 4, 8, 16, 32, 64...

(Note that the geometric progression has the early lead but the exponential soon proves a swift victor.)

― Chris P (Chris P), Thursday, 29 May 2003 23:50 (twenty-two years ago)

Andrew you kick ass.

I'm gonna write a program of my own to do this tomorrow. Prolog seems the obvious choice but bleh if I can remember how to work it and double-bleh if I have a compiler handy anymore.

Java is the next obv choice (since thats what I mainly use at work) but hell if perl won't be more fun I think.

― Sterling Clover (s_clover), Friday, 30 May 2003 04:20 (twenty-two years ago)

i more exponential than you.

― hard boiled egg, Friday, 30 May 2003 04:28 (twenty-two years ago)

Both of which add up to 13 and multiply up to 36

Where did 36 come from?

― Kenan Hebert (kenan), Friday, 30 May 2003 04:36 (twenty-two years ago)

Oh, never mind. Jeez. I get it.

― Kenan Hebert (kenan), Friday, 30 May 2003 04:36 (twenty-two years ago)

I think my brain just exploded.

― Christine 'Green Leafy Dragon' Indigo (cindigo), Friday, 30 May 2003 04:55 (twenty-two years ago)

That's the idea. It's too late for this sort of thing.

― Kenan Hebert (kenan), Friday, 30 May 2003 04:59 (twenty-two years ago)

Yes, Andrew's solution is right. This is the program I used to check it. It prints out the sum and product, rather than the original numbers.

#!/usr/bin/perl sub c {$t=shift;foreach $v (@{shift()}){return 1 if ($t==$v)}return 0} for ($i=2;$i<=200;$i++){next if ($ar[$i]);$j=$i;$p=0;while ($p<=200){$ar[$p=$i*$j]=1;$j++}push @p,$i} TP: for ($i=4;$i<=200;$i++) { for ($j=2;$j<=int($i/2);$j++) {next unless (c($j,\@p));next TP if (c($i-$j,\@p));} push @q,$i} foreach $q (@q){$p{$q}=[];for ($i=2;$i<=int($q/2);$i++){push @{$p{$q}},$i*($q-$i)}} foreach $p (keys %p){$r{$p}=[]; U:foreach $q (@{$p{$p}}){foreach $r (keys %p){next if ($p==$r);next U if (c($q,$p{$r}))}push @{$r{$p}},$q}} foreach $r (keys %r){next if (scalar @{$r{$r}}>1);foreach $s (@{$r{$r}}){print "$r $s\n"}}

― caitlin (caitlin), Friday, 30 May 2003 09:08 (twenty-two years ago)

Hardcore.

― Andrew Farrell (afarrell), Friday, 30 May 2003 10:19 (twenty-two years ago)

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