The Old Joel on Software Forum: Part 4 (of 5)

Screw math...

Who in their right mind would take the square root of -1?

So damn confusing. No one in the real world ever takes the root of -1. I talked with this CPA friend of mine, and he had no idea what he was talking about. Business apps don't use root of negatives - exponents and powers are primarily used in interest bearing terms so toss all this irrational nonsense.

An how come that "PIE" number has so many digits? Three point nerdy-nerdy-nerdy-in-come-free. What's up with that? PIE should come in slices: Quarters, Eigths, Sixths, something a business app can use.

hoser
Tuesday, May 25, 2004

e^iπ + 1 = 0

People are expected to understand that? Yuck. I bet if we got rid of those 5 numbers, and the numbers derived from them, math would be much easier.

Danil
Tuesday, May 25, 2004

Now that's an interesting feature.

Um, Joel? How was I supposed to guess that Greek isn't supported? I seem to have forgotten.

Danil
Tuesday, May 25, 2004

Hear, hear...bring back base 12 for currency whilst we're about it.

a cynic writes...
Tuesday, May 25, 2004

You're right! Let's all, please, dumb ourselves down to be exactly as intelligent as the average business person. We need to all leverage some synergy right away! And paradigms! We need to think outside the box and rematrix our latent cross-fertilization value creation initiatives!!!

anon
Tuesday, May 25, 2004

Yeah, what anon said! Er, wrote!

The waterboy
Tuesday, May 25, 2004

a = b
a^2 = ab
a^2 - b^2 = ab - b^2
(a + b)(a - b) = b(a - b)
a + b = b
b + b = b
2b = b
2 = 1

Capn' Kirk
Tuesday, May 25, 2004

And that is why division by zero is an error.

If a=b then a-b=0, so you cannot divide by (a-b) going from line 4 to 5...

O Canader
Tuesday, May 25, 2004

Gee, spoil the fun so early...

Capn' Kirk
Tuesday, May 25, 2004

Sorry, at the university I went to, they remove the sense of humour when they give you the BSc in Mathematics -- but only for pure math, not Applied Math.

O Canader
Tuesday, May 25, 2004

-1 . you mean "i". Business appls dont need this. PI is the ratio of circumference to diameter. thats why it has so many digits.
when pythagoras existed, E Business applications did not exist, FYI

Karthik
Tuesday, May 25, 2004

You know math does have other uses besides business...

Bill Rushmore
Tuesday, May 25, 2004

Haha. I'll bite on the troll bait.

Yes square root of minus one does seem kinda wierd. BUT. Really, talking about the square root of two is no less wierd, in a way.

Working with the rational numbers (which were the only kind of numbers we really had for a large portion of history), there's no such thing as square root of two. Just doesn't exist, is meaningless to talk about it. No rational number squares to give two.

But then some bright spark comes along and thinks... hm. Well, yeah, there isn't any number that squares to give two, but it would sure be handy if there was, if we could talk about these kind of numbers. Pythagoras might throw us out of the boat for pointing this out, but hey.

So, they think, lets just pretend there /is/ such a number, and see if we can extend the kinda of algebra we're used to doing with rational numbers, to work with these new 'square root' numbers too. And turns out we can, of course. At some point along the line sqrt(2) stopped becoming some kind of silly impossible thing that doesn't exist, and started becoming something we work with alongside all the other numbers. And the exact same thing goes for square root of minus one too.

It's just a little bit harder to relate i to things in real life, like you can relate root two to the diagonal of a square. But algebraicly speaking there's not much of a difference.

Trying to think of uses of complex numbers in business and I have to admit I'm stumped. I guess complex analysis is often used to prove some big theorems in statistics (central limit theorem anyone?) which have a lot of business applications. But really. Who gives a shit. The things are way cool.

Matt
Tuesday, May 25, 2004

"Sorry, at the university I went to, they remove the sense of humour when they give you the BSc in Mathematics -- but only for pure math, not Applied Math."

Something similar happens to us CS grads. We get a new sarcasm/cynicism plugin called 'WiseA++'.

Capn' Kirk
Tuesday, May 25, 2004

>>Who in their right mind would take the square root of -1?

There is much more out there than the narrow view of business applications.

According to Wikipedia
http://en.wikipedia.org/wiki/Complex_number#Applications

"The complex number field is also of utmost importance in quantum mechanics since the underlying theory is built on (infinite dimensional) Hilbert spaces over C."

as far as quantum mechanics is concerned.
http://en.wikipedia.org/wiki/Quantum_mechanics#Applications

"Much of modern technology operates under quantum mechanical principles. Examples include the laser, the electron microscope, and magnetic resonance imaging."

Furthermore.

"Many of the phenomena studied in condensed matter physics are fully quantum mechanical, and cannot be satisfactorily modeled using classical physics. This includes the electronic properties of solids, such as superconductivity and semiconductivity. The study of semiconductors has led to the invention of the diode and the transistor, which are indispensable for modern electronics."

double_dark
Tuesday, May 25, 2004

As the first forms of writing we have were about selling donkeys and oxen and how many bushels of wheat could be expected and this was before Greek mathematics (which was partially derived from Babylonian anyway), I think we can safely say that business processes have been around as long as there were two people and one had something the other needed.

Sometimes I am more than reminded of a planet founded by a lot of hairdressers, waiters and PR assistants.

Simon Lucy
Tuesday, May 25, 2004

Matt, imaginary numbers are invaluable in EE for power analysis, where you have to deal with voltage and phase.

And, since I was the target of this thread, then let me point out that phase angles in AC are often measured in degrees. :-P

Philo

Philo
Tuesday, May 25, 2004

Well yeah, I know they're used all over the place in physics and engineering. Another example would be fourier series analysis and the FFT used for pretty much any kind of digital signal processing (audio codecs, voice over IP, whatever).

I was trying to think of something directly business-related though, as opposed to just 'facilitates some broad base of technology used in business'... hm.

Matt
Tuesday, May 25, 2004

Well, it was fun. :) Trolling for Philo. Gonner make me some spanikopita once I get enuf.

Philo, did they ever let you drive the boat?

hoser
Tuesday, May 25, 2004

Phase angles in degrees? I cringe at the thought of such ugliness ;-)

But yes, lots of jokes about mathematicians being at Pi/2 to reality...

Matt
Tuesday, May 25, 2004

If by "boat" you mean "aircraft carrier" then yes - I qualified as both helmsman and Officer of the Deck on the USS Midway (CV 41)

Philo

Philo
Tuesday, May 25, 2004

aircraft carrier? pfft. why don't you go get yourself a REAL boat?

jackass
Tuesday, May 25, 2004

---"Hear, hear...bring back base 12 for currency whilst we're about it. "----

All in favour, as long as we have base 12 for everything else.

Or base 8

Whoever designed hands and feet clearly wasn't a mathematician.

Stephen Jones
Tuesday, May 25, 2004

Matt,

FFTs and related transforms (DWT) are used in business applications to analyze timeseries data. There are probably more physicists working on Wall Street than in physics research. So those who work in "business" have a use for i as well.

Rob VH
Tuesday, May 25, 2004

Also, signal strength should always be expressed and calculated as a number from 0 to 10, not "dB" which are just stupid. Nobody thinks "I'm going to turn my iPod up to 72 dB," they think "I'm going to set it at 3 or 4" or "I'm going to turn it all the way up."

Ron
Tuesday, May 25, 2004

There is a web page going around about base 12, and they point out that you could on one hand in base 12, as long as you use all of your finger knuckles.

And for those math doubters, I worked on a project that dealt with radio wave propigation simulation, and it had intermediate results that involved complex numbers. I've also had to deal with trig for some light 3-D graphics work, and now have to deal with e and ln some for finance.

Keith Wright
Tuesday, May 25, 2004

...to 11?

Philo

Philo
Tuesday, May 25, 2004

One’s intellectual and aesthetic life cannot be complete unless it includes an appreciation of the power and the beauty of mathematics. Simply put, aesthetic and intellectual fullfillment requires that you know about mathematics.

– J. P. King

Tapiwa
Tuesday, May 25, 2004

"...to 11?"

That would be considered 10 + (1)i

Ron
Tuesday, May 25, 2004

Aha! yep I'm should have thought of all that time series analysis of financial derivatives and what have you. I'm sure it's just dripping with i's. Never took any of the financial mathematics courses myself as they tend to sound boring as fuck, although a lot more lucrative than abstract algebra I'm sure, so hmm...

Nice quote there by the way, I'd recommend 'A mathematician's apology' by GH Hardy (one of the greats of early 20th Century English mathematics) as a great justification for laymen wondering what pure/abstract maths is all about and why people get so excited about the beauty of it. Slim little book, very readable, little to no formulas or anything and doesn't require you to understand any of the maths yourself in order to 'get' it...

Matt
Tuesday, May 25, 2004

The Indiana State Legislature once debated a bill that would've established the value of pi as 3.2, 4 or 9 (they weren't sure either).

But any branch of science that allows both irrational and imaginary numbers seems to be pushing it.

Tom H
Tuesday, May 25, 2004

Philo, were you ever involved in combat ops? (Midway OOD is damn impressive.)

Tuesday, May 25, 2004

Played tag with three other aircraft carriers in the Persian Gulf during Desert Storm. :-)

Philo

Philo
Tuesday, May 25, 2004

>> "But any branch of science that allows both irrational and imaginary numbers seems to be pushing it."

Ummm... Mathematics is NOT a branch of science. Math is deductive, sciece is inductive.

anon
Tuesday, May 25, 2004

I'd be interested in seeing a description of algebraic calculus that could be considered as deductive.

That's the same kind of thinking that says, all is known other than the next series of decimal points.

Maths, when it involves things other than counting your change or trigonometry and such, is a language. One of the ways we recognise languages is that we can say things in them that have not yet been said, and could not be deduced to be said in the future.

Simon Lucy
Tuesday, May 25, 2004

">> "But any branch of science that allows both irrational and imaginary numbers seems to be pushing it."

Ummm... Mathematics is NOT a branch of science. Math is deductive, sciece is inductive."

So how would you classify quantum mechanics? It's deductive - pure math is used to give results that correlate staggeringly well with observables. Just curious.

anon not the same one
Tuesday, May 25, 2004

When you say 'I'd be interested in seeing a description of algebraic calculus that could be considered as deductive'... what exactly are you looking for there? I'm not even quite sure what you mean by 'algebraic calculus', but any typical real analysis textbook will start out with the axioms for a complete ordered field and proceed to prove all of the basic properties of calculus from them very formally and deductively. If you were a sadist you could start off with just the ZF axioms for set theory and build everything up from there in a formal logic... any way you look at it you can build up mathematics completely deductively from suitable axioms.

Perhaps your issue is with how we choose the axioms in the first place? that one really is more a matter of intuition...

Matt
Tuesday, May 25, 2004

Matt

Didn’t Gödel show that you couldn’t build even elementary number theory up on the basis of deduction from a set of axioms?

as
Tuesday, May 25, 2004

imaginary numbers are invaluable in EE for power analysis...

... and to GM's, Finance Officers, Marketing... um... dudes...

Jack of all
Wednesday, May 26, 2004

There is numeric integration and algebraic integration, for example.

Calculus was invented, not discovered, almost simultaneously by Newton and Liebnitz as a way of describing relative motions or changes. You might say that it was purely deductive reasoning that enabled them both to come up with similar solutions, or you might also say that what they realised (and Galileo or Tycho didn't) was that they could use symbols combined in such a way as to reasonably describe complex systems.

Simon Lucy
Wednesday, May 26, 2004

> Played tag with three other aircraft carriers in the Persian Gulf during Desert Storm. :-)

That must have been a pretty mighty sight. Well done, by the way.

Wednesday, May 26, 2004

Well, he showed that there are statements that are undecidable (can't be proven or disproven) in any formal system that can model arithmetic. They're very contrived and peculiar examples though. Kind of analagous to handing someone a piece of paper that says 'this sentence is false' and acting like it makes the world of language fall apart.

When I said 'all of mathematics' I meant more the bulk of theorems that have been proven, the mathematics we use (and even more that we don't, but hey). So perhaps I should have said 'all the mathematics we use can be derived deductively from axioms'.

Matt
Wednesday, May 26, 2004

Simon:

Well yes there are definitely a lot of inductive, intuitive thought processes that go into /discovering/ mathematics, I wasn't denying that. Newton and Leibniz obviously had their respective flashes of inspiration about calculus. Seeing patterns, figuring out ways to prove things, drawing analogies between things, realising which axioms are useful to work with and which aren't, which definitions are useful things to use, which abstractions lead to beautiful theorems etc, are all somewhat inductive/intuitive processes...

But all that intuitive thought has to be chased home with deductive reasoning to actually prove the theorems, or else they just aren't considered a part of mathematics. That's what I meant. Mathematics is built up deductively. Choosing in which direction to build it up, is more of an intuitive process, but that's kinda different (although clearly very important too).

Sometimes there are things mathematicians talk about which can't be reasoned about deductively, for example the Church-Turing thesis that states (roughly) that a system that's able to perform any kind of rule-based mechanical calculations, must be equivalent to a turing machine. It's called a thesis not a theorem and isn't formally considered a part of mathematics, because nobody can find a way to state, let alone prove it in a formal context, but it's widely believed to be true on intuitive grounds. I guess that makes an interesting example...

Matt
Wednesday, May 26, 2004

The grammar of mathmatics (and there can be many grammars), describes the solutions its designed for. There may be deductive reasoning going on, undoubtedly there is, but it is nothing much to do with axioms, other than a kind of a-priori knowledge of the domain.

So, developing a new grammar, calculus, opened up a whole new set of solutions, not out of any deductive reasoning but because the language enabled the solution to be described.

Similarly, it was the use of topology applied to black holes that gave Hawking the insight as to how a Big Bang could happen.

The answer was not deduced, it arose out of applying a grammar to a different domain.

Deduction is something we do after the event to check what we did, or explain to others how rigorous we have been. It must have been this way, because of X and Y. Whereas in all reality describing the problem gave the solution, in whatever grammar, in whatever domain.

However, it isn't helping me solve this problem with a third party piece of software intended to update its accounting data with my data and its just sulking when it worked perfectly well last week. I obviously have mistaken the grammar it uses for one I thought I knew.

Simon Lucy
Wednesday, May 26, 2004

Simon:
I see where youre coming from now and I think we essentially agree... I was just differentiating between the mental processes involved in doing mathematics and solving problems (mostly intuitive), and the actual mathematical edifice that's been built up over time, the body of theorems, which all have to follow from previous work in logical steps, deductively. There's a tendency to try and sweep all the intuitive thought under the carpet once you've managed to prove something, but it's very important.

And yes the particular grammar and axioms you choose to work with are somewhat arbitrary. There's the school of thought which says that playing games with axioms and formal logic is essentially all there is to mathematics, and that which says mathematical objects exist independently of any attempts to formalise them. With calculus you can just look at the way things behave in the physical universe and tie it down to that, but with a lot of pure mathematics you don't have that convenience.

Whichever way you look at it doesn't make the necessary steps in a proof any less deductive anyway, or mean that you can just start saying things without proving them and call it mathematics. So, yeah.

I should do some work too. blahblahblah

Matt
Wednesday, May 26, 2004

And, no I wouldn't really consider Newton or Leibniz's work on calculus to be a part of Mathematics, in the strict sense I'm using the word, until it was made rigorous by others. They had the inspiration, but they didn't quite manage to chase it home all the way. They deserve most of the credit for it as it's the inspiration that really matters, but still.

Of course there's more than one way of being rigorous about it, infinitessimal analysis which formally defines actual infinitessimals in a framework of hyperreal numbers is much more fun than fiddling about with limits :-)

Matt
Wednesday, May 26, 2004

Matt, would you consider the continuum hypothesis contrived? It's leaky, by Gödel.

M
Wednesday, May 26, 2004

"An how come that "PIE" number has so many digits?"

As they used to say in Indiana...

"PIE are squared? No, PIE are round. Corn bread are square."

5v3n
Wednesday, May 26, 2004

Hmm... good point CH is less contrived than the usual example. But any mathematics that depends on it is pretty much by definition so abstract as to be useless as anything but formal symbolic manipulation on your (by that point pretty arbitrarily-picked) axioms for set theory. Certainly no constructivist mathematics would touch it with a bargepole.

I think you're missing the point though, when I said maths is built up deductively I didn't mean 'every possible statement can be proven or disproven'.

This is all getting a bit wanky anyway, I'm sure we're boring the pants off most of the software developers here...

Matt
Wednesday, May 26, 2004

The point about pure research is you never know when it will turn out to be useful.

TaKe the birthday theorem; how many people must be in the same room on average before you have more than a 50% chance of meeting somebody with the same birth date. The math on this mignt seem pointless but it becomes very useful in chips when you need to know the possiblity of two bits of memory flipping randomly and thus destroying your parity protection.

Stephen Jones
Thursday, May 27, 2004

Oh yeah probability and combinatorics like that is useful all over the place. Believe me pure maths gets a lot more useless than that :-) Set theorists love proving things about weird and wonderful classes of infinity that are much bigger than anything needed for practical mathematics (for most of pure maths, even)... find a practical use for those sorts of things and you'll surprise/impress a lot of people.

I think once maths gets to a certain point of wierdness and detachment from the real world (or from our intuitive idea of number) the philosophical justification for it being anything more than games with axioms and proofs starts to run thin. Normally if you need weird-ass 'optional' axioms things like the axiom of choice or the continuum hypothesis to prove a theorem, the theorem is then pretty much guaranteed to be useless for any applied purposes... but maybe someone can prove me wrong on that too. Things that are only practically used in a weaker form that doesn't need Choice don't count ;)

Matt
Thursday, May 27, 2004

All numbers are abstract concepts. While the set of natural numbers may be, well "natural", things get more abstract when you progress to integers (what does it mean to have -1 of anything?), rationals, and reals (including irrationals). Complex numbers may be one further step along the chain, but if you're comfortable with the concept of -1 then it's not a huge leap to accept an algebra in which the square root of -1 is defined. I think the terms "imaginary" and "real" are poorly chosen, giving people the impression that the square root of -1 is somewhat less real than the square root of 2. It's just that the algebra defined for the set of real numbers needs to be extended to cater for roots of negative numbers.

Hopefully no one will introduce the original poster to quaternions .

Appleologist
Thursday, May 27, 2004

The square root of -1 is perhaps the best example of lateral thinking, where one expands from a _number line_ to a number _plane_

1) A Definition can be only one of two: It has to be constructed arbitarily from one's whim and fancy OR deducted from an already established definition or observation.

2) Mathematical proof is solely dependent on consistency within the given framework.

Definitions:
1: {Arbitary} i is point on the real plane such
that it is a distance y, 90 degrees from the point x.

Further, x is identified with the point (0,0) and y with point (0,1)

2: {Derived from an existing entity viz. POWER} The nth root of A is that which when raised to power n results in A.

Axioms:

1) In a plane; (x,y) * (u,v) = (xu - yv, xv + yu)

Using Definition One:
i = (0,1)

Using Axiom One:
i * i = (0,1) * (0,1) = (-1, 0)

Ergo: i^2 = -1

Using Definition Two on the above result
n = 2 ; A = -1, ergo: Square Root of -1 = i

KayJay
Friday, May 28, 2004