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Mathematics: stupid and clever questions for people who understand - Comments

AtheistEgbert's Avatar Comment 1 by AtheistEgbert

My favourite two questions:

1) What is the squareroot of minus one?

and

2) What is the result of any number divided by zero?

Have fun with those.

Mon, 07 May 2012 20:01:31 UTC | #940380

Quine's Avatar Comment 2 by Quine

The first one is easy, it's ln(cos(π))/π.

Mon, 07 May 2012 20:16:45 UTC | #940385

Neodarwinian's Avatar Comment 3 by Neodarwinian

@ Quine

No. ( though cos(pi) = - 1 )

1) What is the squareroot of minus one?

= i ( a complex number )

2) What is the result of any number divided by zero?

= Undefined

What is the first derivative of X cubed? ( a warm up )

Mon, 07 May 2012 20:26:37 UTC | #940391

Quine's Avatar Comment 4 by Quine

@ Quine

No.

But oh, it is so. Type this into Google and see what you get: (ln(cos(pi))/pi) * (ln(cos(pi))/pi)

Mon, 07 May 2012 20:47:52 UTC | #940404

VrijVlinder's Avatar Comment 5 by VrijVlinder

@Quine:

Type this into Google

lol try using your computers calculator, Mine even though it says it is a scientific calculator, It tells me "Not A Number" If it is not a number what the hell is it??? Undefined ....ugh!

Logarithms, the supreme language for the unexplained.. How about this one: In+1 = In + WILL or ei*pi + 1 = 0

"The formula, ei*pi + 1 = 0 expressed a world of thought, of truth, of poetry, and of the religious spirit ‘God eternally geometrizes.’"

"Not A Number".....

The God Equation?

Mon, 07 May 2012 21:06:18 UTC | #940413

Jos Gibbons's Avatar Comment 6 by Jos Gibbons

These examples have reminded me that quite a lot of common maths confusions, both by modern laypeople and by historical mathematicians, stem from wanting to know “what is” the entity with a notable set of properties rather than what it does. For example, consider this question:

What is the squareroot of minus one?

One way to answer this is to say, “there’s a quantity we call i, or alternatively j, characterised by its square being -1 and its having ordinary interactions with real numbers (those on the familiar number line) in their arithmetic. Obviously, -i also squares to -1, so there are two square roots of -1, just as 5 and -5 both square to 25. Indeed, one convention is that j=-i, so i and j are the two roots of -1.” Note this definition tells you how it all works, e.g. you infer that (1+2i)(3-4i) = 3+2i-8i*i = 11+2i. But what “is” i in a metaphysical sense? Tough luck with that one; mathematical objects are defined “implicitly” by how they interact rather than “explicitly” in the way you’d say “cats are these things over here”. Indeed, if you replace i with –i everywhere in true statements, they remain true, e.g. (1-2i)(3+4i) = 11-2i. The point is not that we’ve “found” an i whose square is -1; the point is that you can claim there is at least one such i, and from that many other things readily follow, such as the examples above.

Several times in the history of mathematics, problems we couldn’t solve were solved by saying “here are some new numbers with the right properties; let’s see what that implies”. So mathematicians said, “but what are they?” Only about 100 years ago did they reach a consensus that that’s the wrong question to ask. It seemed plausible; positive real numbers are lengths; negative numbers help in accounting, and statements including them can be carefully restated to avoid them; rational numbers can be defined with things like slices of cake, and so on. But with complex numbers it gets a lot weirder. Sure, they have applications, e.g. in quantum mechanics; but you can’t seem to trip over i times something in the real world, in the way you can 3.2 kg of objects on the floor.

i is an interesting example because the first reason people disliked square roots of negative numbers was because, at the time, they weren’t even sold on negative numbers! In those days, there were four kinds of monic quadratics. You didn’t say, x squared + ax + b = 0 was the general case; you said x squared = ax + b and x squared + ax = b were 2 different kinds of quadratic. Only positive roots (or a root of 0) would “count”. If a quadratic equation with real coefficients has complex non-real roots, nowadays we easily find them with the quadratic formula, but in those days they’d say “those roots don’t exist”. They didn’t mind saying, “the reason you can’t find 2 roots to every quadratic is because not every quadratic has 2 roots”.

But then something weird happened. Cardano invented a way to solve any cubic equation. But it turned out that even when all three roots were positive real numbers, complex numbers had to be temporarily worked with to get the roots; their role was irreducible. This was called the casus irreducibilis, and ultimately motivated the concession that complex numbers “count”.

Another problem with phrases like “root(-1)” is other systems exist with such roots obeying different rules. For examples, quaternions have infinitely many numbers whose square is -1, because they don’t commute, i.e. when you multiply quaternions the order matters. Wherever you find “square roots of -1”, the important thing is, what are the rules that are obeyed and what can you infer from that?

One last potential problem if you introduce new numbers is it might make the enlarged theory inconsistent. We can show that, if the theory of real numbers is consistent, so is the theory of complex numbers. But now look at this question:

What is the result of any number divided by zero?

Division is an inversion of multiplication, so “a over b = c” means “the unique root of bx = a is x = c”. (If commutativity fails then care is needed concerning whether we prefer to instead define division with the equation xb = a.) So if I say “a over 0 = c” that means “the unique root of 0x = a is x = c”. But if a = 0 any x is a root, while if a isn’t 0 no x is a root, so it’s never unique. But let’s suppose I introduced “z = 1/0”. I can’t keep all the old rules. Look at this argument: “0z = 1, 2 = 2(0z) = (20)z = 0z = (30)z = 3(0z) = 3, a contradiction”. Here it is assumed “z = 1/0 implies 0z = 1” and “a(bc) = (ab)c” (associativity of multiplication). You can invent z if you want, but you’ll have to drop at least one of those rules. One example of an area where we invent division by 0 is wheel theory.

Quine playfully answered “what is root(-1)” with:

The first one is easy, it's ln(cos(π))/π.

Let me explain why that’s right (in a way). cos pi = -1. (Quine, please tell me how you did an actual pi!) To define ln(-1), we use the fact that ln is the inverse of exp, rather like the division-multiplication relationship. But for complex numbers exp doesn’t have a unique inverse, so ln(-1) could be ipi(2n+1) for any integer n. One common convention implies, among other things, that we take n = 0. The argument is now complete; dividing by pi gives i. I hope that makes sense to Neodarwinian. S/he could reply, “ln(-1) is ambiguous, so maybe you have to divide by 3*pi, for example”. Well, true. But Quine’s convention defines ln(z) as the so-called principal natural logarithm of z.

What is the first derivative of X cubed?

3 X squared, if you differentiate with respect to X. An old joke:

Full of himself, e to the x meets a scared 7 running round the corner. “A derivative is after me! He’ll destroy me by turning me into 0.” “Well, he can’t change me however often he mist me,” sniggers e to the x. “Hello,” he says, introducing himself to the derivative as he arrives. “I’m e to the x.” “Hello,” replies the derivative. “I’m d/dy.”

Mon, 07 May 2012 21:06:35 UTC | #940414

ShadowMind's Avatar Comment 7 by ShadowMind

Dividing by zero could also be taken as lim(x->0) of 1/x, which would imply that the result is infinite, but then you get into questions about which infinite (think about the graph of y=1/x). But that's one of the good things about science - some solutions just result in more questions!

Mon, 07 May 2012 21:20:04 UTC | #940419

Quine's Avatar Comment 8 by Quine

(Quine, please tell me how you did an actual pi!)

There are many ways, but the easiest is just to copy and paste from places like this one.

Mon, 07 May 2012 21:47:06 UTC | #940428

godzillatemple's Avatar Comment 9 by godzillatemple

How can you possibly have "orders" of infinity? Infinity is not an actual number -- it's a concept that means that there is no end to numbers. So how can you multiply infinity times itself in the first place, and how could the result be anything other than infinity in the second place?

Mon, 07 May 2012 21:54:44 UTC | #940429

HardNosedSkeptic's Avatar Comment 10 by HardNosedSkeptic

Comment 6 by Jos Gibbons :

Quine, please tell me how you did an actual pi!

You can include Greek letters in your posts by making use of HTML entities. These are strings of characters which are interpreted by the browser as a single character. They always begin with an ampersand symbol (&) and end with a semicolon.

For pi (π), simply type π

You can include lots of mathematical and other symbols in the same way. For example, to include a “squared” symbol (e.g. 3x²) type ².

See here for a comprehensive list of HTML entities. I cannot guarantee it will be possible to use all of them on this website – you will have to use trial and error to a considerable extent.

Mon, 07 May 2012 21:56:03 UTC | #940432

VrijVlinder's Avatar Comment 11 by VrijVlinder

right alt+p gives you π using your keyboard. also left alt+5 gives you ∞ symbol here is a list of Alternative key codes for your keyboard. They may vary on a windows keyboard but you can figure out where they are. by pressing right or left alt along with a letter or number . HTML does not work here some stuff gets filtered and * takes it's place. But ASCII Characters work best ASCII Character Codes Chart even though you can just use your keyboard if you want to.

Mon, 07 May 2012 22:19:17 UTC | #940441

delToro87's Avatar Comment 12 by delToro87

Comment 9 by godzillatemple :

How can you possibly have "orders" of infinity? Infinity is not an actual number -- it's a concept that means that there is no end to numbers. So how can you multiply infinity times itself in the first place, and how could the result be anything other than infinity in the second place?

Believe it or not, there are different sizes of infinity! For example, the set of all integers and the set of all real numbers both have infinite cardinalites, but the cardinality of the reals is greater than that of the integers. There are in fact an infinite number of sizes of infinity, the question is - which size of infinity is there an infinite size of infinities?

Mon, 07 May 2012 23:32:57 UTC | #940451

mmurray's Avatar Comment 13 by mmurray

Comment 9 by godzillatemple :

How can you possibly have "orders" of infinity? Infinity is not an actual number -- it's a concept that means that there is no end to numbers. So how can you multiply infinity times itself in the first place, and how could the result be anything other than infinity in the second place?

If you use the correct notion of two infinite sets being the same size then it turns out that there are infinite sets that are not the same size. For example the infinite set we know best perhaps is the natural (counting numbers).

1 , 2 , 3 , 4 , ...

Defining when two infinite sets are the same size is a bit tricky so let me skip that. Let's instead call a set countable if you can list it out like the natural numbers. So you can (in theory!) count it. This means we have defined a notion of what it means for a set to be the same size as the set of natural numbers. Of course you will never finish counting but I just want a rule for counting that will get to every element of the set if I count for long enough and which never counts anything twice. It's basically what we do when we count the letters of the alphabet and find there are 26 of them. We tap A and say 1, tap B and say 2, until we get to Z. If we have counted everything at least once and nothing twice we will find there are 26 letters.

Consider now the set of all real numbers strictly between 0 and 1. We are going to do the famous Cantor diagonalisation argument. Imagine we have counted all the real numbers between 0 and 1 and the list is

a1 , a2, a3 , ...

Write out their decimal expansions as an array

a1 = 0. a11 a12 a13 ...
a2 = 0. a21 a22 a23 ...
a3 = etc etc

Now manufacture a new number by writing down its decimal expansion

b = 0 . b1 b2 b3 b4

Make the digit b1 the next digit after a11 (do it cyclically so if a11 is 2 make b1 = 3, if a11 is 9 make b1= 0 etc)

So maybe our list looked like

a1 = 0. 0 1 6 4 5 ...
a2 = 0. 5 8 8 9 ...
a3 = 0. 2 4 6 0 7 ...

so b = 0. 1 9 1 ...

Notice that b is a real number between 0 and 1 so it should be in the list. But its decimal expansion is different to the decimal expansion of everything in the list as it disagrees with ai in the ith decimal place. So it isn't there. So we were wrong in thinking we could count all the real numbers between 0 and 1. We say that the real numbers are uncountable.

It doesn't stop there as we then find there are uncountable sets which are not all the same size.

Infinite sets fail some basic things we know about finite sets. For example if you have a bag with 10 coins in it and I take out 10 coins from the bag the bag is empty. But if you have a bag with the natural numbers in it and I take out 2, 4 , 8 I get infinitely many things and you have infinitely many things left. Google the Hilbert hotel for more things like this.

Michael

Tue, 08 May 2012 00:10:47 UTC | #940455

Neodarwinian's Avatar Comment 14 by Neodarwinian

But oh, it is so. Type this into Google and see what you get: (ln(cos(pi))/pi) * (ln(cos(pi))/pi)

My TI-84 says, - 1! ( I trust Texas to make instruments, though not sense! )

3 X squared, if you differentiate with respect to X. An old joke:

No doggerel, just algorithms!

Tue, 08 May 2012 00:19:41 UTC | #940458

Sample's Avatar Comment 15 by Sample

Hmm, not sure I can add much to this discussion, however I do know that 3 out of 2 Americans don't understand statistics. (Credit: Mark Crislip/Science-based Medicine).

Mike

Tue, 08 May 2012 01:40:05 UTC | #940469

mmurray's Avatar Comment 16 by mmurray

Comment 14 by Neodarwinian :

But oh, it is so. Type this into Google and see what you get: (ln(cos(pi))/pi) * (ln(cos(pi))/pi)

My TI-84 says, - 1! ( I trust Texas to make instruments, though not sense! )

3 X squared, if you differentiate with respect to X. An old joke:

No doggerel, just algorithms!

Or feed it into www.wolframalpha.com

Michael

Tue, 08 May 2012 02:25:11 UTC | #940477

Chomolungma's Avatar Comment 17 by Chomolungma

You can think of i as representing a rotation of 90 degrees in the complex plane.

One thing I have problems visualizing is how quaternions define a rotation in 3d space. I get Euler angles (yaw, pitch, roll) and I can see how a 3x3 matrix can define a rotation/scaling transformation (it's just 3 vectors defining 3 axes defining a new co-ordinate system to map a point onto) but quaternions just don't make intuitive sense. I get how there have to be 4 components, because you can define any arbitrary rotation using an axis of rotation (a 3 component vector) and a rotation angle around it (scalar), and I know how to make a quaternion by plugging those things into a formula, but I just don't understand what results. How is the rotation somehow smeared across the other 3 components, to end up with 4 components that are each some weird hybrid of part axis, part rotation? Does anyone know an intuitive way to visualize this?

Tue, 08 May 2012 03:19:20 UTC | #940479

Jos Gibbons's Avatar Comment 18 by Jos Gibbons

How can you possibly have "orders" of infinity? Infinity is not an actual number -- it's a concept that means that there is no end to numbers. So how can you multiply infinity times itself in the first place, and how could the result be anything other than infinity in the second place?

Further to the previous excellent discussions of infinite sizes of sets, or transfinite cardinals as they’re technically known, there are two other kinds of infinity in mathematics.

One of these is to do with set theory; in particular, sets whose members, members’ members etc. are also all sets. (It’s all built from the empty set originally. We say it’s set theory without urelements.) Those sets which are transitive (their members are subsets of them) and well-ordered by membership (so any non-empty subset of them has a member which has none of the other members of the subset as members, and given any two distinct members of the original set one is a member of the other but not vice versa) are called ordinals. The class of all ordinals is itself transitive and well-ordered by membership, which proves it’s not a set (this “Burali-Forti paradox” was the first paradox in naïve set theory, i.e. the first proof not all classes are sets). The “first” ordinal with a property at least one ordinal has is thus always well-defined, provided the ordering relation with which we define “first” is membership. For example, the first transfinite ordinal, i.e. ordinal with infinitely many members, is ω and all subsequent ordinals are also transfinite. Not all transfinite ordinals differ in cardinality, e.g. ω+1 and ω have the same cardinalities. In situations like these, while bijections between the sets exist, order-preserving bijections don’t; ordinals are used to define order types, which are more specific than mere cardinalities.

If the axiom of choice is assumed, all sets have the same cardinality as at least one ordinal, so all sets have a well-ordering (because a bijection between the set and the ordinal can be used to define one), and all well-orderings have the same order type as exactly one ordinal’s ordering relation under membership, although ordering relations other than well-orderings don’t correspond to any ordinal’s ordering relation under membership. If the negation of the axiom of choice is assumed, some sets don’t have the same cardinality as any ordinal, and these sets cannot be well-ordered.

The other has nothing to do with either set-theoretic notion of infinity, but is probably easier to understand. Real numbers are extended to include ∞, or +∞ & -∞, so as to get either the affinely or projected extended real number system. We define the arithmetic they obey to preserve as many of the old rules as possible, e.g. ∞×∞= ∞ and so it’s a root of x² = x as you suggested. This is also true of transfinite cardinals. One key difference between this kind of infinity and a transfinite cardinal is that the former satisfies 2 to the x = x and the latter satisfies 2 to the x > x (Cantor’s theorem). Extended reals have associated limit-theoretic rules, e.g. regarding what the phrase "limit as x tends to @#8734; f(x) = a" means when (i) a is real and (ii) a is infinite.

How is the rotation somehow smeared across the other 3 components, to end up with 4 components that are each some weird hybrid of part axis, part rotation? Does anyone know an intuitive way to visualize this?

I do know one way to make sense of quaternions’ characterisation of rotations, but I have to first enlarge the theory from 3 dimensions of space to 1 of time and 3 of space as per special relativity. Did you know quaternions are especially useful in special relativity because of the way Lorentz transformations work? A rotation of the spatial axes alone is expressible with a Lorentz transformation, so my explanation will begin by associating quaternions with Lorentz transformations, then looking at rotations specifically.

The four-vector (a, b, c, d) is mapped to the quaternion a+bi+cj+dk. Lorentz transformations are linear maps on the set of four-vectors. Linear maps on real or complex numbers are quite boring; you just multiply by a constant. Quaternions are a less trivial case, however, because the breakdown of commutativity means the most economic irreducible way to state all linear maps on quaternions is to say what are the 1,i,j,k components of the values it maps 1,i,j,k to. Indeed, thinking about what arbitrary linear maps on the set of four-vectors do in matrix terms makes this 16D phenomenon, and this way to parameterise it, intuitive. Another way to do it is by writing the map as a linear combination of the terms x, ix, jx, kx, xi, xj, xk, ixi, ixj, ixk, jxi, jxj, jxk, kxi, kxj, kxk. If you want an exercise, you can work out how to interconvert these two approaches.

Of course, only some linear maps are Lorentz transformations, so they correspond to only some linear maps on quaternions. This is where I get to enjoy those Greek-symbol letters various people have given me on this thread! If η is the familiar metric from special relativity, a Lorentz transformation is a matrix Λ iff ΛTηΛ = η (that T is meant to be a superscript; does anyone know how to get arbitrary superscripts rather than 1, 2 or 3?), T denoting taking a transpose. Another way to define them is that, if A=ΛA’, B=ΛB’ then ATηB = A’TηB’. That means we can write it in terms of how the map affects quaternion multiplication. Bear in mind that (a+bi+cj+dk)(e+fi+gj+hk)=(ae-bf-cg-dh)+(ch-dg)i+(df-bh)j+(bg-cf)k. In other words, if u, v are the 3D vectors (b,c,d), (f,g,h), and we treat quaternions as paravectors http://en.wikipedia.org/wiki/Paravector (scalar-vector sums), we can write (a+u)(b+v) = (ab - u.v) + u×v. That formula is even prettier when you realise its scalar part is the Minkowski inner product of the associated four-vectors.

Rotations are the Lorentz transformations in which time is unaffected and space rotates on its own, and so the matrix has first row and first column (1,0,0,0). All these quaternion calculations are heavy work and hard to summarise on here, but if you go through them all as an exercise hopefully the way quaternions correlate with rotations will “make more sense” afterwards. Let me know if it does.

Tue, 08 May 2012 06:56:51 UTC | #940494

Helga Vieirch's Avatar Comment 19 by Helga Vieirch

Great idea, Jos.

There is a nice website here, that has a lot of links and information, and a good resource for math questions of all sorts. The guy behind this website is Albert Bartlett, someone I really admire because he has used his retirement to become really active in getting people informed about the major issues facing humanity in the near future. He re-analysed M.King Hubbert's data (he sent me a PDF, so if anyone is interested, let me know and I can pass it on). This, and concern about general concern with current policies favouring continued economic growth, motived him to start giving his famous set of lectures on the exponential function and why failing to understand it is the greatest shortcoming of the human race.

And for math fun with artistic intent, check this stuff out!

I don't have any math questions, but I do have a question about what people think are the important uses of mathematics. Is it just a skill that is needed to get into fields where it is applied to practical problems in jobs involving engineering, statistics, and so on? I know how beautiful and calming it is - like a refuge from the whole of a chaotic world. I remember this from when I was a little kid - here was this whole other language that was totally logical and predictable. It was magical and I loved it. But it is not just a cool place in your head or a nice skill needed to do certain jobs. It is an essential tool of all scientific analysis… with enormous power to predict the future of our climate, our species, our planet, and our universe.

And, as Dr. Barlett's effort clearly illustrates, general ignorance of math and math phobia could really hurt us in the future.

Tue, 08 May 2012 08:20:57 UTC | #940500

Anaximander's Avatar Comment 20 by Anaximander

What is the result of any number divided by zero?

It is 5. Maybe not the correct one, but a result anyway.

Tue, 08 May 2012 09:22:31 UTC | #940506

Premiseless's Avatar Comment 21 by Premiseless

What is the minimum distance from which you begin to feel threatened when:

1) A shady looking person stands next to you in a)the supermarket? b) an empty street on a dark night?

2) A same sex individual who looks like a perfect specimen is in the same room as you a) in the toilet? b) when you are stood with your partner? c) when you are looking at someone you are attracted to?

In view of the above and other similar questions, what can be said about the mathematics of your emotional measurements?

P.S. Do we also have religious measurements that we need to consider?

Tue, 08 May 2012 09:38:47 UTC | #940510

AtheistEgbert's Avatar Comment 22 by AtheistEgbert

Infinity has driven many mathematicians mad. Or were they mad to begin with?

Tue, 08 May 2012 21:25:14 UTC | #940615

septiman444's Avatar Comment 23 by septiman444

What about (1/x).sin(1/x) when x ---> 0 ?

Tue, 08 May 2012 21:43:25 UTC | #940619

All About Meme's Avatar Comment 24 by All About Meme

I'm still laughing at how d/dy outfoxed e to the x.

(cough)

Although it may be tangent to the actual discussion topic, Jos, I would be very interested in your take on how Isaac Newton discovered the derivative. For instance, what was the mathematical world like just prior to the year 1665, with respect to problems that are now child's play for the Calculus? Legend has it that an apple inspired the discovery of gravity, was there a similar "Eureka" moment that led to the birth of "fluxions"?

And please consider this request a compliment to your astounding erudition in a wide range of fields. I don't think many could attempt to divine the mind of an acknowledged genius like Newton, if you'll pardon the expression.

Tue, 08 May 2012 23:33:16 UTC | #940632

Jos Gibbons's Avatar Comment 25 by Jos Gibbons

I don’t know whether septiman444 knows the answer to his/her own question, but I’m sure some people won’t, so I’ll answer it. Its answer boils down the same was as “what’s 1/0?” (except that, as far as I’m aware, no non-standard theory solves this one in the way wheel theory solves that).

What about (1/x).sin(1/x) when x ---> 0 ?

As with division by 0, the “what is it?” (in this case, what is the limit) question shouldn’t be worded so as to assume the existence of a quantity that, it turns out, doesn’t exist. In this case, it would mean, “for which a is it true that, for any δ>0, there’s an ε such that, if |x-0|<δ |(1/x) sin (1/x)-a|<ε?” (The wording is different when a=∞, but that’s not “the limit” here either.) Needless to say, the problem is no a has that property. For the special case x = 1/(2n+½)π sets (1/x)sin(1/x) to (2n+1/2)π, which diverges as n goes to ∞ (equivalent to x going to 0), whereas by contrast if x = 1/nπ we get 0 instead.

Although it may be tangent to the actual discussion topic

Stealth pun!

I would be very interested in your take on how Isaac Newton discovered the derivative. For instance, what was the mathematical world like just prior to the year 1665, with respect to problems that are now child's play for the Calculus? Legend has it that an apple inspired the discovery of gravity, was there a similar "Eureka" moment that led to the birth of "fluxions"? And please consider this request a compliment to your astounding erudition in a wide range of fields. I don't think many could attempt to divine the mind of an acknowledged genius like Newton, if you'll pardon the expression.

I see you pick and choose which puns to point out. Anyway, I’ll give it a shot.
The first thing to understand is that, when mathematicians solve a new maths problem, historically it’s usually because a new practical problem requires it. The second thing is to understand that, while the derivatives of a handful of specific functions were needed for one approach to calculating the areas and volumes of certain shapes, which is why even the ancient world did some very rudimentary “calculus” as we would now identify it, to motivate a differentiation algorithm for arbitrary functions required mathematicians to care about how quickly arbitrary functions change, which basically meant it had to wait for Newton to care about motion. By contrast, Leibniz was a polymath who wanted to one day see calculations solve every problem regardless of field, so essentially he invented calculus because he noticed it didn’t exist yet. (Well, technically Newton beat him by 9 years; there’s a big controversy regarding whether Leibniz really knew this, but at any rate his emphasis differed from that of Newton, because Leibniz was all about the integrals.)
The next thing to realise is that calculus has quite a detailed history. As I said, developments came because we wanted them. Eureka moments are so named because Archimedes allegedly yelled Eureka when, upon noticing his body’s displacement of water, he worked out how to solve a problem that had been assigned to him. I don’t think we know, for any of the specific developments in the history of calculus, that there’s an associated “He worked it out like Dr House in such and such a coincidental moment” story. And, of course, both the bath story and the apple story are now considered myths. A weaker requirement, which Wikipedia calls the Eureka effect, is just that you “suddenly work it out”, which it seems is quite common in mathematics. Or so we think. In my experience, it’s more like, “we realise we could solve the problem step by step in this way, and one hard part (possibly the only one) is here”, and that one hard part has a Eureka moment, or possibly gets broken down again etc. I suppose there has to be something discrete in how the problems are solved, but a gradual breakdown might seem unworthy of the term “Eureka moment”.

Wed, 09 May 2012 06:05:37 UTC | #940680

The Jersey Devil's Avatar Comment 26 by The Jersey Devil

(3) Does anyone know what fields future discussions could address?

Eventually, you could revisit Physics or Biology for people that are new to the site or for people that have learned since then and are now stumped on more advanced topics.

Wed, 09 May 2012 21:39:04 UTC | #940779

Chomolungma's Avatar Comment 27 by Chomolungma

Jos Gibbons

math stuff

I have to confess that a lot of what you wrote there is above my level of understanding, but those are some interesting connections I didn't know about and suggests further avenues of study/exercise. Thanks!

Thu, 10 May 2012 07:51:35 UTC | #940832

Zeuglodon's Avatar Comment 28 by Zeuglodon

(3) Does anyone know what fields future discussions could address?

Chemistry would be an excellent topic. That subject deserves more attention given how important it was in shaping the Industrial Revolution, and it connects physics and biology. Or, unless you count it as a subset of mathematics, maybe a statistics thread? Or probability?

If you count stats as a subset of mathematics, then maybe you could explain why a large standard deviation does not make two sets of results useless when their averages are compared and diverge a little. You raised this point with me on another thread, but though you described it as a common intuitive error, I can't see why. I still have this image of a scatter graph in my head which suggests a weak to non-existent correlation.

If you need context, it was in that study about how The Thinker lowered people's confidence in god's existence. Do you remember it?

Comment 6 by Jos Gibbons

Your description of how the square root of -1 was accepted in mathematics was interesting, but wasn't it based effectively on a shortcut - i.e. on pragmatism? I may be misunderstanding this, but couldn't you arrive at the same answer using real numbers? I mean, to what real-world applications do imaginary numbers apply?

Thu, 10 May 2012 14:24:14 UTC | #940863

Jos Gibbons's Avatar Comment 29 by Jos Gibbons

unless you count it as a subset of mathematics, maybe a statistics thread? Or probability?

They seem like subsets of it enough that I’d hope people with questions about them raise such questions here, but maybe they’d rather wait for a thread like that later. You’ve brought up an example or two below:

maybe you could explain why a large standard deviation does not make two sets of results useless when their averages are compared and diverge a little. You raised this point with me on another thread, but though you described it as a common intuitive error, I can't see why. I still have this image of a scatter graph in my head which suggests a weak to non-existent correlation.

The tricky thing here is “variance” (square of “standard deviation”) is an ambiguous term. I’ll need to put all this into a disambiguating context (or at least a context which elucidates how much ambiguity is in need of disambiguation) by using the central limit theorem. Given n IIDs (independent identically distributed random variables), e.g. measurements of a real-world quantity that has a probability distribution, if each of those IIDs x_1, …, x_n has common mean μ and common variance σ², their mean x has mean μ and variance σ²/n. What is more, when n is large, the distribution of x is almost exactly Gaussian or “Normal”. That lets you approximately calculate the probability distribution of the “z-value” of x, i.e. how many of its standard deviations it is away from its mean (z < 0 when x < μ). When statisticians talk about a 95 % confidence interval, they verified μ lies in the interval with probability 0.95 on this Gaussian approximation. Note that the standard deviation is much smaller for the mean than for the individual x_i.

Now consider a more general case, where the x_i still have a common mean but different variances, which means they’re no longer IIDs. An unbiased weighted mean w of the x_i is equal to the sum over all i of (a_i)(x_i), where the a_i are non-negative reals of sum 1. The usual mean takes each a_i to be the same, i.e. each = 1/n. For any choices of the a_i, you can calculate the variance of w from the variances of the x_i, provided the x_i are still independent. There’s a standard answer to the question, “which choice of the a_i, in terms of the σ_i, minimises the variance of w?” Needless to say, statisticians have gotten pretty adept at making standard deviations as small as possible so that gaps between means can be large by comparison.

So now let’s say you want to know whether xs have a greater mean than ys, and so you’re interested in the mean and standard deviation of the variable x-y. To be sure, a standard deviation mustn’t be too large a percentage of that mean’s modulus, or the result will have little statistical significance. But which standard deviation needs to be small? That of the sample rather than individual values. I remember explaining this once to a poster who didn’t see how satellite readings could confirm temperature changes smaller than their temperature errors. As for the experiment to which you refer, the fact is I can’t say how significant the final results were without knowing a lot more about the numbers quoted, including which types of “standard deviation” that term referred to. (Indeed, it’s more complicated than I’m implying here; for example, there are some contexts in which a variance should be divided not by n, but instead by n-1.)

On with complex numbers. (Did you know the guy in charge of Conservapedia refuses to embrace complex numbers? He comes from a field that needs them!)

wasn't it based effectively on a shortcut - i.e. on pragmatism?

Not entirely. For starters, the complex numbers are the unique algebraically closed extension of the reals. If you want the set S of permitted coefficients in polynomials to include all roots of such polynomials, complex numbers are the way to go. It gives a sense of completeness, rather than claiming sometimes there just aren’t all that many roots, which smacks of denial when compared with a system that finds them. To be sure, you can say everything again with real numbers only; in solving quadratic equations with real coefficients, for example, you could seek those real pairs (x,y) with a(x²-y²)+bx+c = (2ax+b)y = 0. You can even restate the complex-coefficients generalisation if you really want to. In the end, however, that’s more a case of deliberately avoiding admitting you’re talking about complex numbers. And if z²+1=0 is rewritten into a statement with a root in your theory, that’s as good as admitting i is there. I will say more on that below.
Getting back to the casus irreducibilis, the only way to work out on the way there what the roots are is to make use of existence theorems on the complex numbers. For example, one key part of solving x³ + px + q = 0 relies on noting there must be complex (possibly real) u, v satisfying u+v = S, uv = P, regardless of your choice of S, P. The proof takes S = x, P = -p/3. As for getting three different roots, later you obtain formulae for u³, v³ from which you can only get u, v to within factors which are complex cubic roots of 1. Once you have the final result, you could say, “Theorem: these are the roots” and then verify they are through substitution, and in the case where all roots are real your proof of their validity wouldn’t need to use complex numbers. But why bother with all of that hassle?
Another famous example of uses of complex numbers is to calculate a cosine-based formula C or a sine-based formula S by first making sure you have one of each, very similar to one another, and then use results like de Moivre’s theorem and the properties of geometric series to obtain C+iS and hence C and S, so you get two results for the price of 1. But the “even if you don’t need complex numbers for this, they make things a lot easier” examples keep coming. Of the many examples from Euclidean geometry alone, my personal favourite is Napoleon’s theorem, so named because legend has it Napoleon Bonaparte found a proof of this theorem (although I don’t think the legend says he used complex numbers):

Given a triangle ABC, erect in the plane on its sides equilateral triangles (i) outwardly and (ii) inwardly. In case (i), the centres of these equilateral triangles are themselves the vertices of an equilateral triangle. In case (ii), this is true too. These two bonus equilateral triangles’ areas differ by the area of triangle ABC. (You can also express the bonus triangles’ energies as linear combinations of the original triangle’s area and the sum of the squares of its side lengths.)

couldn't you arrive at the same answer using real numbers? I mean, to what real-world applications do imaginary numbers apply?

You probably know complex numbers appear in the axioms of quantum mechanics, because the states are vectors in a complex-valued Hilbert space. You can rewrite the entire thing using only real numbers, but it’s inadvisable. You end up making the “inner product” more like a Minkowski product in special relativity, which means things like the non-negativity of norms and Cauchy-Schwarz inequalities need a fundamental overhaul, and you’re better off just proving the underlying theorems about complex numbers. (In fact, even special relativity is sometimes instead written using complex numbers to make its equivalents of those results easier to prove.) Because that’s all the jump from the quantum axioms to the quantum theorems is – it’s proof of a theorem about complex numbers which states that that jump is logically valid.

What real-world applications do imaginary numbers have, indeed? Assuming you mean ordinary 2-part complex numbers, they unavoidably come up in quantum mechanics and make many calculations easier in such diverse fields as optics, electrical engineering and fluid physics (including its applications such as in climatology, hydraulics or aerodynamics, and possibly also where you need to understand vibrations, e.g. designing bridges that will handle the wind properly). “Hypercomplex” numbers such as quaternions have even more esoteric links to reality, but it’s all useful nonetheless.

Thu, 10 May 2012 16:45:46 UTC | #940889

Jos Gibbons's Avatar Comment 30 by Jos Gibbons

Correction: the Minkowski space gives a modified form of the triangle inequality; that's what I meant to talk about.

Thu, 10 May 2012 17:26:51 UTC | #940898