# Irrational, Imaginary, and Transcendental

( 3 , 4 , 5 ) ( 5, 12, 13) ( 7, 24, 25) ( 8, 15, 17) ( 9, 40, 41) (11, 60, 61) (12, 35, 37) (13, 84, 85) (16, 63, 65) (20, 21, 29) (28, 45, 53) (33, 56, 65) (36, 77, 85) (39, 80, 89) (48, 55, 73) (65, 72, 97) (20, 99, 101) (60, 91, 109) (15, 112, 113) (44, 117, 125) (88, 105, 137) (17, 144, 145) (24, 143, 145) (51, 140, 149) (85, 132, 157) (119, 120, 169) (52, 165, 173) (19, 180, 181) (57, 176, 185) (104, 153, 185) (95, 168, 193) (28, 195, 197) (84, 187, 205) (133, 156, 205) (21, 220, 221) (140, 171, 221) (60, 221, 229) (105, 208, 233) (120, 209, 241) (32, 255, 257) (23, 264, 265) (96, 247, 265) (69, 260, 269) (115, 252, 277) (160, 231, 281) (161, 240, 289) (68, 285, 293)

The more mathematically minded among you may recognize the above as a list of pythagorean triples. If you’ve forgotten what that means, I offer the below animated .gif to jog your memory.

Of course most right triangles don’t have sides that can be represented by whole numbers; fractions abound, and many commonly encountered right triangles in construction, fabrication, and other enterprises have hypotenuses that cannot even be expressed as a ratio of integers.

For example, a right triangle with two sides that are equal has a long side (hypotenuse) of a length that cannot be expressed as the ratio of two integers. This number is a familiar and indispensable friend to photographers. There are other examples these sorts of numbers that have proved to be useful; the Golden Ratio, Pi, and others.

You’d think that the discovery of these sorts of very useful numbers would have been met with great acclaim, but in fact legend says that the first person to prove/discover these sorts of numbers, Hippasus of Metapontum, was either murdered or exiled by his fellows.

Discouraging.

—

My nephew is a smart lad. I know this from talking to him, but it is confirmed by the fact that he is presently pursuing a Ph.D in economics at MIT.

I saw him over Thanksgiving, and because of my earlier posts and the ensuing comment threads, I asked him if his work ever called for the use of *i*.

“I hate *i*!” came his reply without a moment’s hesitation.

This puts my nephew in good company. René Descartes was suspicious of *i*, and the numbers that that spill forth from *i*.

And my experience as a math teacher suggests that discomfort with *i* is not confined to the brilliant. Average students often (usually) find *i* discomfiting.

Poor *i*.

—

In reading up to make this post, I came across a category of numbers I’d never heard of before: Transcendental Numbers. As this is new information to me, I hope I’m correct when I say that transcendental numbers are numbers that are not algebraic numbers. Pi is a transcendental number; so it *e*. 4 is not. The ratio of the hypotenuse of a right triangle with two equal shorter sides to those shorter sides is not a transcendental number.

Color me not well informed about what transcendental numbers are, or why this property of transcendentalism is important.

—

Thanksgiving morning, as I was doing my research, I made a tweet or two, as mental markers, and as it happens, an iFriend who is a mathematician saw my tweets and that that prompted a short conversation where I learned (if I understood him correctly) that the aspects of mathematics that seem most self-evident, as in 1,2,3,4… is actually the least well-founded.

Today, as I talked about this with my wife on our long drive back from Flatbush to Montauk, I tried to conceive of a race of beings who apprehended the ratio of a circles diameter to its circumference, ratio of the hypotenuse of a right triangle with two equal shorter sides to those shorter sides as fundamental concepts, and have to build upon such a foundation novel ideas like whole numbers. I doubt it’s an original thought, but my brain couldn’t take the idea any further.

—

As mentioned in the comment thread of a previous post, the Help Wanted post has yield a hire. We’ve managed to poach a young artist and craftsman away from the Smithsonian and soon he and I and the rest of our build team will be making practical application of the properties of triangles; right, left, and other.

I’ve also just learned that before he was a US Supreme Court Justice, Oliver Wendell Holmes Jr. practiced admiralty law. In relation to the concept of property and property rights and how and why we build the social systems we build, I’ve been thinking about Ecuador and the “tuna wars” of the 1970s. More on that soon.

Quantum physics uses complex numbers a lot.

PS, Dont mix complex numbers with matrix algebra. Your mind just breaks.Report

murali,

surely it isn’t pretty, but hardly brainbreaking.

The only reason quantum mechanics requires complex numbers is because our current formulation of math is ineffectual and poor.

someone ought to rewrite the whole subject, one of these days…Report

If you don’t correctly mix complex numbers with matrix algebra, your airplane will <a href=”http://en.wikipedia.org/wiki/Short_period”>crash</a>.Report

There is a hierarchy of infinities.

The smallest infinity is that of the natural numbers (0,1 ,2 ,3, …), the technical term for which is aleph-null. (“Aleph” because that’s the nomenclature used for infinities, “null” because it’s the smallest.)

The size of the integers (natural numbers plus their negatives) is also aleph-null.

The size of the rationals (fractions with integers on top and bottom) is also aleph-null.

The size of the algebraic numbers (all roots of polynomials with integer coefficients) is still aleph-null. This includes all square roots, cube roots, and so on. That ratio of the hypotenuse of a right triangle with two equal legs is the square root of two, thus algeraic. The Golden Mean is also algebraic.

The size of the reals, called “C” [1] is much bigger than aleph-null. Thus the transcendentals, which are all the real numbers that are not algebraic, is also much bigger than aleph-null. (if you remove the algebraic numbers from the reals, they leave such a tiny gap you’d hardly notice they’re missing. The technical term for this is that they have “measure zero”.) Some of them have names like e and pi, but almost all do not, for the simple reason that all the possible names in all languages totals up only to aleph-null.

This is strange stuff. When I first learned it, it seemed completely unintuitive and the proofs seemed like hand-waving. Having understood it for going on forty years now, it makes perfect sense, and I cannot grasp why it violated everything I thought I understood about number.

1, C may be the same as aleph-one, which is the second-smallest infinity. Proving this requires an assumption called the Axiom of Choice, which some mathematicians prefer to avoid, and all agree to at least keep track of where it’s needed.Report

C = aleph-one is actually the continuum hypothesis, and is independent of the axiom of choice in fact.Report

You are of course correct. I was mixng up the answers to two famous riddles:

What’s yellow and equivalent to Choice?What’s big and gray and undecidable?(Zorn’s lemon and the continuum hippopotamus respectively.)Report

What’s purple and commutes?

Abelian grapes.ReportThe thing that blows my mind about the transcendentals is that they comprise more or less all the reals (as per Mike S above), we really only regularly encounter two (pi and e), and yet these two transcendentals occur in such a wide variety of applications that are different from their definition. So does this imply the incredible amount of mathematics that we haven’t even scratched? Or are Pi and e really somehow more special than the others.

The answer (it seems to me) is that we have gone looking for these particular numbers as an answer to a question. So . . . what questions are we not asking to find more transcendental numbers . . .Report

It’s reassuring to find out that infinities actually do come in different sizes, especially since I deploy this concept rhetorically with some regularity. (“So long as the scope of one’s efforts are limited to what can be supported in one’s spare time, there is really no limit to what might be attempted!“) Also, in the reading I did ahead of this post I encountered the arabic concept of an “algebraic object”; that was the springboard for imagine a race of beings that understood algebraic objects as being more tangible (for lack of a better word) than whole numbers. I got as far as imagining them throwing own of their own into the sea for suggesting that one is one because it’s one of something the quantity of which is one. People have been murdered for less.Report

It will probably not surprise you to learn that, not only do infinities come in different sizes, but there is no infinity large enough to answer the question “How many distinct infinities are there?”Report

God.

According to Cantor. And he either (ought to know) or (was crazy), both of which are true.Report

One of my mom’s favorite movies is “The Mirror Has 2 Faves”. In it, George Segal seems flumoxed on how to teach infinite sets. I tried with my wife (who has very limited math knowledge) and was able to get the concepts (including subtracting an infinite set from another infinite set) in a matter of minutes.

That part of the movie made me soooooooooooo mad!!!!Report

The size of the reals, called “C” [1] is much bigger than aleph-null. Thus the transcendentals, which are all the real numbers that are not algebraic, is also much bigger than aleph-null.The set of noncomputational numbers is likewise bigger than the algebraics.

In fact, most of the transcendals that we know of (like pi and e) represent a trivial subset of the reals; the set of computable numbers is of order aleph-null.

So most of the things that make up the difference in size between the reals and the natural numbers are in fact inexpressible using the language of finite mathematics.

Which means, in many ways, the “reals” are way more imaginary than most of the complex numbers that people are familiar with.

Because those are, themselves, also countable.Report

The issue with imaginary numbers is that some people simply aren’t able to stop thinking of numbers as countable quantities that translate directly to the real world. Even real numbers are seen as “integers plus a bit”. The only transcendental number is pi, and even that is just “button on the calculator that I push when I’m using an equation I memorized.”Report

When I teach imaginaries I use the idea that “In math we have an answer for everything” and start off with “Remember in 3rd grade and you had the problem 6-10=?”. Back then there was no answer because you can’t take 10 away from 6. You can but then you’re short 4. Thus the negative numbers.

Once the kids firmly remember that they understand how to deal with numbers that “don’t exist” then it’s that much easier to tackle imaginary numbers which are just a variation on that motif of “There must be an answer”.

Now transcendental numbers are a different story but not that complicated: They’re just irrational numbers for which there was no logical creation. Pi is transcendental because it just is an irrational number. The Square Root of 2 is not transcendental because we can write that irrational decimal instead as a 2 under the radical.

The one I did a poor job of teaching this year was “e”. I introduced it to my honors precalc kids but know I did not do a great job of putting it into context. For the curious e is part of the limit you reach when you do complex compounding at an infinite rate. Lemme splain:

You make an investment that pays 10% per year. You can then use a simple formula to compute the total of your investment after 30 years. You’ll have some quantity. Okay wait… calculator….

Okay. $2,000 invested at 10%/ yr for 30 years will grow to $35,878.

Now, you can also do compounded interest where the interest is applied more often. However to balance this you get less interest per application. So suppose you were to have your interest compounded twice a year. That would mean you get paid 5% at each application but you get double the applications. Personally I think this is a scam on the part of the banks to say “look at how much more interest you get” by giving you less interest more often. However it works out well for you:

That $2,000 invested at 10%/yr compounded twice a year will grow to $37,358.

So what if we do it something more crazy? What if we compound it daily?

So $2,000 invested at 10%/yr, compounded daily yields $40,156.

So compare those three values. You get more money the more often you compound but the return on the work of investing goes down. You get less ~increase~ as you expand the rate at which you compound. In fact there comes a point where compounding more often simply doesn’t increase the return any measurable amount. For example if we do hourly compounding the yield is $40,170. That’s compounded 24 times as many times as daily but we only see an increase of $14. The increase is even less if we get down to minute by minute compounding.

So the transcendental number ‘e’ is discovered as the actual ~Limit~ that this value approaches if we go from compounding 2 times a year, to compounding 356 times year to compounding and INFINITE number of times a year. Roughly, e=2.71 but it goes on infinitely and without pattern like pi and all other irrational numbers.

So…. what questions do you have?*

*Never ask “do you have any questions?” because it invites kids to just say “no” which usually isn’t the case. 🙂Report

Yes, the important thing about mathematics is that there is an infinite sea of answers out there and you just have to pick one. That was the point of the Existence And Uniqueness Theorem, which the ODE class teacher just sort of mumbled at us and then never explained further.Report

It’s hard to understand e without understanding the natural logarithm (even with the Pe^rt application), and it’s hard to understand the natural logarithm without understanding dx/dt = kx. (at least for me, that’s when it all clicked)Report

I like this approach. As an adult when I encounter ideas I have trouble with I frequently recount earlier ideas and recollect how it was difficult to understand them before I understood them, and how after I understood them they were easy to understand.

Part of what fueled this post was my reflecting on the comments threads that my earlier posts provoked and the language people used (carelessly or thoughtfully) to make their arguments.

Irrational numbers are not irrational, but the name stuck, and I’m sure it doesn’t make learning about them any easier with such a perjorative nomenclature. Same for imaginary.

That’s what I found it fascinating to learn that what I regard as the most self-evident aspect of mathematics is regarded by mathematicians as being one of the kludgiest.Report

In fairness, I think the rational/irrational terminology comes from “ratio”, not from “reason”.Report

You think? The Pythagoreans dumped poor Hippasus over the side for even suggesting them. Maybe their called “irrational” because that’s how they make people behave!Report

The motivation that I always remember was from the question “Is there a function g(x) such that g'(x) = g(x)?”Report

And while pi is generally first encountered while studying circles, it also arises from “Is there an f(x) such that f”(x) = -f(x)?”. or “How should e**x be defined for complex x?”.Report

http://www.coldbacon.com/writing/borges-tlon.htmlReport

For example, a right triangle with two sides that are equal has a long side (hypotenuse) of a length that cannot be expressed as the ratio of two integers.If the equal sides are rational, yes. But if the equal sides are of length sqrt(2), for example, then the hypotenuse will be 2.Report

I like how the Pythagoreans only believed in rational numbers, but they worshiped as holiest of holies the Golden Ratio, phi, which is irrational.

As for i, it really gets useful in generalizing the use of the natural logarithm to describe oscillatory functions likes sines and cosines. This is very helpful in, for example, studying the stability of linear systems.Report

It is virtually impossible to do electrical engineering without ‘

i“.Report