## Grand Finale (Part IV): A Long Time Coming…

Although to penetrate into the intimate mysteries of nature and thence to learn the true causes of phenomena is not allowed to us, nevertheless it can happen that a certain fictive hypothesis may suffice for explaining many phenomena. — Leonhard Euler

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I will try to be fairly objective with this final blog and let the mathematics speak for itself. Thanks so much for your continued interest and response to these blogs! I greatly appreciate it, and I hope you learned a thing or two about mathematics and its eloquence.

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Recall that a complex number takes the form a + bi = r(cos(x) + i*sin(x)), where x = θ.

Let us focus our attention on cos(x) + i*sin(x).

From last blog, we can rewrite sin and cos as infinite sums.

sin(x) = x – x3/6 + x5/120 – x7/5040 + x9/362880 + … + (-1)nx2n + 1/(2n+1)! + …

cos(x) = 1 – x2/2 + x4/24 – x6/720 + x8/40320 + … + (-1)nx2n/(2n)! + …

Therefore, our expression cos(x) + i*sin(x) becomes the following.

cos(x) + i*sin(x) = 1 – x2/2 + x4/24 – x6/720 + x8/40320 + … + i*(x – x3/6 + x5/120 – x7/5040 + x9/362880 + …)

Let us now distribute i through the parenthesis.

cos(x) + i*sin(x) = 1 – x2/2 + x4/24 – x6/720 + x8/40320 + … + (ix – ix3/6 + ix5/120 – ix7/5040 + ix9/362880 + …)

We can rearrange the right side of the equation by exploiting the commutative property of addition, which states that the order in which numbers are summed does not matter. We will rearrange the equation so that the exponent of x increases as one reads the expression from left to right.

cos(x) + i*sin(x) = 1 + ix – x2/2 – ix3/6 + x4/24 + ix5/120 – x6/720 – ix7/5040 + x8/40320 + ix9/362880 + …

As it is currently written, this infinite sum does not convey much information, but after exploiting a property of the principle imaginary number i, we will be able to see something quite striking.

Recall i = sqrt(-1). Therefore, if we square both sides of the equation, i2 = -1. If we multiply this equation again by  i, we get i3 = -i. If we multiply this equation by i one more time, we get i4 = 1. Notice that, if we multiply the subsequent equation again by i, we come full circle in our answers, and the pattern repeats itself.

To avoid confusion, the pattern is written explicitly below.

i0 = 1

i1 = i

i2= -1

i3 = -i

i4 = 1

i5= i

i6 = -1

i7 = -i

i8 = 1 …

Let us now apply this property to the infinite sum.

cos(x) + i*sin(x) = i01 + i1x + i2x2/2 + i3x3/6 + i4x4/24 + i5x5/120 + i6x6/720 + i7x7/5040 + i8x8/40320 + i3x9/362880 + …

Notice that, if we went through each term we could see that i0 goes to one, i1 goes to ii2  goes to negative one, i3 goes to negative i, i4 goes to one, and so on. Thus, you can see that the expression above is compatible with our previous expression for cos(x) + i*sin(x). We can write this symbolically below.

1 + ix – x2/2 – ix3/6 + x4/24 + ix5/120 – x6/720 – ix7/5040 + x8/40320 + ix9/362880 + … = i01 + i1x + i2x2/2 + i3x3/6 + i4x4/24 + i5x5/120 + i6x6/720 + i7x7/5040 + i8x8/40320 + i9x9/362880 + …

Let us now exploit a property of exponents to group i and x together.

i01 + i1x + i2x2/2 + i3x3/6 + i4x4/24 + i5x5/120 + i6x6/720 + i7x7/5040 + i8x8/40320 + i9x9/362880 + … = (ix)01 + (ix)1 + (ix)2/2 + (ix)3/6 + (ix)4/24 + (ix)5/120 + (ix)6/720 + (ix)7/5040 + (ix)8/40320 + (ix)9/362880 + …

To finish our rearragment, let us call make a quick substitution: ix = u.

(ix)01 + (ix)1 + (ix)2/2 + (ix)3/6 + (ix)4/24 + (ix)5/120 + (ix)6/720 + (ix)7/5040 + (ix)8/40320 + (ix)9/362880 + … = u0 + u1 + u2/2 + u3/6 + u4/24 + u5/120 + u6/720 + u7/5040 + u8/40320 + u9/362880 + …

The right side of the equation should look very familiar to you now, as we derived it in last week’s blog. You may recall the following infortmation.

eu = u0 + u1 + u2/2 + u3/6 + u4/24 + u5/120 + u6/720 + u7/5040 + u8/40320 + u9/362880 + …

We can now substitute ix back in for u to get the following.

eix = (ix)01 + (ix)1 + (ix)2/2 + (ix)3/6 + (ix)4/24 + (ix)5/120 + (ix)6/720 + (ix)7/5040 + (ix)8/40320 + (ix)9/362880 + …

After performing this substitution, let us pause for a moment to reflect on what we have just proven.

eix = cos(x) + i*sin(x)

This equality, known as Euler’s Great Formula, serves as a basis for complex analysis and has many important applications. Right now, we will discuss one of those important outcomes, i.e. the most amazing formula in all of mathematics!

Recall x is just an angle. Let us say, for the sake of discussion, we choose the angle 180 degrees. As you may recall from the first grand finale blog, mathematicians prefer to measure their angles in radians, so 180 degrees really becomes π radians. Let us plug this value into Euler’s Great Formula and see what we find.

e = cos(π) + i*sin(π)

From basic trigonometry, we know cos(π) = -1 and sin(π) = 0.

e = -1

If we add one to both sides of the equation, the following relationship ensues.

e + 1 = 0

This, my friends, is the most amazing formula in mathematics. Titled Euler’s Great Identity, it combines the five fundamental constants with the three fundamental operations and has applications far reaching in mathematics. For example, have you ever thought to take the logarithm of a negative number? Is it really as blasphemous as your math teachers once told you?

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## Grand Finale (Part III): A Crash Course in Calculus with an Important Outcome

“I found a discarded textbook on calculus in a wastebasket and read it from cover to cover.”  — John Pople

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The first fundamental teaching from calculus discusses the effect of changing some independent variable on a given dependent variable. In other words, if I change a variable x and x is related to another variable y (i.e. y is a function of x), how much does y change?

Believe it or not, but you have seen this concept before in your algebra class. Recall that the equation for a line in the xy-plane is expressed as y = mx + b, where m is the slope of the line and b is the y-intercept of the line. Now, say I change x a little bit in this equation such that it takes on the value x + ∆x, where ∆x represents the amount I changed x. If I asked for the change in y as a result of changing x from x to x + ∆x, one could simply evaluate the y values at both x and x + ∆x using the equation for a line and, from there, subtract the two values for y to find the change in y. (For our purposes, we will represent a change in y with the symbol ∆y.)

Evaluating the equation for the line at x yields y = mx +b. Evaluating the equation for the line at x + ∆x yields y + ∆y = m(x + ∆x) + b. Subtracting the two equations (and doing a little pinch of algebra) yields the following relationship ∆y = m∆x. Therefore, we recognize that lines have the amazing property that, if I want to find a change in y as a result of a change in x, I need only multiply the change in x by the slope of the line. (You may recognize this relationship from the definition of slope, which is m = ∆y/∆x.)

Calculus extends this theory to graphs in the xy-plane that are not lines.

In general, we can consider a curve in the xy-plane to be comprised of very tiny (infinitesimal) line segments. We can calculate how much y changes when x changes over these very tiny lines. In other words, we can use the relationship ∆y = m∆x, where m is the slope of the small line segment. To indicate that we are over a very small line segment, we consider our changes in x and y to be infinitesimal, and so we use the notation dx and dy to represent infinitesimal changes in x and y, making our formula dy = mTdx.

Notice I changed m to mT. There is a definite reason for this change. Recall m is the slope of a tiny line segment. Recall also that these tiny line segments are essentially describing some generic curve in the xy-plane. Thus, from the curve’s perspective, m appears to be the slope of the tangent line at one of its points. (Hence, I included the subscript T for tangent.) This may explain why you hear discussion of tangent lines from students taking calculus classes.

As it turns out, mT becomes a function of x that gives you the value of the slope of the tangent line at some point on a curve y = f(x). mT can be tricky to calculate. Here, I will only go over how to find mT for simple curves.

Consider the function f(x) = a*xn plotted on the xy-plane, where a and n are constants. The slope of the tangent line for some generic x on the graph of f is then given by mT = a*n*xn-1. (I will not attempt to prove this rule, for it requires some extensive mathematics beyond the scope of this blog.) To make sure you are comfortable with this notion, however, let us do a quick example. Let us say that f is the graph of the standard parabola in the xy-plane, y = x2. Following from our formula, the slope of the tangent line to f must be equal to mT = 2x. If we consider the point where x = 4 (i.e. (4,16)) on our graph, we should then expect that the slope of the tangent line to our curve at that point equals 8 because mT = 2*4.

There is another very special curve in the xy-plane that we can very easily calculate mT. That is the curve whose y values are equal to its mT values for all values of x. (In other words, it is the graph of a function y = f(x) such that y = f(x) = mT for all values of x.) We can actually use the previous formula to help us discover this curve.

Recall y and mT are both dependent on x. Let us assume x begins at zero but can vary over the real numbers after starting at zero. Therefore, we have the following set of equations when x starts.

y = 0

mT = 0

At this point, you would say that the two values are equal and so we have found our answer, but you would be incorrect. Look at the previous formula mT = a*n*xn-1. If mT = 0, then either a, n, or xn-1 must be zero. In this case, n = 0. Now, if we take a look at the other part of the formula, y = f(x) = a*xn, we can conclude n = 0 and, assuming a = 1, y = 1. Therefore, for this set of equations to hold true, we need to add one to y.

y = 0 + 1

mT = 0

To keep the equality true, however, we must also add one to mT.

y = 0 + 1

mT = 0 + 1

Now, we have mT = 1 = a*n*xn – 1. Notice, there are no x terms in the expression for mT; we just have the value one. This implies that the exponent over x must now be zero. Thus, n – 1 = 0, or n = 1. If n = 1, then we can use the other part of the formula, y = f(x) = a*xn, to discover y = x, assuming a is one. Therefore, our equations become updated once again.

y = 0 + 1 + x

mT = 0 + 1

Again, to keep the equality true, we must also add x to mT.

y = 0 + 1 + x

mT = 0 + 1 + x

Hopefully, you can see that this process is never ending: We keep adding terms to the first equation only to add another term in the second equation that will go back and change the first equation. If you continue this process a few more times, however, you may start to recognize a pattern, which saves us a lot of time. (You can feel free to keep expanding the expressions for y and mT, but it becomes more and more complicated as you expand outwards, especially if you think about it the way I have taught you, i.e. the “non-Calculus” version.) The final result is a curve with the following equation.

y = 1 + x + x2/2 + x3/6 + x4/24 + … + xn/n! + …

The curve of this equation has tangent slope values equal to the y value along its length.

Believe it or not, but this curve can be simply expressed as the more familiar curve y = ex. (I will not show the proof here, for it requires mathematics beyond the scope of this blog.)

Therefore, I can claim the following.

ex = 1 + x + x2/2 + x3/6 + x4/24 + … + xn/n! + …

I can do a similar analysis to the one I have done here for other special curves in the xy-plane like y = sin(x) and y = cos(x). At the end of the day, the following two equations can be obtained.

sin(x) = x – x3/6 + x5/120 – x7/5040 + x9/362880 + … + (-1)nx2n + 1/(2n+1)! + …

cos(x) = 1 – x2/2 + x4/24 – x6/720 + x8/40320 + … + (-1)nx2n/(2n)! + …

We will use the subsequent three equations to transform our understanding of complex numbers next week, wrapping it up nicely with the most amazing formula in all of mathematics. One more blog to go!

Summary Exercise

Approximate f(1) = sin(1) to within a maximum possible error of 0.01.

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## Grand Finale (Part II): Not-so-Complex Numbers

“Out of intense complexities intense simplicities emerge.” — Winston Churchill

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As I alluded to in the very first blog, the way in which we as mathematicians organize numbers is very special: We use what is called the Real Number System to categorize any number into a family of numbers with similar properties. While I will not go into the specific breakdown of the Real Number System here, I do want to mention the biggest family within the system. Consider a number, any number. The chances are high that your number is a member of what we call the real number family, the biggest family in the Real Number System. In fact, any number you see and place practical value to your daily life—such as 1, 0, sqrt(2), -3, π, ¼, or 0.5—is a member of the real numbers.

Now, at first, you may find it superfluous that mathematicians would create a family of numbers wherein every number known to mankind is accepted. You may also find it a tad redundant to call this family the “real” numbers; after all, how can numbers be imaginary? Well, it turns out that, against our better intuition, there are numbers that transcend the real family, and so classifying it is not a total waste as originally thought.

Consider the number sqrt(-1). As it is written currently, sqrt(-1) suggests that there is such a number that, when you multiply it by itself, you will get -1 as a result. Now, if such a number exists, let us assume it is positive. A positive number multiplied by a positive number is always positive. Therefore, our number cannot be positive because we want the result of the multiplication to be negative. The number cannot be zero either because zero times zero is zero, not -1. Therefore, we must now assume that the number is negative. A negative number multiplied by a negative number yields a positive number. Hence, our number cannot be negative either because we need the product with itself to be -1, which is not a positive number.

Thus, it appears that sqrt(-1) lies outside the boundaries of the real numbers, and so we call it an imaginary number. More particularly, we call sqrt(-1) the principle imaginary number, denoting it with the letter i. All other imaginary numbers can be expressed in terms of this principle imaginary number. For instance, consider sqrt(-4). We can rewrite sqrt(-4) as sqrt(4)*sqrt(-1) by properties of exponents. Notice sqrt(4) simplifies to 2. Also notice that sqrt(-1) becomes i. Therefore, sqrt(-4) is equivalent to 2i.

If we now take these imaginary numbers and add to them a real number, we get what is called a complex number. In general, complex numbers take the form a + bi, where a is a real number and bi is an imaginary number. -3 + 5i, for example, is a complex number with a real component -3 and an imaginary component 5i.

Complex numbers can be graphed in what is called the Argand plane. You can think of this plane as a modified version of the familiar rectangular plane, where the x-axis becomes the real axis and the y-axis becomes the imaginary axis. If a complex number is expressed in the form a + bi, then the graph of the complex number in the Argand plane is given by moving a units along the real axis and b units along the imaginary axis.

The diagram above has a lot of variables thrown in it; however, the most important thing to see for now is that the complex number x + yi was graphed by moving x units on the real axis and y units on the imaginary axis.

I will now explain the origin of the other variables you see in the diagram. An important note that you should be made aware of before I do so is that there are many different ways to graph a complex number a + bi in the Argand plane. One such way is to do exactly what we have been doing: Move a units along the real axis and then move b units along the imaginary axis.  Another way to define a complex number in the Argand plane is to give a direction of travel, denoted by an angle θ from the real axis, and then specify the traveling distance required to reach the complex number on the plane, denoted by a radius r stemming from the origin to the complex number’s location.

Notice in the diagram that x, y, and r form a right triangle. Therefore, we can use trigonometric principles that we acquired last time to establish relationships between x, y, r, and θ. From trigonometry, we can determine that x/r = cos(θ) and y/r = sin(θ). If we multiply both sides of each equation by r, we get x = r*cos(θ) and y = r*sin(θ), which now coincides with the extra variables in the diagram.

Hopefully, we can now draw a very important conclusion about complex numbers. Namely, for any complex number of the form a + bi, there is a corresponding form r*cos(θ) + r*i*sin(θ), or alternatively r*(cos(θ) + i*sin(θ)). It is the expression cos(θ) + i*sin(θ) that we will really begin investigating in the coming weeks.

Summary Exercise

Given a complex number a + bi, find the corresponding r and θ values in terms of a and b. (Hint: Look at the right triangle formed by a, b, and r.)

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## Grand Finale (Part I): Basics of Trigonometry

“If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is.”   — John Louis von Neumann

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It would not be fair of me to throw a formula in your face and dub it amazing without supplying the hard work behind it. (You saw this when we discussed the area of a disk.) Therefore, over the course of the next four blogs, I will do just that. I will not, however, be doing it alone. You, too, will join me on my journey to derive the most amazing formula in all of mathematics, and hopefully, you will acknowledge precisely why it acquired such a name. With that in mind, welcome to the grand finale blogs!

Like any good mathematician, we need a solid mathematical foundation before we can derive anything. This blog and the next three blogs will make sure you understand everything needed to prove this formula. Think of these blogs as less of a math lesson and more of a discovery into the unknowns of your mathematical knowledge. Today, we discuss the very fundamentals of trigonometry, the study of triangles. For those of you who have taken the class and have met it with success, you will not need this supplement. This blog is solely for the benefit of those who may not have had the advantage you had.

To begin, triangles are special geometric objects because they have many properties unique to them. (For example, the sum of the angles inside a triangle must add to 180 degrees.) Some triangles, however, are more special than others. A triangle that contains a ninety degree angle in it, called a right triangle, is the most-specialized case. (See below.)

For definition purposes, we as mathematicians call the side opposite the right angle (the longest side) the hypotenuse. The other two sides are called the legs.

For many years, mathematicians sought to describe relationships between the legs and hypotenuse of a right triangle and the angles formed inside that right triangle. What ensued became known as the trigonometric functions. To be honest, there are dozens of these trigonometric functions, but for the purposes of this blog, we will focus squarely on two of them. One is called the sine function; the other is called the cosine function.

Consider an angle inside a right triangle other than the right angle. (It does not matter which angle you choose.) For standard purposes, let us call this angle θ. Now, consider the length of the side of the triangle opposite to θ (the side of the triangle that does not form the angle θ) as well as the hypotenuse of the right triangle (the side of the triangle opposite the ninety degree angle). The sine function, abbreviated sin, of the angle θ is defined to be equal to the ratio of the length of the opposite side of the triangle to the length of the hypotenuse of the right triangle. Symbolically, sin(θ) = opp/hyp.

To reinforce, the sine function takes a value for some angle (not the right angle) in a right triangle and converts it to a useful ratio of lengths between the opposite side and hypotenuse of that right triangle. For instance, in the diagram above, the sine of the indicated angle could be found by measuring the length of the opposite side and dividing it by the measured length of the hypotenuse.

The cosine function is described in a similar manner. Again, consider an angle θ in a right triangle, where θ is not the right angle. Consider the length of the side adjacent to that angle (the side of the triangle that helps to form θ but is not the hypotenuse) as well as the length of the hypotenuse. The cosine, abbreviated cos, of the angle θ is defined to be equal to the ratio of the length of the adjacent side to the length of the hypotenuse. Symbolically, cos(θ) = adj/hyp.

To reinforce, the cosine function takes a value for some angle in a right triangle (not the right angle) and converts it to a useful ratio of lengths between the adjacent side and hypotenuse of that right triangle. For instance, in the diagram above, the cosine of the indicated angle could be found by measuring the length of the adjacent side and dividing it by the measured length of the hypotenuse.

So to recap what we have learned, just remember that the lengths of the sides of a right triangle can be related to the angles formed inside that right triangle through trigonometric functions. Among these trigonometric functions are sine and cosine. For an angle θ (not the right angle), the sine function is defined as sin(θ) = opp/hyp, and the cosine function is defined as cos(θ) = adj/hyp. For the most part, this summarizes all the trigonometry you need to know in order to complete the derivation of the most amazing formula in all of mathematics. One catch to this summary, however, still remains: Mathematicians do not measure their angles in degrees.

Rather than be sensible, mathematicians prefer to describe their angles in a unit called “radians” (abbreviated rad). I will not talk much about radians on this blog other than the fact that they make our lives so much easier as mathematicians. (If you want to see the ideas behind radians, click here.) The conversion equality between degrees and radians is the following: 180° = π rad.

At any rate, the morale of the story is this: Before plugging in values for angles into sine and cosine, convert those angles into radians!

Below I have included a summary exercise to ensure your comprehension of everything I have taught. It is not meant to be impossible, but rather, it is meant to incorporate everything you have learned in a way that leads to your own exploration of mathematics. If you have no idea where to start, just leave a comment, and I will guide you in the right direction. It is imperative that you have a solid understanding of these trigonometric concepts.

Summary Exercise

Prove the following trigonometric identity: sin(θ) = cos(π/2 – θ). (Hint: Recall that the sum of the angles inside any triangle must equal 180 degrees, or π radians.)

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## The Traditional Way to Learn Mathematics: A Mountain, then No Mountain, and then a Mountain Again

“The history of every major galactic civilization tends to pass through three distinct and recognizable phases, those of Survival, Inquiry and Sophistication, otherwise known as the How, Why, and Where phases. For instance, the first phase is characterized by the question ‘How can we eat?’, the second by the question ‘Why do we eat?’ and the third by the question, ‘Where shall we have lunch?’” — Douglas Adams

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I hope over the past few months that I have challenged your perceptions of mathematics. (After all, that is my ultimate goal.) Perhaps, I have challenged you in such a way that you want to continue to explore and pursue the subject on your own accord to a certain extent of formality that suits you best. Be warned, however, that there is a correct way and an incorrect way to learn this subject. What you have done in your math classes so far has been largely incorrect: In school, math concepts do not build on each other nor is there any emphasis placed on the “behind-the-scenes” rigour. Truly, to understand the subject requires an awareness of both, and below, I provide you insight from a great modern mathematician on how to learn this subject in the way it was meant to be learned.

As a side note, we will be drawing on these ideas during the grand finale of “Musings of a Math Nerd,” if that gives you any incentive to continue.

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“There’s More to Mathematics than Rigour and Proofs”

By: Dr. Terence Tao

One can roughly divide mathematical education into three stages:

1. The “pre-rigorous” stage, in which mathematics is taught in an informal, intuitive manner, based on examples, fuzzy notions, and hand-waving. (For instance, calculus is usually first introduced in terms of slopes, areas, rates of change, and so forth.) The emphasis is more on computation than on theory. This stage generally lasts until the early undergraduate years.
2. The “rigorous” stage, in which one is now taught that in order to do maths “properly”, one needs to work and think in a much more precise and formal manner (e.g. re-doing calculus by using epsilons and deltas all over the place). The emphasis is now primarily on theory; and one is expected to be able to comfortably manipulate abstract mathematical objects without focusing too much on what such objects actually “mean”. This stage usually occupies the later undergraduate and early graduate years.
3. The “post-rigorous” stage, in which one has grown comfortable with all the rigorous foundations of one’s chosen field, and is now ready to revisit and refine one’s pre-rigorous intuition on the subject, but this time with the intuition solidly buttressed by rigorous theory. (For instance, in this stage one would be able to quickly and accurately perform computations in vector calculus by using analogies with scalar calculus, or informal and semi-rigorous use of infinitesimals, big-O notation, and so forth, and be able to convert all such calculations into a rigorous argument whenever required.) The emphasis is now on applications, intuition, and the “big picture”. This stage usually occupies the late graduate years and beyond.

The transition from the first stage to the second is well known to be rather traumatic, with the dreaded “proof-type questions” being the bane of many a maths undergraduate. (See also “There’s more to maths than grades and exams and methods.”) But the transition from the second to the third is equally important, and should not be forgotten.

It is of course vitally important that you know how to think rigorously, as this gives you the discipline to avoid many common errors and purge many misconceptions. Unfortunately, this has the unintended consequence that “fuzzier” or “intuitive” thinking (such as heuristic reasoning, judicious extrapolation from examples, or analogies with other contexts such as physics) gets deprecated as “non-rigorous”. All too often, one ends up discarding one’s initial intuition and is only able to process mathematics at a formal level, thus getting stalled at the second stage of one’s mathematical education.  (Among other things, this can impact one’s ability to read mathematical papers; an overly literal mindset can lead to “compilation errors” when one encounters even a single a typo or ambiguity in such a paper.)

The point of rigour is not to destroy all intuition; instead, it should be used to destroy bad intuition while clarifying and elevating good intuition. It is only with a combination of both rigorous formalism and good intuition that one can tackle complex mathematical problems; one needs the former to correctly deal with the fine details, and the latter to correctly deal with the big picture. Without one or the other, you will spend a lot of time blundering around in the dark (which can be instructive, but is highly inefficient). So once you are fully comfortable with rigorous mathematical thinking, you should revisit your intuitions on the subject and use your new thinking skills to test and refine these intuitions rather than discard them. One way to do this is to ask yourself dumb questions; another is to relearn your field.

The ideal state to reach is when every heuristic argument naturally suggests its rigorous counterpart, and vice versa. Then you will be able to tackle maths problems by using both halves of your brain at once – i.e. the same way you already tackle problems in “real life”.

It is perhaps worth noting that mathematicians at all three of the above stages of mathematical development can still make formal mistakes in their mathematical writing.  However, the nature of these mistakes tends to be rather different, depending on what stage one is at:

1. Mathematicians at the pre-rigorous stage of development often makes formal errors because one is unable to understand how the rigorous mathematical formalism actually works, and is instead applying formal rules or heuristics blindly.  It can often be quite difficult for such mathematicians to appreciate and correct these errors even when those errors are explicitly pointed out to them.
2. Mathematicians at the rigorous stage of development can still make formal errors because they have not yet perfected their formal understanding, or are unable to perform enough “sanity checks” against intuition or other rules of thumb to catch, say, a sign error, or a failure to correctly verify a crucial hypothesis in a tool.  However, such errors can usually be detected (and often repaired) once it is pointed out to them.
3. Mathematicians at the post-rigorous stage of development are not infallible, and are also capable of making formal errors in their writing.  But this is often because they no longer need the formalism in order to perform high-level mathematical reasoning, and are actually proceeding largely through intuition, which is then translated (possibly incorrectly) into formal mathematical language.

The distinction between the three types of errors can lead to the phenomenon (which can often be quite puzzling to readers at earlier stages of mathematical development) of a mathematical argument which locally contains a number of typos and other formal errors, but is globally quite sound, with the local errors propagating for a while before being cancelled out by other local errors.  (In contrast, when unchecked by a solid intuition, once an error is introduced in an argument by a pre-rigorous or rigorous mathematician, it is possible for the error to propagate out of control until one is left with complete nonsense at the end of the argument.)  See this post for some further discussion of such errors, and how to read papers to compensate for them.

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## Circular Euclidian Geometry: Overlooked Creativity

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1. There is still more space to fill. Assign all individuals currently residing in the hotel to even hotel room numbers. Assign all individuals who have arrived at the hotel to odd hotel room numbers.

2. There would be no cards left.

3. The wizard would see all his coins return to him, and the mermaid would have no coins. The wizard and the mermaid would have equal amounts of coins. The wizard would lose all his coins, and the mermaid would have all the coins.

If you would like to see how I arrived at these conclusions, just ask.

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“Do not disturb my circles!” — Archimedes

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I was originally planning to discuss infinity in its entirety today, but in wake of Hurricane Sandy, I felt inspired to do a short activity on circular Euclidian geometry. (After all, hurricanes are roughly circular in shape.)

We are all familiar with A = πr2, the formula to find the area of a disk bounded by a circle of radius r. (Notice I said area of a disk and not area of a circle. If you ever hear your friends say “area of a circle,” correct them immediately. A circle is a set of points equidistant to a common point. You cannot find an area of a set of points equidistance to a common point!) Despite being a simple formula, most high school students still have no idea how this amazing conjecture was discovered. Today, we will “prove” the theorem to be correct, and hopefully, you will come to see and appreciate how much creativity can be compacted inside a simple formula like A = πr2. (You will need a piece of paper!)

Draw a circle on your paper. Cut that circle into four equal sectors (i.e. four equal slices of pizza). Remove two of these sectors from the circle and arrange them such that both sectors point upwards. Take the remaining two sectors from the circle and arrange them such that both sectors point downwards. Place these two sectors above the sectors pointing upwards. Naturally, you can see that the two pairs of sectors will fit nicely into each other to form a quasi-parallelogram.

Now, repeat this process by creating another circle and cutting it into eight equal sectors. Remove four of these sectors from the circle and arrange them such that the sectors point upwards. Take the remaining four sectors from the circle and arrange them such that the sectors point downwards. Place these four sectors above the sectors pointing upwards. Again, notice that these two groups of sectors will fit nicely into a quasi-parallelogram. The fit, however, is more refined this time than it was last time.

For clarity purposes, let us repeat the process one last time. Create another circle and cut it into sixteen equal sectors. Remove eight of these sectors from the circle and arrange them such that the sectors point upwards. Take the remaining eight sectors from the circle and arrange them such that sectors point downwards. Place these eight sectors above the sectors pointing upwards. As before, the pieces fit together, and we see the same quasi-parallelogram that we saw in the last iteration. This time, however, notice that the quasi-parallelogram appears more like a rectangle. This is great news because we know the area of a rectangle: A = bh, where b is the base of the rectangle and h is the height of the rectangle.

So now, imagine that we extract infinitely many sectors from a circle. Half of those sectors will point upwards. Half of those sectors will point downwards. If we fit these two groups together in the manner we have been doing, theoretically, we should get a perfect rectangle. Therefore, all we need to know now is the base and height of that rectangle.

The base of the rectangle is a bit hard to see at first, but take notice that it is comprised of tiny arclengths from the circle. These arclengths add together to give half the circumference (“perimeter”) of the circle with which you started! Because we know that the circumference of a circle is given by the equation c = 2πr, where r is the radius of the circle, the base of the rectangle must be half that value, or b = πr. (If you cannot see this, keep looking at your diagram! It will click!)

The height of the rectangle is easy to see because it is given by the lengths of the edges of the sectors from the circle with which you started. Notice that the edges of these sectors have a length equal to the radius of your initial circle. Therefore, h = r.

Substituting both of these values into the formula A = bh, we get A = (πr)(r), which simplifies to A = πr2.

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Mathematician’s Side Note

Note: I am assuming you are familiar with the following concepts.

Another way to find the area of a disk with radius R, called the “Onion Peeling” method, uses basic techniques from calculus. Consider a disk to be nothing more than a series of concentric rings. Each of these rings has a small area associate with it, and if you were to cut one of these rings and stretch it out horizontally, you would see that the ring would actually become a small rectangle.  The area of this rectangle could be given by the product of the rectangle’s height and base. The base of the rectangle would be equal to the circumference of the ring. The height of the rectangle would be given by an infinitely small change in the disk’s radius. In other words, the small area of one ring would be equal to the product of the ring’s circumference and the infinitely small change in the disk’s radius, symbolically (dA) = (2πr)(dr). We now wish to add the areas of all the rings from radius zero to radius R, the disk’s radius. Symbolically, we would write Int(2πr, r, 0, R). Evaluating this integral leaves the result A = πR2, which is what we sought to prove.

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## Logical Reasoning: To Infinity and Beyond

“Logic is not a body of doctrine, but a mirror-image of the  world. Logic is transcendental.” — Ludwig Wittgenstein

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Some scholars argue that mathematics is the fundamental language: That it is both universal and scientific. “One does not understand anything until he or she quantifies it,” these scholars often say.

For the most part, I agree with them, and I hope you do as well. We can qualitatively describe and categorize an object to any extent we want. Therefore, grouping objects together in this fashion is a fairly effective and quick method of organization. However, there is always a chance in our qualitative analysis that two objects have almost the same characteristics, or moreover, they do have the same characteristics. It is herein that our qualitative analysis seems to decompose. We need a system of organization more concrete and more reliable. We need something that never lies. We need quantities.

How is it that quantites can never lie? (How is it that mathematics, in general, can never lie for that matter?) It is in the answer to this question that I argue that mathematics is not the fundamental language. The answer to my question is simple–logical reasoning. Mathematics is constructed on a beautifully intricate and delicate balance of axioms, postulates, constructions, lemmes, theorems, and corollaries. The soul of mathematics is to take old knowledge or knowledge that we know can never be false and, using these balanced logical constructions, discover new knowledge. (Of course, there is one other component to the soul of mathematics–imagination.) This is why mathematics never lies: All knowledge is true because it is based on true knowledge. To find new true knowledge requires logic, the true fundamental language.

If you cannot tell already, logic will be our main focus today, but rather than introduce the concept in the orthodox fashion, which would require you to sit down and read dozens of ancient essays on conditional statements and biconditional statements and contrapositive statements and counter-example statements, I am going to let you explore logic on your own. To do this, I am going to call upon paradoxes which arise from the perfectly logical-illlogical concept of infinity. It will be up to you to answer them using your logical intuition and critical thinking as a guide. (These paradoxes have really neat answers, and I really encourage you to try them–NOT skip over them.)

1. Hilbert’s Hotel

Consider a hypothetical hotel with countably infinitely many rooms, all of which are occupied–that is to say every room contains a guest. One might be tempted to think that the hotel would not be able to accommodate any newly arriving guests, as would be the case with a finite number of rooms. Suppose a new guest arrives and wishes to be accommodated in the hotel. The hotel is full. A countably infinite number of guests arrive at the same time. How can the hotel be arranged to fit the incoming guests?

Cards with numbers 1, 2, 3, … are put in a box in a particular order. One minute before noon numbers from 1 till 10 are put and a number 1 is taken back. 1/2 minute before noon numbers from 11 till 20 are put and a number 2 is taken back. 1/3 minute before noon numbers from 21 till 30 are put and a number 3 is taken back. This pattern continues infinitely many times. How many numbers are in the box at noon?

3. The Wizard and the Mermaid

An infinitely rich wizard has a mermaid in the pond of his garden. He likes to play with her and gold coins. Every minute he throws two coins in the pond and she throws him back one coin after a half minute. When this game is allowed to run forever, how will the money be distributed? We are allowed put a label on all the coins, numbering them 1,2,3,4, etc. This paradox actually has more than one solution.

I will post the solutions to these logic problems next week, and then, we will actually discuss some interesting properties of infinity, which I believe I alluded to a few weeks ago. (We are getting closer to the grand finale I have in store.)

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## Fractal Geometry: Mathematics of the Future (Part II)

“Fractal geometry will make you see everything differently. There is a danger in reading further. You risk the loss of your childhood vision of clouds, forests, flowers, galaxies, leaves, feathers, rocks, mountains, torrents of water, carpet, bricks, and much else besides. Never again will your interpretation of these things be quite the same.” – Michael Barnsley

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As promised in my last blog, we will continue our discussion of dimensions in fractal geometry (i.e. fractional dimensions). Unlike last time, however, we will focus more so on the actual mathematics of fractals as opposed to the intuition behind them.  (If you need a refresher on the definition of a fractal, please go back and reread from last week; it will be to your advantage to do so.) Have paper and a pencil handy: You will definitely need them for this blog!

Last week I ended with the notion of fractional dimensions. It was a tad remiss of me, however, to claim something so profound yet never explain the true definition of a dimension. (Now, when I say “true definition of a dimension,” I mean no insults to intelligence. You are all very bright. I am only here for clarification purposes.) So, what is a dimension? Stop reading and think as broadly as you can about the term “dimension.”

Are you still having difficulty conjuring up a reasonable definition? (I know I did when I first thought about dimensions.) To start, let us simplify the question slightly. With what dimensions can we associate?  Well, over our years of growing mathematical observations, we have all come to associate one-dimension with a line, two-dimensions with a plane, and three-dimensions with a space. To put more rigorously, a line has only one independent path of travel (east-west), a plane has two independent paths of travel (east-west and north-south), and a space has three independent paths of travel (east-west, north-south, and up-down). Therefore, by inductive reasoning, we assume that dimensions can indicate the number of independent paths able to be taken in some region.

This definition makes a lot of sense, and it certainly seems to unify our understanding of dimensions in the first, second, and third degrees. However, if fractals have fractional dimension (as I mentioned last week), how is it possible to have a region with a fractional number of independent paths able to be taken? It simply is not possible! Therefore, our definition of dimensions breaks down. To replace this broken definition requires a new, more inventive interpretation of dimensions. (However, please keep the idea of independent paths of travel in your head. It is very important for the fundamental definition of a dimension.)

Consider a line segment. If we double the length of a line segment, we get two copies of the original line segment. If we triple the length of the line segment, we get three copies of the original line segment. If we quadruple the length of the line segment, we get four copies of the original line segment. Get the picture? For any multiple n of a line segment, we get n copies of the original line segment.

Consider a square. If we double both sides of a square, we get a larger square which can fit four of the original squares inside of it. Similarly, if we triple both sides of a square, we get a larger square which can fit nine of the original squares inside of it. If we quadruple both sides of a square, we get a larger square which can fit sixteen of the original squares inside of it. Therefore, in general, for any multiple n applied to both sides of a square, n2 number of copies of the original square can be produced in the resulting square.

Consider a cube. If we double all sides of a cube, we get a larger cube which can fit eight original cubes inside of it. (If you are having trouble visualizing the reasons behind these observations, consider the unit cube below.)

Likewise, if we triple all sides of a cube, we get a larger cube which can fit twenty-seven of the original cubes inside of it. If we quadruple all sides of a cube, we get a larger cube which can fit sixty-four original cubes inside it. Therefore, in general, for any multiple n applied to all three sides of a cube, n3 copies of the original cube will be produced in the resulting cube.

Let us now organize all this information in a table so that we may be able to understand dimensions more explicitly.

 Figure Dimension Linear Multiple Number of Copies Line 1 2 (2)(1) = 2 Square 2 2 (2)(2) = 4 Cube 3 2 (2)(3) = 8 General Case D L LD = S

Note: The term linear multiple is used by mathematicians to describe the n values that I have described in the thought experiments above. Linear multiples tell us how large we wish to scale each independent path of travel in the figure. For instance, if we wanted to double the distance along a line, the linear multiple would be equal to two because we wish to double the east-west direction of travel on the line. If we wanted to double the distances along a square, the linear multiple would still be two because we wish to double the east-west direction of travel on the square as well as the north-south direction of travel on the square. (If you double the distances along a cube, you would have a linear multiple of two as well: Each independent path of travel—east-west, north-south, and up-down—will be scaled by two.)

Clarifications aside, notice the pattern presented in the table. If we wish to find the number of copies of some generic figure, we simply take the linear multiple we want (how large we want to scale each independent path of travel) and raise that number to the dimension of the figure. Herein, we establish a rigorous definition of dimension: It is a number which quantifies the effect of changing sizes of linear independent paths of travel on the number of original figure copies produced in the resulting figure. (That definition is quite a mouthful to explain, and if you do not quite understand, I suggest rereading the last three paragraphs or so to let the definition sink into your mind.)

Let us use this definition now to calculate the dimension of the second-most famous fractal—the Sierpinski’s Triangle. Get your paper and pencil!

Creating this fractal is actually fairly easy. (I find it easier than creating the von Koch snowflake.) Begin with an equilateral triangle (a triangle of equally measured sides). Emphasize the midpoints (i.e. halfway points) of the three line segments creating this equilateral triangle with modest dots. Connect these dots to form another equilateral triangle. Shade in this triangle.  You should now have a figure with three equilateral triangles (pointed right side up) and a fourth darkened equilateral triangle (pointed upside down). For each of the three equilateral triangles that are not shaded, emphasize the midpoints of the three segments which create the triangle. Draw an equilateral triangle which connects these three points; shade in this equilateral triangle. (Remember to do this process for each of the three equilateral triangles that were created in the first iteration.) You should now have a figure with one large shaded triangle (pointed upside down), three small shaded triangles (pointed upside down), and nine small triangles that are not shaded (pointed right side up). If you are able to see the pattern, I encourage you to continue. If not, this picture you have created is sufficient enough to continue.

For clarification purposes, I have drawn above the next few iterations of Sierpinski’s Triangle.

Now, let us consider a linear multiple of two. (In other words, let us multiply the independent paths of travel in the triangle by two; notice in this figure that there are two independent paths of travel—east-west and north-south.) The result of using a linear multiple of two on Sierpinski’s Triangle is depicted below.

After we multiplied both independent paths of travel by two, how many copies of Sierpinksi’s Triangle did we obtain? Well, it appears we got three original Sierpinski’s Triangles. Recall that we can use this information to establish the dimension of Sierpinski’s Triangle. In other words, recall S = LD. In this case, (3) = (2)D. Notice that if we choose D=1, we undervalue the number of copies of the original Sierpinski’s Triangle we obtained by using a linear multiple of two. Similarly, if we choose D=2, we overvalue the number of copies of the original Sierpinski’s Triangle we obtained by using a linear multiple of two. Therefore, by logical deduction, the dimension of Sierpinski’s Triangle must lie between one and two. It must be fractional or irrational!

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Mathematician’s Side Note

For those of you who are dying to find the exact value for the dimension of Sierpinski’s Triangle, I will explain briefly, but please note that I am assuming you are familiar with certain mathematical operations. As we discovered, S = LD. We would really like to express this equation as D in terms of S and L. In order to isolate D, I will take the natural logarithm of both sides of the equation: ln(S) = ln(LD).  After doing so, I will apply the power property of logarithms to rewrite the equation as ln(S) = Dln(L). Lastly, I will divide both sides of the equation by ln(L): ln(S)/ln(L) = D. This equation now expresses dimension as a function of a linear multiple and the number of produced original figures as a result of the chosen linear multiple. In the case of Sierpinski’s Triangle, we have ln(3)/ln(2) = D or, using a calculator to evaluate the logarithms, D ≈1.5850.

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I also promised in this blog that I would show you the most famous fractal of all time. I will not go into the mathematics behind it, which requires an extensive knowledge of complex numbers and complex analysis. Instead, I leave it below for you to ponder. The name of the fractal is the Mandelbrot Set. Based on the information I have provided to you over the last two blogs, you should be able to know how the fractal received that name.

I also wanted to shed light on a question asked by Lewis last week. Fractals are everywhere in nature, Lewis. You just have to go looking for them. This would be a great place to start.

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## Fractal Geometry: Mathematics of the Future (Part I)

“Why is geometry often described as ‘cold’ and ‘dry?’ One reason lies in its inability to describe the shape of a cloud, a mountain, a coastline, or a tree. Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line. Nature exhibits not simply a higher degree but an altogether different level of complexity.” – Benoit Mandelbrot

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If you thought, even for a second, that any of the mathematical discoveries I have presented to you on this website have been absolutely enlightening, then you have been disillusioned; those pieces of information have been known for centuries. Today, we as mathematicians will break with the obvious, with the pieces of information that we have known to be true for a long time. Rather than concern ourselves with numbers, special constants, and sets, all of which are archaic, we will observe mathematics of the twenty-first century—our generation’s contributions to mathematics.  Today, we will study, more likely peruse, fractal geometry, an idea which was established a mere thirty-five years ago by Benoit Mandelbrot. Have a paper and pencil handy: You will need it as we go through the very basics of this fascinating topic!

To get the ball rolling, I want to start with an activity. (It will require paper and a pencil.) Draw a vertical line segment on the right side of your paper. Erase the middle third of this segment. Replace this missing third with a “carrot” pointing to the right (i.e. draw two line segments to join the top third line segment and bottom third line segment in such a way that these two line segments you are drawing also make an arrow pointed to the right).  You should now have a figure composed of four line segments. Erase the middle third in each of these line segments. Replace those missing thirds with a smaller carrot. You should now have a figure composed of sixteen line segments. Erase the middle third in each of these line segments. Replace those missing thirds with an even smaller carrot. You should now have a figure composed of sixty-four line segments.

Rather than continue to sketch the pattern, which is bound to get very messy, let us stop and think critically. What does this pattern resemble? Well, it should become increasingly obvious that this pattern looks like the edges of a snowflake (and just in time for that season to begin).  As it turns out, the figure you have drawn is known as the von Koch snowflake. It is, perhaps, the third most famous fractal in all of mathematics. (Do not worry: I will show you the second and first most famous fractals, respectively, in the next blog.) For further purposes of discussion (and doubts in my ability to instruct effectively), I have posted the von Koch fractal below.

So, we have generated the fractal above quite simply, but how do we define fractals in general? As it turns out, the true mathematical definition is way beyond our comprehension (and unnecessary for this blog’s purposes).  Figuratively, however, we can postulate a definition based on just this one example. Notice: No matter how closely we try to observe the snowflake, we always see the same figure. For instance, on the leftmost picture of the snowflake, if I chose to look at one of those tiny segments that comprise the entire figure under a microscope, the return in my microscope would be the exact same figure displayed by the leftmost snowflake at normal magnification. To put simply, we can never reach the fundamental components of a figure that is a fractal; we can only observe smaller and smaller “worlds” (patterns in the figure that continue for infinity). For our purposes, therein lies our definition of a fractal. It is a geometrical object comprised of infinitely many “worlds.”

Now, I want to shed light on the significance of our definition for a fractal. In standard mathematics, we focus primarily on principles of Euclidian geometry, a study which each one of us has experienced in some form at the grade-school level. For example, we have studied that two lines parallel to each other will never intersect or that three points in space will establish a plane. A direct corollary to this study comes from a completely different study in standard mathematics—the calculus. (I do realize at this point that some readers have not studied calculus, and so I will try to be as conceptual as possible.)

A major component of calculus deals with change. In effect, how does something change over an infinitely small interval? To somebody who has not studied calculus, this question seems to have an impossible answer, but right now, I will show you that it is not so impossible. Draw any generic curve on your paper. Cut that curve into smaller and smaller intervals. Notice: As you make smaller and smaller intervals, your curve becomes a collection of small lines! Therefore, over any given small interval, a curve can be represented by a line in that small interval. The change in the curve over that small interval is directly related to the steepness of the line that represents the curve in that small interval. This same process can be done for a surface, and the conclusion is that a surface becomes a collection of small planes. Now, how does this idea of small intervals of change relate to Euclidian geometry and fractals?

Imagine any object you want in three-dimensional space. Note the surface of that object over small intervals of area. Over these small intervals, the surface can be represented by a plane, correct? More generally, any surface of a three-dimensional object can be thought of as a collection of small planes, true? Because planes are defined by Euclidian geometry, we conclude that three-dimensional objects are Euclidian when their surface can be comprised everywhere of small planes. If we use the same logic, we can say something similar for two-dimensional objects: Two-dimensional objects are Euclidian if their curves can be comprised everywhere of small lines.

Doing all this explanation just to come up with these definitions probably seems tedious, but there is a method to my madness.

Consider the fractal you drew earlier. Is that figure Euclidian? Well, so far, we only know how to prove shapes to be Euclidian if they are of two or three dimensions. In this case, I will logically assume the von Koch snowflake to be a two-dimensional figure because it exists in the plane of my paper. With this assumption, what do all two-dimensional Euclidian figures have in common? Their curves can be broken into very small intervals, and when they are divided into these intervals, collections of small lines represent the curves. Here, we run into a problem. Recall our definition of a fractal: It is a figure where infinitely many worlds exist. The smaller we examine the figure, the more worlds we see (because the pattern imbedded within the figure reiterates itself to infinity). Therefore, the von Koch snowflake is not comprised of lines; rather, it is composed of worlds within worlds within worlds within worlds. Therefore, we come to the conclusion that it is not a Euclidian figure.

This is a very odd result. If you draw any other curve in the plane of your paper, it will surely be Euclidian. What makes this fractal so different? How can it seem to be a two-dimensional curve yet not be Euclidian? It defies standard mathematics!

Well, it turns out that the von Koch snowflake, and any other fractal for that matter, is not one-dimensional, two-dimensional, three-dimensional, or even four-dimensional. Fractals have fractional dimension (hence the name fractal). We will discuss fractional dimension next week and observe the strange life beyond Euclidian objects.

Welcome to the mathematics of the future, our generation’s contribution to mathematics. Be prepared for the freshest take on mathematics you have experienced since counting!

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## The Mathematician: Georg Cantor

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