CHAPTER 10
A New Spin on Euler’s Formula
People often invest new meanings in great works of art over time, breathing new life into them. The same thing happens in mathematics. A few years after Euler died, an obscure Norwegian surveyor with a gift for math conceived an innovative way of thinking about imaginary numbers that threw new light on Euler’s formula. It’s called the geometric interpretation of complex numbers.
The Norwegian innovator was Caspar Wessel, an amateur mathematician who always had trouble making ends meet with his wretchedly low-paid surveying work. An acquaintance described him in a letter as having “a bright, but very slow head, and when he sets out to study something, he can have no peace before he completely understands it.” Wessel also struck those who knew him as distinctly meek: “If he had been in possession of more courage and assurance when it comes to trying unaccustomed work, then with his insight and talent, he could have done a lot for the benefit of the community as well as for himself,” a fellow surveyor once wrote of him.
Wessel might never have published his important new ideas if mathematician Johannes Nikolaus Tetens, a prominent member of the Royal Danish Academy of Sciences, hadn’t encouraged him. In 1797, Tetens, acting on behalf of his shy protégé, read a treatise by Wessel at a meeting of the academy. It was the only math paper Wessel wrote, and it laid out the geometric interpretation.
Unfortunately, Tetens failed to follow up by drawing wider attention to Wessel’s paper, which was modestly, and somewhat mystifyingly, titled (when translated into English) “On the Analytical Representation of Direction.” As a result it languished in obscurity for nearly a century despite its publication in the Danish academy’s journal in 1799. Finally, in 1895, translations of it became widely available, showing that Wessel, who died in 1818, deserved credit for first developing the geometric interpretation.
A few years after Wessel’s big idea appeared in the Danish journal, another amateur mathematician, Jean-Robert Argand, a Paris accountant, independently came up with the same advance. In a remarkable coincidence, his brainchild was also nearly lost to history. For some reason Argand decided to tell the world about it in an anonymous essay he self-published in 1806. It apparently made little impression on the few who saw it when it first came out. Luckily, seven years later French mathematician Jacques Français became aware of the work and was struck by its originality. Using a bit of detective work, he managed to get in touch with its author and alerted the world to his innovation—Argand’s name has been prominently associated with the geometric interpretation ever since.
FIGURE 10.1
In the geometric interpretation, the imaginaries are assigned to their own number line, called the i axis, which is drawn vertically, along with a real number line, called the x axis, which is drawn horizontally intersecting the i axis. As shown in Figure 10.1, the two intersecting number lines inhabit a flat space that looks very much like the xy plane, except that the y axis is replaced by the i axis.
The diagram shows why it makes sense to say the imaginaries hail from a different dimension: the i axis forms one of the two dimensions of a 2-D plane; the plane’s other dimension is formed by the familiar real-number line, which is called the x axis in this context. This 2-D space is known as the complex plane, and the points on it are associated with two-part real-plus-imaginary numbers called complex numbers. Just as grade-school math’s number line suggests how to think of the real numbers as points in 1-D space (lines have one dimension), the complex plane enables complex numbers to be mapped onto points in 2-D space.
Recall that points in the xy plane are designated with pairs of numbers called coordinates. Two numbers also are used to specify each point on the complex plane—a real one marched off horizontally along the x axis, and an imaginary one marched off vertically along the i axis. They’re usually written, as shown in the diagram, in the form of sums, such as 2 + 3i or 1 + −3i (the latter complex number is the same as 1 − 3i). Gauss, the German mathematician, is credited with coming up with this standard a + bi format for writing complex numbers in the early nineteenth century, although some historians trace it back to Cardano, the sixteenth-century Italian mathematician who regarded doing arithmetic with imaginary numbers as mental torture.
Note that any real number can be viewed as a complex number whose imaginary part is 0 times i. The real number 2, for example, is essentially just a stripped-down version of 2 + 0i. Since pure real complex numbers such as 2, or 2 + 0i, have zero imaginary components, their points all lie on the real-number x axis in the complex plane. This makes sense because complex numbers’ imaginary components specify how far away from the x axis their points are in the plane. That distance is zero when the imaginary parts are 0i, and so the points for pure real complex numbers are zero distance away from the x axis.
Similarly, imaginary numbers such as 2i can be viewed as complex numbers with real-number parts equal to 0. The points for all such pure imaginary complex numbers lie along the i axis in the complex plane.
COMPLEX NUMBERS REPRESENT a major upgrade of the number concept. A complex number such as 2 + 3i incorporates two quantities, and thus it effectively holds twice as much information as a real number such as 2, or an imaginary number such as 3i. That means complex numbers can do things that one-part numbers can’t. You’ve already seen one example of this: while a number like 2 can be used to locate a point along a line, a complex number like 2 + 3i can be used to locate a point in 2-D space. Thus, the upgrade expands the concept of numbers from the realm of yardsticks to the richer world of maps. (Which, not coincidentally, is the world in which surveyor Caspar Wessel spent his career.) A real number could tell us how far we’ve come down a road, while a complex number could tell us exactly where we are.
The two-ness of complex numbers also enables them to simultaneously represent moving objects’ speed and direction of motion. To see how this works, picture a field several acres in size. Now envision an archer at the field’s center, and imagine that she shoots an arrow in a northeasterly direction at a certain speed—say 200 feet per second. To model the arrow’s (bow-release) speed and direction with a single complex number, you could think of her as standing at the origin of the complex plane shown in Figure 10.2, with the i axis running north and south, and the x axis running east and west. By drawing an arrow on the plane extending from the origin in the direction of the point 1 + i, you could represent the real arrow’s direction, northeast. And if you set the arrow’s length to 200 units, you could represent the real arrow’s speed by means of that length.
FIGURE 10.2
All the information contained in the arrow that you’ve drawn (the speed and direction of the real-world arrow) is encapsulated by the complex number for the point at its tip, which is 100√2 + 100√2i. (100√2 means 100 times the square root of 2.) In fact, if you were given that complex number and asked to figure out the speed and direction of the arrow, you could use a ruler to plot the point for the number based on its real and imaginary parts. (The parts specify marching off 100√2 units along both the x and i axes—that’s about 100 × 1.414, or 141.4 units.) Then you could approximate the real arrow’s initial speed by using the ruler to measure the distance between the origin and the point. Alternatively, you could use the venerable Pythagorean theorem to calculate the distance between the origin and the point.
The theorem* states that the sum of the squared lengths of the two shorter sides of any right triangle is equal to the square of the hypotenuse’s length. (You might remember this as something like x2 + y2 = z2, where x, y, and z stand for the lengths of right triangles’ sides.) The right triangle of interest here has two sides of length 100√2. Its hypotenuse is the arrow, and you can picture one of its sides lying along the x axis. By the Pythagorean theorem, (100√2)2 + (100√2)2 equals the square of the hypotenuse. Writing that as an equation and rearranging terms, we have (100 × 100 × √2 × √2) + (100 × 100 × √2 × √2) = H2, where H is the length of the hypotenuse. That means, using the fact that √2 × √2 = 2 and further rearranging terms, that H2 equals (2 × 1002) + (2 × 1002) = 2 × (2 × 1002) = (2 × 100)2, which is 2002. Thus, H2 = 2002, which implies that H = 200.
One conclusion you might draw from all this is that such diagrammed arrows can be thought of as representing the same information as the complex numbers associated with the points at their tips. Thus, it’s reasonable to regard the arrows as visual representations of the associated complex numbers. Such arrow-like complex number representations are called vectors. They’re a key component of the geometric interpretation that Wessel pioneered, and, as we’ll see, they can be used to extract implications of Euler’s formula that even Euler himself never recognized, at least not explicitly.
Notice that when you envision a complex number as a vector, you’re translating a two-part number concept into 2-D geometric terms, leading you to picture the abstract number as a visible thing. (Especially if you’re an archery fan.) Such visualizations can make it wonderfully easy to carry out basic mathematical operations with complex numbers, effectively translating abstract concepts into concrete images that our brains are naturally equipped to deal with.
In the mid-1800s, Irish mathematician William Rowan Hamilton expanded the concept of numbers even further by introducing four-dimensional numbers called quaternions and working out how to do calculations with them. Such numbers are used today in everything from computer graphics to aircraft navigation systems. Physicists have a thing for many-dimensional numbers too. Einstein pictured the universe as having four dimensions—three spatial ones plus a fourth dimension for time. And infinite-dimensional spaces are used in quantum physics to model properties of elementary particles. (Don’t worry if that doesn’t mean anything to you—even the White Queen would need a dozen or more pre-breakfast sessions to get her mind around it.)
Thinking of 4-D numbers as vectors would come in handy if you were asked by a Zen master to compose a zero-word autobiography. You could nod sagely and offer a concise depiction of your life to date as a vector in 4-D space. Based on a somewhat arbitrary, earth-centric frame of reference, the vector would extend from a point representing the time and location of your birth (the time might be expressed as estimated number of seconds that had elapsed since Buddha’s birth, and the location might be represented by numbers for latitude, longitude, and altitude) to another point similarly representing the current time and your present location. You wouldn’t have to figure out how to draw the 4-D vector. Instead, you could effectively delineate it by simply designating its two endpoints in the form (a,b,c,d), where the letters represent the salient numbers for location and time. Of course, this version of your life story would omit all its zigzags and have a very thin narrative element. But the master would like its extreme simplicity. He (or she) might even let you hear the sound of one hand during a standing ovation.
AS SUGGESTED ABOVE, picturing complex numbers as vectors sets the stage for representing addition, multiplication, and other operations with complex numbers in a geometric way. Wessel was the first to work out how to envision such calculations geometrically.
Let’s take a look at the geometric interpretation of complex-number addition. Before seeing how such addition is imagined with vectors, however, you should know the rule for doing it arithmetically—that is, how complex numbers are added without reference to vectors. Rafael Bombelli, the Italian mathematician we met in the chapter on imaginary numbers, devised this rule in the 1500s, more than two centuries before the geometric interpretation was introduced. It’s easy: you sum two complex numbers by simply adding their real and imaginary parts separately. For example, here’s how the complex numbers 3 + 1i (which can also be written 3 + i) and −1 + 2i are added:
(3 + 1i) + (−1 + 2i) = (3 + (−1)) + (1i + 2i) = 2 + 3i.
The geometric rule for addition of complex numbers entails constructing a parallelogram using the vectors for the numbers as two of its nonparallel sides. A parallelogram is a four-sided figure whose opposite sides are parallel, as shown in Figure 10.3.
FIGURE 10.3
Drawing a parallelogram to add two vectors effectively defines a third vector that extends diagonally through the parallelogram. This vector represents the sum.
To see what this verbal description means, take a look at Figure 10.4 (next page), which shows the geometric addition of the two complex numbers that were added arithmetically above: 3 + 1i and −1 + 2i. Note that the summation vector drawn diagonally through the parallelogram—the arrow pointing at 2 + 3i—represents the complex number that’s produced by adding the numbers arithmetically.
FIGURE 10.4
I won’t go into all the rules of vector math here; this isn’t a textbook. But please take note of an important feature they have in common: they yield results that invariably agree with calculations using complex-number arithmetic, as was just shown with the vector parallelogram for adding 3 + 1i and −1 + 2i. This consistency between the geometric and non-geometric calculations is critical. If vector-based math yielded something different, the geometric interpretation would be little more than a minor curiosity—like an English translation of War and Peace whose sentences were weirdly mangled versions of the original Russian ones. Maintaining strict consistency is crucial in mathematics to prevent its intricate, interconnected towers of logic from collapsing into heaps of contradictions.
Vector addition is intuitively inviting. In particular, it suggests visualizing the addition of complex numbers as the trajectory of an object as it’s simultaneously pushed by two forces. (The forces in such visualizations are pictured as the vectors being added together, and the trajectory is the summation vector.)
But the translation of multiplication into vector math is the geometric interpretation’s cleverest, most fruitful stroke. In order to keep things simple, I’ll focus on a key example of such multiplication: the vector version of multiplying the complex number 0 + i times other complex numbers. And I’ll lead into that topic with a related example involving only real numbers: the translation of multiplication by −1 into geometric terms.
Since we’re accustomed to thinking of real numbers as points on a number line, it’s not hard to picture them as vector-like arrows along the number line. Thus, as shown in Figure 10.5, the number 4 can be represented by a four-unit-long arrow extending to the right from 0. (This arrow looks a lot like the vectors defined above, but it’s actually a different thing. Vectors in the complex plane are 2-D things, because the plane is a two-dimensional space. The arrow representing 4 is a 1-D thing, because the number line is a one-dimensional space.)
FIGURE 10.5
We know from basic arithmetic that −1 × 4 = −4. This idea can be neatly captured in geometric terms by the following operation: when you multiply −1 times 4, you rotate the arrow representing 4 by 180 degrees (with the point for 0 acting as the axis of rotation) to reverse its direction, turning it into the arrow for −4.
Multiplying −1 times the arrow for any real number can be thought of as producing the same kind of 180-degree rotation.* For instance, multiplying −1 times −5 can be pictured as rotating the arrow for −5, which points to the left from 0, by 180 degrees so that it winds up pointing to the right and thus represents 5. (This rotation operation for “times −1,” by the way, yields results that always agree with the enemy-of-my-enemy rule for multiplying a negative number times another negative number.)
Now let’s extrapolate this rotation idea from the 1-D number line to the 2-D complex plane. To do that, we’ll assume that multiplying a complex number times −1 + 0i, or −1, is geometrically interpreted as causing the vector for the number to rotate counterclockwise by 180 degrees (with the origin acting as the axis of rotation). If this “times −1” rotation rule works the way it’s supposed to, it should yield results that are consistent with those obtained by multiplying complex numbers without the use of vectors.
Here’s an example we can use to check for the desired consistency: multiplying −1 times i. We know from the earlier chapter on imaginary numbers that −1 × i can be more compactly written as −i. Expressed as an equation, that gives us −1 × i = −i, and if we replace the numbers in the equation with the complex-number counterparts, we obtain (−1 + 0i) × (0 + i) = 0 − i.
Now for the vector version, shown in Figure 10.6. As you can see, it involves rotating the vector representing 0 + i counterclockwise by 180 degrees in accordance with our rotation rule for the multiplication operation “times −1” (which, recall, is synonymous with the complex-number multiplication operation “times −1 + 0i.”) This rotation leaves us with the vector representing 0 − i. Thus, the vector and arithmetical versions of this multiplication are indeed consistent.
FIGURE 10.6
This one example doesn’t prove much, but it does suggest that we’re on the right track when we interpret complex-number multiplication as vector rotation. And that, in turn, suggests that it may also make sense to geometrically interpret “times i” in terms of vector rotation. Let’s consider, for example, how to interpret i2 = −1 along these lines. Expressed as i × i = −1, this true-by-definition equation can be written in complex-number terms as (0 + i) × (0 + i) = −1 + 0i. (Keep in mind that “(0 + i) ×” is synonymous with “times i,” and that the vector we’re multiplying times i in this equation is also 0 + i, or i.) The equation indicates that when “times i” is applied to the vector for 0 + i, we should wind up with the vector for −1 + 0i. And that implies that “times i” should be geometrically interpreted as inducing a 90-degree counterclockwise rotation. The previous diagram can help you confirm that is true—if you picture rotating the vector for 0 + i counterclockwise by 90 degrees, you can see that it becomes the vector for −1 + 0i.
But if we assume that “times i” is always associated with a 90-degree vector rotation, will we invariably get vector-based results that are consistent with non-geometric calculations? I don’t want to get sidetracked here developing a formal proof that this is the case, or showing lots of examples suggesting that it’s true. (Trust me, it is.) But I can’t resist mentioning one other case in which the 90-degree rotation rule works like a charm: evaluating i3, or i cubed. When translated into vector rotations, this calculation represents a simple mathematical model of the old Byrds’ song, “Turn! Turn! Turn!” (The song, which is based on a Bible passage, was composed by Pete Seeger. But I best remember the Byrds’ version.)
To bring out the turn-turn-turn encapsulated by i3, it helps to rewrite it as i × i × i × 1. This product calls for rotating the vector for 1 (that is, 1 + 0i) by 90 degrees counterclockwise three times in a row. That leaves the vector pointing in the six o’clock position, which represents the complex number 0 − i, or −i. This checks out with what we know arithmetically, for i3 = i × i × i = i2 × i = −1 × i = −i. The triple turn is portrayed in Figure 10.7.
To reiterate, the multiplication operation “times i” is envisioned as a 90-degree counterclockwise vector rotation in the complex plane. Although the line of reasoning I’ve offered to support this idea differs from Wessel’s thinking as he pioneered the geometric interpretation, it leads to the same 90-degree rotation rule for “times i” that follows from his work. The association of multiplication with vector rotation was one of the geometric interpretation’s most important elements because it decisively connected the imaginaries with rotary motion. As we’ll see, that was a big deal.
FIGURE 10.7
AT THIS POINT WE’VE NEARLY covered enough of the geometric interpretation to translate Euler’s formula (eiπ + 1 = 0, or eiπ = −1) into vector math and thus look at it in a revealing new way. That is, we know how to represent the constants 1, −1, and 0 as vectors: 1 is represented by the vector for 1 + 0i, −1 by the vector for −1 + 0i, and 0 by an extremely short vector consisting of a single point in the complex plane associated with complex number 0 + 0i, the origin. We’ve also covered how to interpret the “+” in Euler’s formula in terms of vector addition—recall the parallelogram rule.
But to finish up interpreting the formula in geometric terms, we must confront a more complicated challenge: devising a vector representation of raising a real number to an imaginary-number power. Specifically, we must find a way to represent eiπ in vector-ese.
Fortunately, we already have a strong clue about what raising e to an imaginary-number power should do in vector math: It should yield results that are consistent with those obtained by raising e to imaginary-number powers in the equation eiθ = cos θ + i sin θ. (We know this equation is true based on Euler’s three non-geometric derivations.) In other words, the vector for eiθ should be interpreted in such a way that it always turns out to be the vector for cos θ + i sin θ when an angle—any angle—expressed in radians is plugged in for θ in these expressions. If that weren’t the case, the consistency between geometric and non-geometric calculations that has worked so nicely for us as a guiding principle would be fatally compromised.
It follows that if we can figure out how to translate cos θ + i sin θ into the visual language of vectors, we’ll also know how eiθ should be translated. Fortunately, we’ve already gone over most of the concepts needed to solve this problem.
In Figure 10.8 (next page), I’ve reformulated an illustration from the trigonometry chapter to help you see how to apply what you already know about trig to the problem at hand. It depicts adding the vector representing cos θ + 0i, which is a pure real complex number (since cos θ is a real number), to the vector for i times sin θ, which is a pure imaginary complex number,* using the parallelogram rule for vector addition. (In this case, the parallelogram’s sides meet at right angles, making it a rectangle.)
The summation vector represents the complex number cos θ + i sin θ. This vector sum, by the way, is consistent with the sum obtained by adding the complex numbers arithmetically:
(cos θ + 0i) + (0 + i sin θ) = (cos θ + 0) + (0i + i sin θ)
= cos θ + i sin θ.
FIGURE 10.8
As you can see in the diagram, the vector for cos θ + i sin θ closely resembles the angle sweeper envisioned in the trig chapter. Remember that the coordinate pair of the point at the sweeper’s tip is (cos θ, sin θ). As you may have already realized, cos θ + i sin θ is the complex-plane version of this coordinate pair. The definitions of the unit-circle-based trig functions guarantee that the point associated with the complex number cos θ + i sin θ will lie on the unit circle in the complex plane, just as (cos θ, sin θ) always does on the unit circle in the xy plane. Further, it will be located at a horizontal distance of cos θ from the i axis, and at a vertical distance of sin θ from the x axis.
Borrowing another idea from the trig chapter, we can think of cos θ + i sin θ as a sort of device for controlling a radius-like angle-sweeping vector, as suggested in the previous diagram. And in keeping with the angle-sweeper action pictured in the trig chapter, when a number (expressed in radians) is plugged in for θ in cos θ + i sin θ, the angle-sweeping vector can be envisioned as rotating counterclockwise from the three o’clock position by that number of radians.
Don’t look now (actually, do look—at Figure 10.9), but we’ve just arrived at a plausible geometric interpretation of eiθ. To wit: eiθ can be geometrically interpreted as an angle-sweeping vector that rotates within the unit circle in the complex plane to sweep out the angle θ.
The geometric interpretation of eiθ represents a wonderfully concise, convenient way to represent the sweeping out of angles. Notice how the trig-function clutter has been dispensed with in the diagram because it’s no longer needed, conceptually speaking.
FIGURE 10.9
Now let’s imagine what happens when we plug in π/2 for the θ in eiθ. Since π/2 radians is the same as 90 degrees, the vector version of eiπ/2 can be interpreted as the angle-sweeping vector after it has carried out a 90-degree counterclockwise rotation, from the three o’clock to the high-noon positions. That means it winds up representing the complex number 0 + i. To check that this vector operation is correct, we can plug π/2 into cos θ + i sin θ (which, remember, is equivalent to eiθ) and treat it as a complex number. Since cos π/2 = 0 and sin π/2 = 1, we have cos π/2 + i sin π/2 = 0 + (i × 1), or 0 + i. That, of course, is what we wanted to see.
Similarly, when π radians is plugged into eiθ, the angle-sweeping vector representing it rotates counterclockwise by 180 degrees (π radians) to wind up in the nine o’clock position. Note that after sweeping out π radians, the eiθ-based angle sweeper winds up representing −1 + 0i. If we translate this statement into one about complex numbers, we have eiπ = −1 + 0i, or, more simply, eiπ = −1. In other words, we’ve just arrived at the most beautiful equation in a geometric way.
What about the geometric interpretation of Euler’s formula expressed in the usual way, eiπ + 1 = 0?
This equation’s left side can be interpreted geometrically as adding the vector for eiπ, which we established above is identical to the vector for −1 + 0i, to the vector for 1 + 0i. This vector sum, in turn, can be imagined as the trajectory of an object, initially positioned at the origin, as it’s simultaneously pushed by two forces. (As noted earlier, this force-pushing analogy for vector addition is suggested by the parallelogram rule for summing vectors.) Since the vectors being summed in this case resemble equal-length arrows pointing in opposite directions, the hypothetical dual forces pushing on the object exactly cancel each other out, resulting in a trajectory consisting of a non-moving point, the origin. And the origin, of course, is 0 + 0i—the complex-number version of the pure real number 0, which is the equation’s right side.
So here’s the main message to take home from this chapter: Raising e to an imaginary-number power can be pictured as a rotation operation in the complex plane. Applying this interpretation to e raised to the “i times π” power means that Euler’s formula can be pictured in geometric terms as modeling a half-circle rotation.
BEFORE MOVING ON to the implications of the take-home message, which is covered in the next and last chapter, let’s briefly consider how the geometric interpretation enables mental shortcuts.
As we just saw, plugging in π/2 radians for θ in eiθ can be pictured as making a three-o’clock-pointing vector within the unit circle rotate counterclockwise by 90 degrees, and thus eiπ/2 = 0 + i, or, more concisely, eiπ/2 = i. Notice that the geometric interpretation immediately handed us this equation without our having to fiddle with trig functions or carry out other math machinations. This conceptual efficiency is due to the fact that we’re now thinking about 2-D things (complex numbers) that have been reformulated as vectors that we can simply rotate in our minds to perform calculations. This mental efficiency has helped make eiθ very useful in engineering and science.
Now for a last variation on the theme. Let’s plug in 2π radians for θ in eiθ, causing the sweeper to rotate by 360 degrees and wind up back where it started, in the three o’clock position pointing at 1 + 0i. That gives us
ei2π = 1 + 0i
which, by subtracting 1 from both sides and using 0 for 0 + 0i, becomes
ei2π − 1 = 0.
This equation has been known as long as Euler’s formula, and although it doesn’t have the cachet of eiπ + 1 = 0, I regard it as suitable for framing. It features all five of the very important numbers seen in the famed formula as well as another VIN: the number 2, “a couple,” the fundamental number for romance. I like to call it Alicia’s formula, after my wife, a former mathphobe now in recovery. (She kindly served as a test reader of this book as it was written.)
THE GEOMETRIC INTERPRETATION of eiπ is rich with emblematic potential. You could see its suggestion of a 180-degree spin as standing for a soldier’s about-face, a ballet dancer’s half pirouette, a turnaround jump shot, the movement of someone setting out on a long journey who looks back to wave farewell, the motion of the sun from dawn to dusk, the changing of the seasons from winter to summer, the turning of the tide. You could also associate it with turning the tables on someone, a reversal of fortune, turning one’s life around, the transformation of Dr. Jekyll into Mr. Hyde (and vice versa), the pivoting away from loss or regret to face the future, the ugly duckling becoming a beauty, drought giving way to rain. You might even interpret its highlighting of opposites as an allusion to elemental dualities—shadow and light, birth and death, yin and yang. Math historians Edward Kasner and James R. Newman once observed that Euler’s formula “appeals equally to the mystic, the scientist, the philosopher, the mathematician.” It seemingly might also appeal to those with a poetic turn of mind, for it suggests that when three of the most elemental numbers are combined, they somehow spring to life and speak of ducklings and dancers, transformations and farewells.
* This theorem is so famous that it was mentioned in the 1939 movie The Wizard of Oz, but not in a way that math teachers would approve of: when awarded a diploma by the wizard, the scarecrow recited an absurdly bungled version of the Pythagorean theorem to demonstrate his newfound braininess.
* In case you hadn’t noticed, the idea of rotating a 1-D vector by 180 degrees is actually a bit strange. When you rotate a line segment in your mind as described, it moves off the 1-D number line and pirouettes through 2-D space before returning to the number line. Thus, the concept of rotating a 1-D vector by 180 degrees so that it points backward is something like a 3-D spaceship tunneling through 4-D space to reverse direction. Nonetheless, this sci-fi-like concept works nicely as a geometric representation of multiplying numbers by −1.
* The sine of θ, or sin θ, is a real number, just as is cos θ. Thus, a vector representing sin θ by itself would lie along the x axis in the complex plane, as do the vectors for all pure real complex numbers. But as we saw earlier in this chapter, multiplying a complex number times i effectively makes the vector representing that number rotate by 90 degrees. Therefore, multiplying i times sin θ (written i sin θ), when interpreted geometrically, rotates the x-axis-aligned vector for sin θ by 90 degrees so that it winds up aligned with the i axis. In fact, i sin θ can be pictured as a vector lying right on top of the i axis.