複數的直觀入門

We’ll start off our journey by attempting to solve the systems of equations provided below:

At first this resembles an ordinary system of two equations – but if we just spend a small amount of time attempting to plug in regular whole / real numbers, you’ll be able to see a bit of a dilemma. Let’s try plugging in 3 for both x and y:

Let’s try plugging in 4 and 2 now to see if this gets us any closer:

Playing around a bit more should get use to realize that there is something awfully wrong here: it doesn’t’ seem possible to solve the above equations using our regular number system – no matter what we try to plug in, our answers always seem to sway far apart from one another. Attempting to come up with feasible solutions using negative numbers doesn’t help either since the multiplication step would require 2 negative numbers and no 2 negative numbers can add up to a number greater than 0.

So – what can we do?

Well – let’s try solving our equation algebraically instead of going through our heuristic approach above. We rearrange our first equation to solve for y:

We substitute this into our 2^nd equation to obtain:

We now expand the left hand side:

Let’s move all our terms to the left hand side to get our equation into standard quadratic form:

Finally, we multiply every term by -1 so the squared term is positive:

Taking a glance at the above equation provides us with a bit of a mystery: it doesn’t seem to cross 0 (on our x-axis) at all when we attempt to graph it:

It thus appears that our equation doesn’t have a solution – but let’s try the algebraic approach now to do a quick double check – to do so, we can use the quadratic formula to provide a solution to the above equation. As a reminder, a quadratic equation in the form provided below:

has a solution given by the formula:

From our quadratic formula:

we know that a = 1, b = -6, and c = 36, and we can plug this in to obtain:

This is where we once again encounter a big mystery: the solution to our original quadratic formula appears to take the square root of a negative number (-108) but this is not possible using our regular number system composed of real numbers. We’ve hit a wall.

This exact mathematical wall involving the seemingly impossible square root of a negative number is precisely where the Italian mathematician Gerolamo Cardano found himself in 1545. While developing his groundbreaking general solutions to cubic equations, Cardano noticed something bizarre: even when an equation yielded a completely real, tangible answer, the algebraic path to get there forced him to temporarily calculate the square root of a negative number. For instance, in some of his working expressions, quantities would arise that required manipulating roots like the square root of -15 alongside other terms.

Cardano discovered that if you stubbornly followed the standard rules of algebra with these forbidden quantities—multiplying and simplifying them—the imaginary parts would eventually vanish, leaving behind a perfectly accurate, real-world solution. In fact, in 1545 he published Ars Magna, one of the most important algebra books of the Renaissance. Among its many problems, Cardano asked his readers to divide 10 into two parts whose product was 40. In modern notation, he was searching for two numbers u and v such that:

This should look familiar since it very much resembles our original problem involving x and y. Using a similar approach which we used to turn our original equations into a quadratic one, Cardano simplified the above into the equation shown below:

Applying the quadratic formula leads to a result that would have looked impossible to almost every mathematician of Cardano’s time:

At this point, every instinct from ordinary arithmetic says to stop - you should not be able to take the square root of −15. There is no real number whose square is negative, so the two parts would have to be:

But now something almost magical happens: if we ignore this impossibility for a moment and just apply the usual rules of algebra:

The end product is:

Yet, despite watching the math work out cleanly in front of him, Cardano couldn’t bring himself to believe it. He viewed the whole maneuver as a mere “convenient artifice”— a mathematical magic trick with no actual substance or practical meaning behind it. French philosopher René Descartes (1596–1650) strongly agreed with that skepticism — looking at these strange numbers, Descartes dismissed them as nonsensical fabrications and coined the term imaginary for the roots of negative quantities—a derogatory label meant to mock them as products of pure fancy, which unfortunately stuck around for centuries.

Throughout the 18th century, however, mathematicians found themselves unable to ignore these numbers. They kept popping up as indispensable shortcuts in calculus, algebra, and physics. It was Leonhard Euler who finally gave them a proper home and standard notation, introducing the symbol i to represent the square root of -1:

Even with Euler’s notation, a lingering question remained: what do these numbers actually represent? If real numbers live on a 1-dimensional number line stretching from negative infinity to positive infinity, where do imaginary numbers live?

The answer arrived near the turn of the 19th century when Jean-Robert Argand and Caspar Wessel independently figured out the missing piece of the puzzle: a geometric model. They realized that multiplying by -1 rotates a number 180 degrees on the number line:

Therefore, multiplying by the square root of -1 (or i) shouldn’t just push you further left or right on a one-dimensional line—it represents a 90-degree rotation into a completely new, perpendicular dimension. This is a natural product of i² being equal to -1 (a 180 degree rotation). Since i represents half of this multiplicative step, it can naturally be mapped to 90 degrees (a halfway 180 degree rotation). By turning our 1D number line into a 2D coordinate plane (where the horizontal axis holds real numbers and the vertical axis holds imaginary numbers), every “complex” number suddenly corresponds to a physical point on a flat plane:

In the above image, the horizontal axis now represents the real number line (-2, -1, 0, 1/3, 3, etc…) while the vertical axis represents a brand new perpendicular dimension representing the real part of our number (-2i, -1i, 0, 1/3i, 3i, etc…). The curved blue line shows what happens when we multiply by i: multiplying +1 by i swings it a quarter turn upward, landing precisely on +i which is a 90 degree rotation.

To put things simply: our new dimension gives us a way to turn our 1-dimensional real number system into a 2-dimensional one. To do this, we allow the regular x-dimension to model the real part of our new number system while the vertical y-dimension now represents the new ‘imaginary’ dimension.

We can visualize this intuitively by taking a look at the diagram below, where a is the real part while b is the imaginary one:

This model provides a natural correspondence between complex numbers and points in the plane. The horizontal axis is called the real axis, the vertical axis the imaginary axis, and an arbitrary complex number z = a + ib appears as the point with coordinates (a, b) on the plane.

We also note that using our new system of dimensions, multiplication by i simply represents a 90 degree rotation:

In the early 1800s, many strict mathematicians rejected the Argand / Wessel plane because it felt like a cheap visual trick. They argued that just because you can draw a picture of the square root of 1 as a vertical line doesn’t prove it logically exists as a number.

In 1833, Hamilton published a paper that removed all geometry and “imaginary” mystery completely. He defined a complex number strictly as an algebraic ordered pair of real numbers, written simply as (a, b). He then defined precise rules for how to add and multiply these pairs:

We can visualize the addition component easily since it maps to regular vector addition:

We can also visualize multiplication (albeit it is a bit less intuitive than addition) by looking at some of the visuals below - in the below visual we multiply an imaginary number by i (to produce a 90 degree rotation):

Another more complex example is provided below where we multiply (2, 1) and (1, 2) in our new imaginary system:

In the above image, we can see an example showing:

Both factors have real and imaginary components. Their directions add together to produce:

and their lengths multiply:

Therefore, the product lands at (0, 5).

This geometric breakthrough was originally popularized and cemented by Carl Friedrich Gauss. Gauss utilized the complex number system to construct the proof of the fundamental theorem of algebra, which demonstrated that every non-constant polynomial equation has at least one root in the complex number system. As an example – even though the graph for the equation below never crosses 0 – we can still find four roots in the complex plane which all appear symmetrically on the unit circle:

That definitive proof finally convinced the wider mathematical community of the profound importance and absolute validity of complex numbers—transforming them from Cardano’s torturous artifice into one of the most powerful tools in all of mathematics and science.

Finally, taking a cue from Cardano and Euler, let’s take that bold leap ourselves and apply this concept to our own unsolved mystery. We left off with the troubling fraction below:

Instead of stopping, we can pull apart that -108. We know that -108 is the exact same as 36 multiplied by -3:

We can evaluate the square root of 36 and pull it outside as a 6:

Here is where Euler’s i comes to the rescue. We can separate out the square root of -1 from the -3:

Substituting in our imaginary unit i and dividing by 2 provides our ultimate, two-part solutions:

Since x + y = 6, the corresponding y-values are:

We write the 2 ordered solutions as:

Let’s check the product:

The two middle terms are equal and opposite, so the imaginary part disappears. The final i² becomes -1, turning the last subtraction into addition. We can visualize this as adding our 2 angles (resulting in a vector of 0 degrees since they are opposite in direction) and scaling our resulting vector (of length 6) by 6 across the x-axis resulting in 36:

We can also easily verify that x + y = 6 since the i components cancel each other out and our real-parts simply add up to equal to 6:

To sum it up – complex numbers are not all that complex: all we’re doing is transforming the number system represented in 1 dimension into a 2-dimensional number. A real number needs only one coordinate, so we draw it on a regular line; a complex number needs two coordinates – a real part and an imaginary part - so we draw it on a plane like the one shown below:

The x-axis above represents our real part while the y-axis represents the imaginary component. Addition now looks like ordinary movement in the plane and we add the real parts horizontally and the imaginary parts vertically.

Multiplication contains an even deeper idea: multiplying by i transforms and the point a + bi into and rotates our 2-dimensional point (or vector) by 90 degrees counter-clockwise:

Multiplying by i a second time turns it another 90 degrees, producing a total rotation of 180 degrees. That is why i squared equals -1: two quarter-turns point us in exactly the opposite direction on the real axis.

Thus, multiplication also maps nicely into this new world (especially when multiplying by i).

Before we explore this connection, it is worth taking a moment to understand what the trigonometric functions actually represent.

At first, sine and cosine are usually introduced as rules involving the sides of a right-angled triangle. Given an angle θ, we define:

Although these definitions are useful, they can make sine and cosine feel like little more than tools for solving triangles. Their deeper meaning becomes much clearer when we place the triangle inside a circle.

Imagine drawing a circle with a radius of 1 and then rotating a line from the positive horizontal axis through an angle θ. The endpoint of that line has two coordinates: a horizontal position and a vertical position. The horizontal coordinate is given by cos θ, while the vertical coordinate is given by sin θ:

Thus the pair:

identifies a point on the unit circle:

As θ changes, the point moves around the circle and the values of sine and cosine keep track of its vertical and horizontal positions.

To put it simply, sine and cosine transform an ANGLE into two directions of movement. Cosine tells us how far we travel horizontally, while sine tells us how far we travel vertically. If our rotating line has a length r instead of 1, these two components become:

This should already look remarkably similar to the complex-number system. A complex number also contains two components: a real component measured horizontally and an imaginary component measured vertically. Trigonometry therefore gives us another way of describing a complex number—not only by stating its horizontal and vertical coordinates, but by stating its LENGTH and ANGLE.

This connection becomes especially powerful when we begin multiplying complex numbers. In the early 18th century, French mathematician Abraham De Moivre noticed a striking similarity between complex multiplication and the addition formulae of the sine and cosine functions from trigonometry. Given that:

it turns out that these two seemingly separate formulae fit together perfectly inside a single complex number. To see why, we first need to look at our complex plane in a slightly different way.

So far, we have described a complex number z = a + bi by giving its horizontal position a and its vertical position b. However, there is another equally useful way to locate the same point. Instead of asking how far right and how far up we travel, we can ask two new questions: how far is the point from the origin, and through what angle must we turn to face it?

We call the distance from the origin r and the angle θ (theta). From the Pythagorean theorem:

and from ordinary trigonometry:

Multiplying both equations by r gives:

We can now substitute these expressions into z = a + bi:

Nothing mysterious has been added here. We have simply changed the language used to describe the point. The form a + bi tells us its horizontal and vertical components, while the form r(cos θ + i sin θ) tells us its length and direction. The first is like giving a street address; the second is like giving a distance and a compass. If r = 1, our point lies on the unit circle and the expression becomes especially simple:

The cosine supplies the horizontal coordinate while the sine supplies the vertical one. As θ changes, the complex number travels around the unit circle. In this form, an angle is no longer merely attached to a complex number—it has become part of the number itself.

Now let’s multiply two unit complex numbers:

Multiplying them in the ordinary algebraic way gives:

Since i² = −1, we can collect the real and imaginary parts:

At first, this result looks a bit cumbersome. But take another look at the two trigonometric addition formulae we started with. The real part is exactly cos(α + β), while the imaginary part is exactly sin(α + β). Therefore:

This is the striking connection which de Moivre uncovered. Complex multiplication automatically performs the angle-addition rules of trigonometry. The large collection of terms created by ordinary algebra is not random at all—it is the sine and cosine addition formula hiding in plain sight.

To put it simply, complex multiplication performs two geometric actions at the same time:

1. It MULTIPLIES the lengths: r₁ × r₂.

2. It ADDS the angles: α + β.

Let’s check the picture with an exact algebraic example. Consider:

The trigonometric rule predicts a product with length 2 × 2 = 4 and angle 30° + 60° = 90°:

Ordinary algebra confirms exactly the same thing:

The real terms cancel, leaving a number pointing straight upward on the imaginary axis. The algebra and the geometry are not two different explanations—they are two views of the same multiplication.

Once this pattern is visible, repeated multiplication becomes almost effortless. Multiplying a complex number by itself twice doubles its angle. Multiplying it by itself three times triples its angle. In general, for any whole-number power n:

This is now known as De Moivre’s formula. If the original complex number has a length r, that length is also multiplied by itself n times:

For example, let’s calculate (1 + i)⁴. In rectangular form, 1 + i has:

Therefore:

Applying de Moivre’s formula gives:

Instead of expanding four factors and trying to keep track of every i, we simply raise the length to the fourth power and multiply the angle by four. The number rotates from 45° to 180° while its length grows from √2 to 4.

Another great and simple example which takes the cube of a complex number is provided below:

So De Moivre’s theorem is not just a formula; it is a compact description of rotation. Raising a complex number to the nth power rotates it by n times its original angle.

The same idea can also be run backwards. Powers multiply angles, so roots divide them. However, complex angles contain one final twist: turning through 360° brings us back to the same direction. Thus 0°, 360°, 720°, and every other complete turn all describe the positive real axis.

Suppose we want to solve:

The number 8 has length 8 and angle 0°, but it can also be described using the angles 360° and 720°. Taking the cube root divides the length and each of these angles by three:

This gives three different solutions:

The three answers are equally spaced around a circle. Their angles differ by 120° because 360° ÷ 3 = 120°. More generally, the n different nth roots of a non-zero complex number are evenly spaced around a circle, like the vertices of a perfectly regular polygon. This is one of the reasons complex roots display the beautiful symmetry which we saw earlier in the equation z⁴ + 1 = 0.

We can now see why sine and cosine belong so naturally inside the complex number system. They are not decorative additions placed on top of complex arithmetic. They are the coordinates of rotation itself. Cosine keeps track of the horizontal component, sine keeps track of the vertical component, and complex multiplication combines them in precisely the way required to add angles.

To sum it up – complex numbers were not invented because mathematicians wanted to make arithmetic more complicated – they were invented because ordinary algebra kept producing patterns that the real-number system could not satisfy. What first appeared to be an impossible square root turned out to be evidence that the number line was only a slice of a larger world. Once we allow that world to have two directions instead of one, the mystery begins to disappear: a complex number is not an imaginary answer pretending to exist — it is an ordinary point in a richer mathematical space.

To put it simply, the world of 1-dimensional real numbers turns into a ‘complex’ world involving pairs of reals numbers which are endowed with their own arithmetic. The rectangular form a + bi tells us where a complex number is, while the trigonometric form r(cos θ + i sin θ) tells us how far it reaches and which way it points. Once we use this second language, multiplication becomes scaling plus rotation, powers become repeated rotation, and roots become evenly spaced directions around a circle. What looked like a complicated two-part arithmetic has quietly turned into the geometry of length and angle.

Complex analysis and the use of complex numbers might seem like an abstract idea, but the reality is quite the opposite. Because they are the natural language of rotation, scaling, and oscillation, complex numbers are uniquely equipped to describe the cycles and waves that permeate our physical world. Far from being a mathematical parlor trick, they are an essential tool for engineers, physicists, and computer scientists.

Signal processing and communications are built on the same idea. Any reasonable signal—sound, an image, a radio wave—can be decomposed into a sum of complex exponentials.

The Fourier transform does this, and the fast Fourier transform makes it fast enough for everyday devices. This is how MP3 and JPEG compression work, how WiFi and 5G encode and decode data, how noise-canceling headphones separate unwanted sound, and how radar, GPS, and medical imaging reconstruct information. In each case, complex numbers turn difficult time-domain problems into simpler frequency-domain operations.

Control systems use complex analysis to keep machines stable. Drones, autopilots, robots, cruise control, and thermostats are modeled with Laplace transforms and transfer functions. The poles and zeros of those functions live in the complex plane. Their locations tell engineers whether a system will settle down, oscillate, or become unstable. A pole in the left half-plane usually means stability; a pole on the imaginary axis means sustained oscillation; a pole in the right half-plane means instability. That abstract picture directly determines whether a real machine behaves safely.

Physics and engineering fields rely on complex analysis even more deeply. In electromagnetism, fluid flow, and heat conduction, complex potentials and conformal mappings solve two-dimensional field problems. They are used in aerodynamics to design airfoils, in electrostatics to shape electric fields, and in civil and mechanical engineering to model stresses and flows. Quantum mechanics goes further: the Schrödinger equation contains i explicitly, and wavefunctions are complex-valued. Without complex numbers, modern descriptions of atoms, semiconductors, lasers, and magnetic resonance imaging would be impossible.

So complex numbers are not a strange detour from reality. They are a compact language for rotation and oscillation, and complex analysis is the calculus of that language. The imaginary unit i is not imaginary in the sense of unreal; it is a 90-degree turn. From the AC in our walls to the signals in our phones, from the stability of a drone to the images inside an MRI, the mathematics of complex numbers quietly runs much of the everyday world. What began as an impossible square root became one of the most useful and unifying ideas in science and engineering.

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