For example, the definition of the Discrete Fourier Transform (DFT) is as follows:
; This is the result. Equation (1) clearly contains pi (π), Napier's number (e), which is the base of the natural logarithm, and
It contains the number "i" (the imaginary unit, i = √-1) which, when squared, equals "-1".
Of course, this is because we use Euler's formula when deriving equation (1), but in this way, light,
For engineers who work with waves, such as those in electromagnetism, vibration, and acoustics, "Euler's formula" is essential.
The semicolon (;) is an important identity that makes calculations easier. Substituting π for θ in equation (2), we get cos π = -1 and sin π = 0 (by moving -1 to the left side);
; This gives us Euler's identity, which includes the most fundamental numbers in mathematics (π, e,
The numbers 1, 0, and the imaginary unit i) are connected by a single equation, which is considered one of the most beautiful mathematical formulas in the world.
In the novel "The Housekeeper and the Professor" (by Yoko Ogawa), which was made into a movie a few years ago, the subtle relationships between the characters are also depicted.
It was used in situations that suggested a relationship between them. Incidentally, the symbols e and i are Euler (18th century)
It is said that the great mathematician Kii was the first to use it.
It is constructed from the imaginary unit i (= √-1), which, when squared, equals -1, and two real numbers x and y;
The semicolon (;) is called a complex number, where x is the real part of z (Re z) and y is the imaginary part of z (Im z).
The complex number z can be easily understood by relating it to a point P(x, y) on the xy-plane, and in this case the xy-plane is complex
This is called the elementary plane (or Gaussian plane) (Figure 1).
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Figure 1: Polar form and complex conjugate in the complex plane.
In equation (4), if we set y = 0, z becomes a real number, and the real axis on the complex plane becomes a number line of real numbers.
Thus, complex numbers include all real numbers. Also, for z in equation (4):
This is called the absolute value of z, and geometrically, it represents the distance OP between the origin O and point P on the complex plane.
If we consider OP as a vector z, then we can consider it as its length (norm). In equation (4), z
If we map the corresponding point P(x, y) to polar coordinates (r, θ), then z is in polar form;
This is expressed as (Figure 1). Here, r is the absolute value of z, and θ is called the argument of z, and the figure
It is essentially the angle (in radians) that vector OP makes with the positive direction of the real axis (x-axis).
Applying Euler's formula from equation (2) to equation (6):
; and the expression becomes even simpler. (9) The major advantage of using the expression is that it can be divided geometrically.
In addition to becoming easier to understand, it also simplifies the calculation of multiplication and division between complex numbers.
I will omit the concrete calculation examples, but multiplication is by multiplication of absolute values (real numbers) and the arguments are added, and division is
Division of absolute values (real numbers) and subtraction of argument are performed.
Representing complex numbers geometrically in polar form (vector length and argument) helps to better understand the meaning of the imaginary unit i.
Yes, you can. Multiplying a complex number z by i means rotating the vector z counterclockwise by 90 degrees (π/2).
This is equivalent to rotating in the positive direction. Because;
;that's why;
This is because the argument increases by π/2 (Figure 2).
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Figure 2: The meaning of multiplying a complex number by i
Using the same logic, multiply a positive point (real number) on the real axis (x-axis) by i twice (i² = -1, therefore -1).
Multiplying by (x) results in a 180-degree rotation, which becomes a negative point on the number line.
Furthermore, negative numbers can also be explained in a unified way by using imaginary numbers.
For the complex number z in equation (4), the number obtained by reversing the sign of the imaginary part is called the complex conjugate of z;
; This represents a position P'(x, -y) that is symmetrical to the real axis (x-axis) geometrically (Figure 1). Complex conjugate
An important property of this is that the sum <(11)> and the product <(12)> are real numbers. That is;
If an algebraic equation with real coefficients has complex solutions, then its complex conjugate must also be a solution.
In fact, this can also be explained by the properties of complex conjugates. Equations (11) and (12) are the "solutions" of quadratic equations with real coefficients.
This represents the relationship between the coefficients. Equation (1) shows the result of DFT on a real signal x (n).
The complex Fourier spectrum X(k) of the fruit is also generally a complex number, but it contains negative frequency components and positive ones.
Frequency components are always related to complex conjugates.
Historically, the imaginary unit i explicitly appears in the solutions to quadratic equations (for example, x² + 1 = 0).
When we are looking for this, we previously considered it to have no solution, but in the case of a cubic equation, the solution is real.
Even when the result is a number, using the quadratic formula (Cardano's formula) may sometimes yield imaginary numbers.
For example, in the case of x³ - 15x - 4 = 0:
Therefore, there are three real solutions to x = 4, -2 + √3, and -2 - √3, but using Cardano's formula, one of the solutions to equation (13) is:
This is how it can be solved. Of course, Cardano (16th century, contemporary of Leonardo da Vinci)
In that era, imaginary numbers were not recognized, and numbers with negative values inside their square roots were not accepted.
So it was probably met with surprise.
However, the number inside the cube root of (14) is clearly a complex number, but if you look closely inside the two cube roots, you will see that they are conjugates.
This reveals that the relationship is related to complex numbers. Even when taking the cube root, the relationship with the complex conjugate remains unchanged.
Therefore, (14) should be a real number.
Far away;
Therefore, the value of (14) can be found using equation (11);
; Thus, we can find the complex number
Exponents and roots can be calculated very easily.
Here's another example of a calculation using the argument of a complex number. Equation (16) for calculating pi (Hah
There are formulas that can be expressed as Ton's formula and formula (17) (Machin's formula).
If we consider the left-hand side as the sum of the arguments of two complex numbers, then we can find the value (angle) on the right-hand side from the product of the complex numbers.
It can be calculated. For example, in equation (16), the left side represents the argument of (4 + i)³ (99 + 5i).
When you calculate this, you get 4913 + 4913 i, and you can see that the argument is π/4.
Similarly, equation (17) becomes (5 + i)4/(239 + i), and calculating this gives 2 + 2i.
Finally, let's consider Euler's formula in (2). The left side of equation (2), e iθ, is a complex exponential function and
It is a periodic function that has the same period as trigonometric functions (in this case, 2π), and is also a complex function.
This expresses two pieces of information. Furthermore, a major advantage of exponential functions is that "differential operations are simplified."
It also has the characteristic that "the form of the function does not change even when differentiated simply by the derivative."
As shown in Figure 3, the complex exponential function e iθ has the property that differentiating it four times returns it to its original form, and equation (2)
The same differentiation operation can be performed on the trigonometric functions on the right-hand side and similarly return to the original value (Figure 4), but the complex exponents in Figure 3...
You can see that the function is a much cleaner approach.
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Figure 3 Differentiating eiθ four times
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Figure 4 shows the four-fold differentiation of sinθ.
These properties are such that, as shown in Figure 5, multiplying a complex number z by i four times and completing one revolution around the circumference returns to the original value.
I understand that they correspond to each other.
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Figure 54 Multiply a complex number by i four times.
This time we covered the basics of complex numbers, but next time we'll discuss their applications in signal processing.
I hope we can talk about it.
Finally, I've listed the references I used, so please feel free to refer to them if you're interested.
○ References
- "The Story of Imaginary Numbers" by Paul J. Nayn (translated by Junji Koda, Seidosha)
- "The Emotion of Imaginary Numbers" by Takeshi Yoshida (Tokai University Press)
(Excerpt from the email newsletter issued on January 22, 2009)
