User:Aleksander Stos/ComplexNumberAdvanced: Difference between revisions
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''This is an experimental draft. For a brief description of this project click [[User_talk:Jitse Niesen#Essentials|here.]]'' | ''This is an experimental draft. For a brief description of this project click [[User_talk:Jitse Niesen#Essentials|here.]]'' | ||
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Complex numbers may be naturally represented on the ''complex plane'', where <math>z=x+iy</math> corresponds to the point (''x'',''y''), see the fig. 1. | Complex numbers may be naturally represented on the ''complex plane'', where <math>z=x+iy</math> corresponds to the point (''x'',''y''), see the fig. 1. | ||
{{Image|Complex_plane3.png|right|250px|Fig. 1. Graphical representation of a complex number and its conjugate}} | |||
The modulus is just the distance from the point <math>z=(x,y)</math> and the origin. More generally, <math>|z_1-z_2|</math> is the distance between the two given points. Furthermore, the conjugation is just the symmetry with respect to the x-axis. The sum of two given complex numbers <math>z_1</math> and <math>z_2</math> can be geometrically determined as a vertex of a parallelogram determined by the points 0, <math>z_1</math> and <math>z_2</math> (i.e. the fourth vertex given these three, see the fig. 2). | The modulus is just the distance from the point <math>z=(x,y)</math> and the origin. More generally, <math>|z_1-z_2|</math> is the distance between the two given points. Furthermore, the conjugation is just the symmetry with respect to the x-axis. The sum of two given complex numbers <math>z_1</math> and <math>z_2</math> can be geometrically determined as a vertex of a parallelogram determined by the points 0, <math>z_1</math> and <math>z_2</math> (i.e. the fourth vertex given these three, see the fig. 2). {{Image|Complex_sum_2.png|right|250px|Fig. 2. Graphical representation of the sum of two complex numbers}} | ||
==Trigonometric and exponential form== | ==Trigonometric and exponential form== | ||
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:<math> (cos\theta+i\sin\theta)^n = \cos(n\theta)+i\sin(n\theta)</math> | :<math> (cos\theta+i\sin\theta)^n = \cos(n\theta)+i\sin(n\theta)</math> | ||
{{Image|Graphical_multiplication1.png|right|200px|Fig 3. Multiplication by <math>i</math> amounts to rotation by 90 degrees.}} | |||
Graphically, multiplication by a constant complex number <math>z=re^{i\theta}</math> amounts to the rotation by <math>\theta</math> and the [[homothety]] of ratio ''r''. In particular, the multiplication by ''i'' amounts to the rotation by the right angle (counter-clockwise), see Fig. 3. | Graphically, multiplication by a constant complex number <math>z=re^{i\theta}</math> amounts to the rotation by <math>\theta</math> and the [[homothety]] of ratio ''r''. In particular, the multiplication by ''i'' amounts to the rotation by the right angle (counter-clockwise), see Fig. 3. | ||
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where ''z'' is the variable and ''a'' a non-zero constant has exactly ''n'' solutions. They are called ''n''<sup>th</sup> (complex) ''roots'' of ''a''. If ''a'' is written in the exponential form, <math>a=re^{i\theta},</math> then the ''n'' roots of ''a'', denoted as <math>z_0,z_2,\ldots,z_{n-1}</math>, are given by | where ''z'' is the variable and ''a'' a non-zero constant has exactly ''n'' solutions. They are called ''n''<sup>th</sup> (complex) ''roots'' of ''a''. If ''a'' is written in the exponential form, <math>a=re^{i\theta},</math> then the ''n'' roots of ''a'', denoted as <math>z_0,z_2,\ldots,z_{n-1}</math>, are given by | ||
:<math> z_k = \sqrt[n]{r}\cdot\exp\left (i \left(\frac{\theta+2k\pi}{n}\right)\right),\quad k=0,1,\ldots,n-1. </math> | :<math> z_k = \sqrt[n]{r}\cdot\exp\left (i \left(\frac{\theta+2k\pi}{n}\right)\right),\quad k=0,1,\ldots,n-1. </math> | ||
{{Image|Complex_roots.png|right|300px|5th roots of unity}} | |||
One may observe that | One may observe that | ||
:<math> z_{k} = z_{k-1}e^{i\theta/n};</math> or, equivalently, <math> z_{k} = z_0 e^{ik\theta/n},\quad k=1,2,\ldots,n-1.</math> | :<math> z_{k} = z_{k-1}e^{i\theta/n};</math> or, equivalently, <math> z_{k} = z_0 e^{ik\theta/n},\quad k=1,2,\ldots,n-1.</math> |
Latest revision as of 02:19, 22 November 2023
The account of this former contributor was not re-activated after the server upgrade of March 2022.
This is an experimental draft. For a brief description of this project click here.
Definition
Complex numbers are defined as ordered pairs of reals:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{C}= \{ (a,b) \colon a,b\in \mathbb{R} \}.}
Such pairs can be added and multiplied as follows
- addition: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a, b) + (c, d) = (a + c, b + d)}
- multiplication: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a, b)(c, d) = (ac - bd, bc + ad)}
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \mathbb{C}} with the addition and multiplication is the field of complex numbers. From another of view, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle \mathbb{C} } with complex additions and multiplication by real numbers is a 2-dimesional vector space.
To perform basic computations it is convenient to identify numbers of the form Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{ (a,0):\;\;a\in\mathbb{R} \}} with the usual real line Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (\mathbb{R})} and to introduce the imaginary unit, i=(0,1).[1] The imaginary unit has the fundamental property Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \scriptstyle i^2=-1.} Indeed, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (0,1)\cdot(0,1) = (-1,0) = -1.} Any complex number Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z=(a,b)} can be written as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z=a+bi} (this is often called the algebraic form) and vice-versa. The numbers a and b are called the real part and the imaginary part of z, respectively. We denote Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a=\Re (z)} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b=\Im (z).} Remark that two complex numbers are equal if and only if their real and complex part are equal, respectively. Notice that i makes the multiplication quite natural:
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a + bi)(c + di) = (ac - bd) + (bc + ad)i. }
We define the modulus of z, denoted by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |z|} ,
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |z|=\sqrt{a^2+b^2}.}
We have for any two complex numbers Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z_1} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z_2}
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\bar z| = |z|;}
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |z_1\cdot z_2| = |z_1| \cdot |z_2|;}
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\frac{z_1}{ z_2}| = \frac{|z_1|}{|z_2|},} provided Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z_2\not =0.}
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \big| |z_1| - |z_2| \big| \le |z_1+z_2| \le |z_1| + |z_2|}
For Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z=a+bi} we define also Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bar z} , the conjugate, by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bar z= a-bi.} Then we have
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bar{(\bar z)} = z}
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bar z_1 \pm \bar z_2 = \overline{(z_1 \pm z_2)};}
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bar z_1 \cdot \bar z_2 = \overline{(z_1\cdot z_2)};}
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overline{\left( \frac{z_1}{z_2} \right)} = \frac{\bar z_1}{\bar z_2},} provided Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z_2\not =0;}
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z\bar z = |z|^2.}
Geometric interpretation
Complex numbers may be naturally represented on the complex plane, where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z=x+iy} corresponds to the point (x,y), see the fig. 1.
The modulus is just the distance from the point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z=(x,y)} and the origin. More generally, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |z_1-z_2|} is the distance between the two given points. Furthermore, the conjugation is just the symmetry with respect to the x-axis. The sum of two given complex numbers Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z_1} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z_2} can be geometrically determined as a vertex of a parallelogram determined by the points 0, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z_1} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z_2} (i.e. the fourth vertex given these three, see the fig. 2).
Trigonometric and exponential form
As the graphical representation suggests, any complex number z=a+bi of modulus 1 (i.e. a point from the unit circle) can be written as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z=\cos \theta + i\sin\theta} for some Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta\in [0,2\pi).} So actually any (non-null) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z\in\mathbb{C}} can be represented as
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z=r(\cos\theta + i\sin \theta),} where r traditionally stands for |z|.
This is the trigonometric form of the complex number z. If we adopt convention that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta \in [0,2\pi)} then such Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta} is unique and called the argument of z.[2] The equality of two complex numbers Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z_1=r_1e^{i\theta_1}} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z_2=r_2e^{i\theta_2}} is equivalent to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r_1=r_2} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta_1=\theta_2+2k\pi} for certain integer k. Graphically, the number Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \theta} is the (oriented) angle between the x-axis and the interval containing 0 and z.
Closely related is the exponential notation. Let us define the complex exponential as
The sum is convergent for any and by a comparison with the series of and it may be seen that
This is known as Euler's formula. As a particular example we get
In turn, the Euler's formula may be used to determine and :
In fact, these formulas are valid not only for but also for any
Another consequence is that any (non-zero) can be written as
- with the same r and as above.
This is called the exponential form of the complex number z.[3] It is well-adapted to perform multiplications. Indeed, for any and we have
- provided
The following particular case of complex multiplication is well-know as the de Moivre's formula [4]
Graphically, multiplication by a constant complex number amounts to the rotation by and the homothety of ratio r. In particular, the multiplication by i amounts to the rotation by the right angle (counter-clockwise), see Fig. 3.
Complex roots
Any non-constant polynomial with complex coefficients has a complex root. This result is known as the Fundamental Theorem of Algebra. Consequently, any complex polynomial of degree n has exactly n roots (counted with multiplicities). In particular, the equation
- ,
where z is the variable and a a non-zero constant has exactly n solutions. They are called nth (complex) roots of a. If a is written in the exponential form, then the n roots of a, denoted as , are given by
One may observe that
- or, equivalently,
Since that multiplication by the exponential represents a rotation by , the above formula interpreted geometrically means that the roots form a regular n-sided polygon centred at the origin; the vertices of the polygon belong to the circle of radius (see fig. 4).
Particularly important are the roots of unity, i.e. solutions of . The cubic roots of 1 (with n=3) are
and for n=4 we have
See also the fig.4 for the 5th roots of unity.