As you know, the imaginary number is defined as: where a \in \mathbb{R}
And therefore complex number c \in \mathbb{C} is imaginary number plus a real number.
Imaginary number relates to rotation. For detailed intuitive understanding of imaginary number, see this article
Different forms of complex numbers:
a + bi
r(\cos \theta + i \sin \theta)
re^{i \theta}
Euler's identity: e^{i\pi} = -1 is a result of power representation of complex numbers.
There is no natural total order of complex number as \mathbb{C} is not an ordered field and there is no ordering, but one can artificially construct a total order (but not well-order):
first compare its argument and then modulus
first compare its modulus and then argument
Similar to \mathbb{R}, the well-order of \mathbb{C} might be possible to construct by making use of Axiom of Choice, but that's hard.
However, this is not a well-ordering
Complex number addition is the same as vector addition.
For detailed intuitive understanding of complex number, see this article
Notice complex number multiplication is different from vector multiplication:
whereas in vector multiplication for \vec{i}, \vec{j} unit vector:
To visualize complex multiplication, we view it as vector scale, rotation, and addition:
We can also visualize it after expanding the formula:
Because of the scaling, we see that complex-multiplication preserves the ratio of angle:
We can also see multiplication as multiplying modulus and adding the argument using exponential form:
For detailed intuitive understanding of complex multiplication, see this article
Complex Conjugate Pair is defined as following:
When complex number and its conjugate are multiplied together, they becomes a real number:
When complex number and its conjugate are added together, they becomes a real number:
Note that only a complex number's conjugate have both two properties above.
When the solution to quadratic equation involves complex root c \in \mathbb{C}, then c^* \in \mathbb{C} is also a root.
For z = a + bi \in \mathbb{C}, we define modulus and argument.
Modulus: the magnitude of the complex vector
Argument: the angle of the complex vector
We can also write the complex number in trignometry expression (modulus-argument form) after knowing a = r\cos \theta, b = r\sin \theta:
However, for a point P in Argand Diagram, we have infinite many z = r(\cos \theta + i \sin \theta) correspond to P because there are infinite many \theta to choose from. So we define principal argument as a convention of choosing the angle:
We get arithmetic property for z_1, z_2 \in \mathbb{C}:
|z_1z_2| = |z_1| |z_2|
|\frac{z_1}{z_2}| = \frac{|z_1|}{|z_2|}
\arg(z_1z_2) = \arg(z_1) + \arg(z_2)
\arg(\frac{z_1}{z_2}) = \arg(z_1) - \arg(z_2)
The loci is just a set of 1D points generated by complex-functions in Argand Diagram.
We observe complex-functions:
because complex addition is vector addition, the above formula generates a circle of radius r around (x_1, y_1) for z_1 = x_1, iy_1.
because complex addition is vector addition, the above formula generates a perpendicular bisector of line segment between z_1, z_2.
This generates a half-line starting from z_1 and shoot at angle \theta to infinity.
The region is just a set of 2D points generated by complex-functions in Argand Diagram.
We have relationship to sinusoids:
We have properties of complex series, for w, z \in \mathbb{C}:
Rotational Property:
Roots of Unity (de Moivre number): any complex number that yields 1 when raised to some positive integer power n \in \mathbb{Z}
The n-th root of unity if n \in \mathbb{Z}^+, \forall k \in \{0, 1, ..., n - 1\} is:
The above set of roots, if plotted in Argand Diagram, will yield vertices of a regular n-gon.
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