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Assumption is the mother of all screw-ups, Anonymous.
A natural number is a mathematical object. It is a number that occurs commonly in nature. As such, it is a whole, non-negative number, that is, a member of the sequence 0, 1, 2, 3, …, obtaining by starting from zero and adding 1 successively. The set of natural numbers is denoted by ℕ. Hence, ℕ = {0, 1, 2, 3, …}.
An integer number is a whole number that can be positive or negative, but does not have fractions or decimals. Integers are the collection of whole numbers and negative numbers. The set of integer numbers is denoted by ℤ. Hence, ℤ = { ···, -3, -2, -1, 0, 1, 2, 3, ···}.
A rational number is a number that is of the form p/q where p and q are integers and q is not equal to 0. The set or rational numbers is denoted by the doublestruck capital letter ℚ. It is the set of all fractions with integer numerators and non-zero integer denominators. ℚ = {$\frac{p}{q}|~ p, q \in ℤ, q \ne 0$} where p and q are integers (with q not equal to zero). Rational numbers can have either a terminating (e.g., $\frac{1}{2} = 0.5$) or repeating decimal (e.g., $\frac{1}{3} = 0.333...$) representation.
An irrational numbers is any number that is not a rational number, i.e., it cannot be expressed as a fraction ratio of two integers. The decimal representation of an irrational number goes on forever without repeating. The following are examples of irrational numbers: $\sqrt{2} = 1.41421356…, \pi = 3.14159265…, \sqrt{3}$, e (Euler’s number), and φ (the golden ration).
The set of rational and irrational numbers constitute the set of real numbers. It is denoted by ℝ. It includes the positive and negative integers, the fractions made from those integers (or rational numbers) and also the irrational numbers.
Complex numbers are fundamental in various fields of mathematics, physics, and engineering.
A complex number is specified by an ordered pair of real numbers (a, b) and is expressed or written in the form z = a + bi, where a and b are real numbers, and i is the imaginary unit, defined by the property i2 = −1 ⇔ i = $\sqrt{-1}$, e.g., 2 + 5i, $7\pi + i\sqrt{2}.$
Two complex numbers a + ib and c + id are equal if and only if a = c and b = d.
In the complex number z = a + bi: a is called the real part of the complex number, denoted as Re(z) = a. b is called the imaginary part, denoted as Im(z) = b. Example: z = 3 +4i, the real part is Re(z) = 3 and the imaginary part is Im(z) = 4.
We often use symbols z, w, ζ to denote complex numbers. They are the points on the plane, expressed as ordered pairs (a, b), where the real part “a” represents the coordinate for the horizontal axis and the imaginary part “b” represents the coordinate for the vertical axis. The set of complex numbers is denoted or expressed as ℂ = ℝ² = {(a, b) or a +bi: a, b ∈ ℝ}.
a + i·0, 0 + i·b, 0 + i·1, 0 + i·(-1), a + i·(-b) are often abbreviated as a, ib or bi, i, -i, and -ib respectively. We identify a real number a with the complex number a + i·0. In this sense, the real numbers are contained within the complex numbers. Numbers of the form 0 + ib = ib are called purely imaginary numbers.
The complex conjugate of a complex number z = a + bi is denoted by $\bar z$ and is defined as $\bar z = a -bi$, e.g, $\bar 3 + i = 3 -i$, $\overline{\sqrt{2}-\frac{\pi}{3}i } = \sqrt{2} +\frac{\pi}{3}i$. If you plot the complex number on the complex plane, the conjugate is a reflection across the real axis (x-axis), changing the sign of the imaginary component while the real part remains unchanged.
$\bar {\bar z} = z, Re(z) = \frac{z+ \bar z}{2}, Im(z) = \frac{z - \bar z}{2i}$
Suppose z = $\bar z ⇒ b = 0$, z is a real number. A complex number is real if and only if its imaginary part is zero, making it equal to its conjugate.
Similarly, if z = $-\bar z ⇒ a = 0$, meaning z is a pure imaginary number. If a complex number equals its negative conjugate, then it is purely imaginary.
Multiplying a complex number by its conjugate yields a real number: $z·\bar z = (a + bi)(a -bi) = a^2 + b^2$, e.g., z = 3 + 4i, $z·\bar z = (3 + 4i)(3 - 4i) = 3^2 + 4^2 = 9+16 = 25.$
The conjugate of the sum of two complex numbers is the sum of their conjugates: $\overline {z_1 + z_2} = \overline {(a + bi) + (c+di)} = \overline{(a + c) + (b+d)i} = (a + c) - (b+d)i = (a -bi) + (c-di) = \overline {z_1}+\overline {z_2}$ (iv). Similarly, $\overline {z_1 · z_2} = \overline {z_1}· \overline {z_2}$ (v). Besides, $z · \overline {z} = a^2 + b^2 = |z|^2 ↭ |z| = \sqrt{z · \overline {z}}$. The module of a complex number is the square root of the product of the number itself and its conjugate (vi)
$\overline {z}\overline {w} =\text{[vi]}~ \sqrt{zw · \overline {zw}} =\text{[v]}~ \sqrt{z·w·\overline{z}·\overline {w}} = \sqrt{z·\overline{z}·w·\overline{w}} =\text{[vi]}~ |z|·|w|$. The modulus of the product of two complex numbers is equal to the product of their moduli.
Properties: |z| ≥ 0, |z| = 0 ↔ z = 0, |z + w| ≼ |z| + |w|
To divide complex numbers, z2≠0 (0 + i0, that is, x2 and y2 are not simultaneously zero), we eliminate the imaginary unit from the denominator by multiplying both the numerator and denominator by the complex conjugate of the denominator. $\frac{z_1}{z_2} = \frac{x_1+iy_1}{x_2+iy_2} = \frac{x_1+iy_1}{x_2+iy_2}\frac{x_2-iy_2}{x_2-iy_2}=\frac{x_1x_2 + y_1y_2 + i(x_2y_1-x_1y_2)}{x_2²-y_2²} = \frac{x_1x_2 + y_1y_2}{x_2²-y_2²} + i(\frac{x_2y_1-x_1y_2}{x_2²-y_2²})$
Example: Calculate $\frac{2+i}{1-3i}$
$\frac{2+i}{1-3i}$ =[To simplify the expression, multiply both numerator and denominator by the conjugate of the denominator:] $\frac{2+i}{1-3i}·\frac{1+3i}{1+3i} = \frac{-1+7i}{10} =[\text{Thus, the simplified form is:}] \frac{-1}{10}+\frac{7}{10}i$.
A complex number z = a +bi can also be expressed in polar form, which uses its modulus (magnitude) r and argument (angle) θ:
|z1-z2| is the distance between these two points in the complex plane. |Re(z)| ≤ |z|, |Im(z)| ≤ |z|, $z · \bar z = |z|², |z_1+z_2| \le |z_1| + |z_2|$ and the equality holds if and only if both complex numbers lie on the same half ray through the origin in the complex plane.
Proof:
$|z_1+z_2|² = (z_1+z_2) · \overline{z_1+z_2} = (z_1+z_2)(\overline{z_1}+\overline{z_2}) = z_1·\overline{z_1} + z_2· \overline{z_2} + z_1·\overline{z_2} + z_2· \overline{z_1} = |z_1|² + |z_2|² + z_1·\overline{z_2} + \overline{z_1·\overline{z_2}} = |z_1|² + |z_2|² + 2Re(z_1·\overline{z_2}) \le |z_1|² + |z_2|² + 2|z_1·\overline{z_2}| = |z_1|² + |z_2|² + 2|z_1||\overline{z_2}| = |z_1|² + |z_2|² + 2|z_1||z_2| = (|z_1| + |z_2|)², \text{ hence } |z_1+z_2| \le |z_1| + |z_2|$
By induction, it can be demonstrated that $|z_1 + z_2 + \cdots z_n| \le |z_1| + |z_2| \cdots |z_n|$
Furthermore, $|z_1| = |z_1 - z_2 + z_2| \le |z_1 - z_2| + |z_2| \leadsto |z_1| - |z_2| \le |z_1 - z_2| \leadsto |z_1 - z_2| \ge |z_1| - |z_2|, \text{ and mutatis mutandis } |z_1 - z_2| \ge |z_2| - |z_1| \leadsto |z_1 - z_2| \ge \big||z_2| - |z_1|\big|$
The polar form of a complex number is: z = a + bi = rcos(θ) + rsin(θ)i = r(cos(θ) + isin(θ)) =[This can be written compactly using Euler’s formula], reiθ, e.g., z = 1 + i = $\sqrt{2}e^{\frac{π}{4}i}$. It satisfies ei(θ+β) = eiθeiβ.
Euler’s formula is a fundamental formula in complex analysis. It establishes the mathematical relationship between trigonometric functions and the complex exponential function. It states that eiθ = cos(θ) + isin(θ). This allows us to represent complex numbers compactly in exponential form. ei
Given z = |z|eiθ, w = |w|eiβ, zw = |z|eiθ|w|eiβ =[ei(θ+β) = eiθeiβ] |z||w|ei(θ+β) = |zw|ei(θ+β)
Given a complex number, it can be expressed as z = |z|eiθ1 = |z|eiθ2 ⇒ eiθ1 = eiθ1 ⇒ cos(θ1) + isin(θ1) = cos(θ2) + isin(θ2) ⇒ cos(θ1) + isin(θ1) = cos(θ2) + isin(θ2) ⇒ cos(θ1) = cos(θ2) and sin(θ1) = sin(θ2) ⇒ θ2 - θ1 ∈ 2πℤ
In complex analysis, the argument (arg) of a complex number z = a+bi is the angle θ formed with the positive real axis. However, because angles can differ by multiples of 2π radians, the argument isn’t unique.
It satisfies $\text{arg}(z_1)+\text{arg}(z_2) = {θ_1+θ_2, θ_1 ∈ \text{arg}(z_1), θ_2 ∈ \text{arg}(z_2)} = \text{arg}(z_1z_2)$. This property holds because the argument function $\text{arg}(z)$ considers all possible angles differing by multiples of 2π. For any complex numbers z1 and z2, this property ensures that the sum of their arguments corresponds correctly to the argument of their product, accounting for the periodic nature of angles.
The same does not apply to $\text{Arg}(z_1)+\text{Arg}(z_2)$ which is typically restricted to $-\pi < \text{Arg}(z) \leq \pi$. Since $\text{Arg}(z)$ is confined to a specific range, the sum $\text{Arg}(z_1) + \text{Arg}(z_2)$ can fall outside this range.
If z ≠ 0, $z\bar z = |z|² ⇒ z(\frac{\bar z}{|z|²}) = 1$ ⇒ Hence, the multiplicative inverse of z is $z^{-1} =\frac{\bar z}{|z|²}$, e.g., $(1+i)^{-1} =\frac{1-i}{|\sqrt{2}|²} = \frac{1-i}{2}$. Alternatively, $z = \sqrt{2}e^{\frac{π}{4}i}, z^{-1} =\frac{1}{z} = \frac{1}{\sqrt{2}e^{i\frac{π}{4}}} = \frac{e^{-\frac{π}{4}i}}{\sqrt{2}}$
This last step need to be demonstrated. Let $z = e^{iθ} ≠ 0, z^{-1}=\frac{\bar z}{|z|²} = \frac{cos(θ)-sin(θ)i}{cos(θ)²+sin(θ)²} = e^{-iθ}$. Hence, $z^{-1} = e^{-iθ}$.
z = -1 -i = $\sqrt{2}e^{\frac{5π}{4}i}, z^{100} =[ \text{De Moivre’s Theorem } ] \sqrt{2}^{100}e^{\frac{5π∙100}{4}i} = 2^{50}e^{125πi} = 2^{50}(e^{πi})^{125} = [e^{πi} = -1] -2^{50}$
The equation |z - a| = r represents a circle in the complex plane (the distance between any point z on the circle and the center a is equal to the radius r) where:
This is analogous to the equation of a circle in the real plane: (x - h)² + (y - k)² = r², where (h, k) is the center and r is the radius.
In particular, if a = 0 (i.e., the center is at the origin), the equation simplifies to |z| = r. If a is real (i.e., the center is on the real axis), we get |z - a| = r.
Let $z_1 = r_1(cos(\theta_1) + isin(\theta_1)), z_2 = r_2(cos(\theta_2) + isin(\theta_1))$. Then, $z_1 · z_2 = r_1r_2(cos(\theta_1 + \theta_2) + isin(\theta_1+\theta_2)), arg(z_1·z_2) = arg(z_1)+arg(z_2)$. In words, the modulus of the product is the product of the moduli and the argument of the product is the sum of the arguments (Figure 1).
In particular, $z = r(cos(\theta) + isin(\theta)), z² = r²(cos(2\theta)+isin(2\theta)).$ Let’s prove that $z^n = r^n(cos(n\theta)+isin(n\theta))$
Case base. It is obviously a true statement for n = 0 and 1.
Suppose that it holds for n = k, $z^k = r^k(cos(k\theta)+isin(k\theta)) \leadsto z^{k+1} = z^k·z = r^k(cos(k\theta)+isin(k\theta))r(cos(\theta)+isin(\theta)) = r^{k+1}[cos(k\theta)cos(\theta)-sin(k\theta)sin(\theta) + i(cos(k\theta)sin(\theta) + sin(k\theta)cos(\theta))] = r^{k+1}(cos((k+1)\theta) + i(sin(k+1))\theta).$ ∎
n = -1, $z^{-1}=\frac{1}{z} = \frac{1}{r(cos(\theta)+isin(\theta))}= \frac{1}{r(cos(\theta)+isin(\theta))}\frac{cos(\theta)-isin(\theta)}{cos(\theta)-isin(\theta)} = \frac{cos(\theta)-isin(\theta)}{r} = r^{-1}(cos(\theta)-isin(\theta)) = r^{-1}(cos(-\theta)+isin(-\theta))$
$z^{-n} = (z^n)^{-1} = \frac{1}{z^n} =[\text{By previous demonstration, the equality holds } \forall n \in \mathbb{N}] \frac{1}{r^n(cos(n\theta)+i(sin(n\theta)))} = r^{-n}\frac{1}{cos(n\theta)+i(sin(n\theta))}\frac{cos(n\theta)-i(sin(n\theta))}{cos(n\theta)-i(sin(n\theta))} = r^{-n}\big(cos(n\theta)-i(sin(n\theta))\big) = r^{-n}\big(cos(-n\theta)+i(sin(-n\theta))\big)$
De Moivre’s Theorem. $z^n = r^n(cos(n\theta)+isin(n\theta))$ and the equality holds $\forall n \in \mathbb{Z}.$
Alternatively, given any arbitrary real number x and a integer n, (cos(x)+isin(x))n = cos(nx) + isin(nx). One can derive de Moivre’s formula using Euler’s formula and the exponential law for integer powers, (eix)n = einx.
z = -3 = 3(cos(π)+isin(π)) = 3(cos(2kπ + π)+isin(2kπ + π)) = 3eiπ, $\forall k \in \mathbb{Z}$. We are interested on calculating $w=z^{\frac{1}{4}} \leadsto w = 3^{\frac{1}{4}}e^{\frac{1}{4}} = 3^{\frac{1}{4}}\big(cos(\frac{2k\pi +\pi}{4})+isin(\frac{2k\pi + \pi}{4})\big), \forall k \in \mathbb{Z}$
For k = 0, 1, 2, and 3, we get all the possible values of w, that is, $w = 3^{\frac{1}{4}}(\plusmn \frac{1}{\sqrt{2}} \plusmn \frac{i}{\sqrt{2}})$.
We embed the complex plane ℂ into ℝ³ as the xy-plane: z = x +iy ↭ (x, y) ↭ (x, y, 0) where (x, y, 0) ia a point in the x-y plane in ℝ³.
Consider the unit sphere in ℝ³, S = {(x, y, z): x² + y² + z² = 1}. The north pole of the sphere is the point N = (0, 0, 1). So this set intersects the x-y plane in the unit circle. When z = 0, x² + y² = 1.
Given any arbitrary number in ℝ³, we can trace a ray to the north pole of the unit sphere that hits the unit sphere S at some point, and this point is unique. The stereographic projection maps a point z = (x, y, 0) in the complex plane to a point Z = (x₁, y₁, z₁) on the unit sphere S by drawing a straight line (a ray) from the north pole N through z until it intersects S. (Figure 2)
The line connecting N(0, 0, 1) and z(x, y, 0) can be parameterized as: Z = {t(0, 0, 1) + (1-t)(x, y, 0), ∀t∈ℝ}. t(0, 0, 1) + (1-t)(x, y, 0) ↭ $\big((1-t)x, (1-t)y, t\big)$. To find the intersection point, we substitute the coordinates of Z(t) into the equation of the sphere: (1-t)²x² + (1-t)y² + t² = 1 ↭ (1-t)²|z|² + t² = 1 ↭[Solving for t] 1 -t² = (1-t)²|z|²
t ≠ 1 since this corresponds to (0, 0, 1), 1 -t² = |z|² + t²|z|² -2t|z|² ↭ t²(|z|²+1) -2t|z|² + |z|² - 1 = 0. Using the quadratic equation, we get $t = \frac{2|z|² \plusmn \sqrt{4|z|⁴-4(|z|²+1)(|z|² - 1)}}{2(|z|²+1)} = \frac{2|z|² \plusmn \sqrt{4z⁴-4z⁴+ 4}}{2(|z|²+1)} = \frac{2|z|² \plusmn 2}{2(|z|²+1)}$ =[We have two solutions for t. The solution t = 1 corresponds to the north pole N itself. Since we are projecting from the north pole, we are interested in the other solution.],$t \ne 1 \frac{|z|²-1}{|z|² +1}$
$z \in S \leadsto (\frac{(|z|²+1-|z|²+1)x}{|z|²+1}, \frac{(|z|²+1-|z²|+1)y}{|z|²+1}, \frac{|z|²-1}{|z|²+1}) = (\frac{2x}{|z|²+1}, \frac{2y}{|z|²+1}, \frac{|z|²-1}{|z|²+1}) = (\frac{z + \bar z}{|z|²+1}, \frac{-i(z -\bar z)}{|z|²+1}, \frac{|z²|-1}{|z|²+1})$
Inverse Projection: Given a point (x₁, y₁, z₁) on the sphere (other than the north pole, where z₁ ≠ 1), we can find the corresponding complex number z in the plane: If we are given a point ($Z \ne (0, 0, 1) \leadsto z_1 \ne 1$) on S, then z on ℂ can be found $z = \frac{x_1+iy_1}{1-z_1}$. Notice that $\frac{x_1 + iy_1}{1-z_1} = (\frac{z + \bar z}{|z|²+1} +\frac{z -\bar z}{|z|²+1})·\frac{1}{\frac{|z|²+1}{2}} = \frac{2z}{|z|²+1}·\frac{|z|²+1}{2} = z$
The stereographic projection establishes a one-to-one correspondence between the complex plane ℂ and the unit sphere S excluding the north pole N. To complete the correspondence, we add a . This extended complex plane ℂ ∪ {∞} is called the Riemann sphere.
The chordal metric (or spherical metric) defines a distance between two points on the Riemann sphere (and therefore between two complex numbers, including ∞).
If z and z’ are finite complex numbers, the chordal distance is the Euclidean distance between their projections on the sphere:
$d(z, z’) = \begin{cases} d(Z, Z’), &z, z’ \ne \infty\\ \frac{2}{\sqrt{1+|z|²}}, &z’ = \infty \end{cases}$
$d(z, z’) = \begin{cases} \sqrt{(x_1-x_1’)²+(x_2-x_2’)²+(x_3-x_3’)²}, &z, z’ \ne \infty\\ \frac{2}{\sqrt{1+|z|²}}, &z’ = \infty \end{cases}$
This correspondence along with the distance is called the stereographic projection. We define the operations: a + ∞ = ∞, a·∞ = ∞ (a ≠ 0), a/∞ = 0, a/0 = ∞ (a ≠ 0), a ∈ ℂ
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