The system of real numbers is insufficient to solve quadratic equations of the form $ax^2 + bx + c = 0$ where the discriminant $D = b^2 - 4ac < 0$. For instance, the simple equation $x^2 + 1 = 0$ has no real solution because the square of any real number is non-negative. To overcome this limitation, the real number system is extended to a larger system called the complex number system.
We denote the square root of negative unity, $\sqrt{-1}$, by the symbol $i$ (iota), such that $i^2 = -1$. This means $i$ is a solution to the equation $x^2 + 1 = 0$.
A number of the form $z = a + ib$, where $a$ and $b$ are real numbers, is defined to be a complex number.
For example, if $z = 2 + i5$, then $\text{Re } z = 2$ and $\text{Im } z = 5$.
Two complex numbers $z_1 = a + ib$ and $z_2 = c + id$ are defined to be equal if and only if their corresponding real and imaginary parts are equal:
$$z_1 = z_2 \iff a = c \text{ and } b = d$$
We define fundamental algebraic operations—addition, subtraction, multiplication, and division—on the set of complex numbers.
Let $z_1 = a + ib$ and $z_2 = c + id$ be two complex numbers.
The addition of complex numbers satisfies the following properties:
Let $z_1 = a + ib$ and $z_2 = c + id$.
Multiplication satisfies the following properties:
The powers of $i$ exhibit a cyclic pattern of length 4:
In general, for any integer $k$:
$$i^{4k} = 1, \quad i^{4k+1} = i, \quad i^{4k+2} = -1, \quad i^{4k+3} = -i$$
Negative powers of $i$ can be computed similarly, e.g., $i^{-1} = \frac{1}{i} = -i$, $i^{-2} = -1$, $i^{-3} = i$, $i^{-4} = 1$.
For any positive real number $a$, the square roots of $-a$ are $\sqrt{a}i$ and $-\sqrt{a}i$. The symbol $\sqrt{-a}$ is defined to represent $\sqrt{a}i$ only.
Critical Caveat: The identity $\sqrt{a} \times \sqrt{b} = \sqrt{ab}$ holds true when at least one of $a$ or $b$ is non-negative. However, if both $a < 0$ and $b < 0$, then:
$$\sqrt{a} \times \sqrt{b} = (\sqrt{|a|}i) \times (\sqrt{|b|}i) = \sqrt{|ab|}i^2 = -\sqrt{ab} \neq \sqrt{ab}$$
Let $z = a + ib$ be a complex number.
For any complex numbers $z, z_1, z_2$:
| Property Type | Mathematical Identity / Formula | Operational Meaning |
|---|---|---|
| Inverse Relation | $z \bar{z} = |z|^2$ or $z^{-1} = \frac{\bar{z}}{|z|^2}$ | Relates a complex number, its conjugate, and its modulus square. |
| Product Modulus | $|z_1 z_2| = |z_1| |z_2| | The modulus of a product is the product of the moduli. |
| QN Modulus | $\left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|}$ (for $z_2 \neq 0$) | The modulus of a quotient is the quotient of the moduli. |
| Product Conjugate | $\overline{z_1 z_2} = \bar{z}_1 \bar{z}_2$ | The conjugate of a product is the product of the conjugates. |
| Sum/Difference Conjugate | $\overline{z_1 \pm z_2} = \bar{z}_1 \pm \bar{z}_2$ | The conjugate distributes over addition and subtraction. |
| Quotient Conjugate | $\overline{\left(\frac{z_1}{z_2}\right)} = \frac{\bar{z}_1}{\bar{z}_2}$ (for $z_2 \neq 0$) | The conjugate distributes over division. |
Standard algebraic identities hold true for all complex numbers $z_1$ and $z_2$:
The Argand plane (or complex plane) is a geometric representation where each complex number $z = x + iy$ is uniquely mapped to an ordered pair $(x, y)$ and plotted as a point $P(x, y)$ in a two-dimensional Cartesian coordinate system.
Mathematical Problem Statement: Find the real values of $x$ and $y$ if $4x + i(3x - y) = 3 + i(-6)$.
Step-by-Step Algorithmic Solution Trace:
- Equate the real parts on both sides of the equation:$$4x = 3 \implies x = \frac{3}{4}$$
- Equate the imaginary parts on both sides of the equation:$$3x - y = -6$$
- Substitute the value of $x$ into the imaginary equation:$$3\left(\frac{3}{4}\right) - y = -6 \implies \frac{9}{4} - y = -6$$
- Solve for $y$:$$y = \frac{9}{4} + 6 = \frac{9 + 24}{4} = \frac{33}{4}$$
Final Evaluation Output: $x = \frac{3}{4}, \quad y = \frac{33}{4}$
Mathematical Problem Statement: Find the multiplicative inverse of $z = 2 - 3i$.
Step-by-Step Algorithmic Solution Trace:
- Identify the real and imaginary parts: $a = 2$, $b = -3$.
- Compute the modulus squared $|z|^2$:$$|z|^2 = a^2 + b^2 = 2^2 + (-3)^2 = 4 + 9 = 13$$
- Find the conjugate $\bar{z}$:$$\bar{z} = 2 + 3i$$
- Apply the multiplicative inverse formula $z^{-1} = \frac{\bar{z}}{|z|^2}$:$$z^{-1} = \frac{2 + 3i}{13} = \frac{2}{13} + i\frac{3}{13}$$
Final Evaluation Output: $z^{-1} = \frac{2}{13} + i\frac{3}{13}$
Mathematical Problem Statement: If $x + iy = \frac{a + ib}{a - ib}$, prove that $x^2 + y^2 = 1$.
Step-by-Step Algorithmic Solution Trace:
- Take the modulus on both sides of the given equation:$$|x + iy| = \left|\frac{a + ib}{a - ib}\right|$$
- Use the quotient property of modulus $|\frac{z_1}{z_2}| = \frac{|z_1|}{|z_2|}$:$$|x + iy| = \frac{|a + ib|}{|a - ib|}$$
- Calculate the moduli:$$\sqrt{x^2 + y^2} = \frac{\sqrt{a^2 + b^2}}{\sqrt{a^2 + (-b)^2}} = \frac{\sqrt{a^2 + b^2}}{\sqrt{a^2 + b^2}} = 1$$
- Square both sides to obtain the final result:$$x^2 + y^2 = 1$$
Final Evaluation Output: $x^2 + y^2 = 1$ (Proved)
Historical Chronicle / Epigraph: Evolution of Imaginary Numbers