Free Grade 10 Complex Numbers: Operations and Applications Practice

**Definition: The Complex Number System** Within the real number system, the square root of a negative number is undefined. To address this limitation, we extend our number system by defining the **imaginary unit**, denoted by *i*, such that *i* = √-1. This allows for the creation of the complex number system. A **complex number** is any number that can be expressed in the standard form `z = a + bi`, where *a* and *b* are real numbers. The term *a* is designated as the **real part** of the complex number, denoted Re(z), and the term *b* is the **imaginary part**, denoted Im(z). This structure allows a single numerical entity to encapsulate two distinct dimensions of information, much like a coordinate on a plane. The algebraic operations of addition and subtraction on complex numbers are analogous to combining like terms in polynomial expressions. To perform these operations, one simply combines the real parts and, separately, the imaginary parts. This principle is fundamental in fields such as electrical engineering, where complex numbers are used to represent alternating current (AC) circuits; the real part may represent resistance, while the imaginary part represents reactance. The combination of signals or impedances is then a direct application of complex number addition or subtraction. **Theorem: Operations on Complex Numbers** Given two complex numbers, z₁ = a + bi and z₂ = c + di: - **Addition:** `z₁ + z₂ = (a + c) + (b + d)i` - **Subtraction:** `z₁ - z₂ = (a - c) + (b - d)i`

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