# [[Magnitude of a Complex Number]] (Mathematics > Complex Numbers) ## Definition The magnitude (or modulus) of a complex number $z = a + bi$, where $a$ is the real part and $b$ is the imaginary part, is the distance from the origin to the point $(a, b)$ in the complex plane. It is denoted as $|z|$ and defined by: $ |z| = \sqrt{a^2 + b^2} $ This gives the length of the vector representing $z$. ## Key Concepts - A complex number $z$ has the form $a + bi$, where $a$ and $b$ are real numbers. - The magnitude is a non-negative real number. - The magnitude represents the distance of the complex number from the origin in the complex plane. ## Important Properties 1. $|z| \geq 0$ for all $z$, and $|z| = 0$ if and only if $z = 0$. 2. $|z_1 z_2| = |z_1| \cdot |z_2|$ (multiplicative property). 3. $|z_1 + z_2| \leq |z_1| + |z_2|$ (triangle inequality). ## Essential Formulas - Magnitude of $z = a + bi$: $ |z| = \sqrt{a^2 + b^2} $ - If $z = r e^{i\theta}$ in polar form, then $|z| = r$. ## Core Examples 1. **Basic Example**: Find the magnitude of $z = 3 + 4i$: $ |z| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 $ 2. **Advanced Example**: Find the magnitude of $z = -1 + i\sqrt{3}$: $ |z| = \sqrt{(-1)^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2 $ ## Related Theorems/Rules - **Triangle Inequality**: For any two complex numbers $z_1$ and $z_2$, the inequality $|z_1 + z_2| \leq |z_1| + |z_2|$ holds. - **Multiplicative Property**: The magnitude of the product of two complex numbers is the product of their magnitudes, $|z_1 z_2| = |z_1| \cdot |z_2|$. ## Common Pitfalls - Confusing the magnitude with the real or imaginary part of the complex number. - Forgetting to square both the real and imaginary parts when calculating the magnitude. ## Related Topics - [[Complex Conjugate]] - [[Polar Form of Complex Numbers]] ## Quick Review Questions 1. What is the magnitude of the complex number $z = 1 + i$? 2. How does the magnitude of a complex number relate to its polar form?