Rectangular form

Definition

Rectangular form of a complex number is expressed as $z = a + bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit. It represents the complex number in terms of its horizontal (real) and vertical (imaginary) components.

5 Must Know Facts For Your Next Test

1. In rectangular form, the real part is represented by $a$, and the imaginary part is represented by $b$.
2. The magnitude (or modulus) of a complex number in rectangular form can be found using $|z| = \sqrt{a^2 + b^2}$.
3. Rectangular form can be converted to polar form using the relationships $r = \sqrt{a^2 + b^2}$ and $\theta = \text{tan}^{-1}(\frac{b}{a})$.
4. Addition and subtraction of complex numbers are straightforward in rectangular form: $(a+bi) + (c+di) = (a+c) + (b+d)i$.
5. Multiplication and division of complex numbers in rectangular form often require converting to polar form for simplification.

Review Questions

• How do you represent a complex number in rectangular form?
• What is the formula for finding the magnitude of a complex number in rectangular form?
• How do you convert from rectangular form to polar form?

Related terms

Polar Form: A representation of a complex number as $z = r(\cos \theta + i \sin \theta)$ or $z = re^{i\theta}$, where $r$ is the magnitude and $\theta$ is the argument.

Complex Number: A number that has both a real part and an imaginary part, typically written as $a + bi$, where $i$ is the square root of -1.

Magnitude: $|z|$, also known as modulus, it denotes the distance from the origin to the point $(a,b)$ in the complex plane, calculated as $|z| = \sqrt{a^2 + b^2}$.