a+bi Form Calculator

Understanding the $a + bi$ Standard Form of Complex Numbers

In algebra and mathematics, any complex number $z$ can be represented in standard rectangular (Cartesian) form as: $$z = a + bi$$ where $a = \text{Re}(z)$ represents the real part, $b = \text{Im}(z)$ represents the imaginary part, and $i$ is the imaginary unit defined by $i^2 = -1$.

On the complex plane (Argand diagram), the real part $a$ corresponds to the horizontal axis (Re), and the imaginary part $b$ corresponds to the vertical axis (Im). Every complex number is a unique 2D coordinate point $(a, b)$ or a vector originating from $(0, 0)$.

How to Convert Polar Form to $a + bi$ Form

When a complex number is given in polar coordinates $(r, \varphi)$ or exponential form $r \cdot e^{i\varphi}$, where $r \ge 0$ is the magnitude (modulus) and $\varphi$ is the argument (phase angle):

1. Calculate the Real Part ($a$): $$a = r \cdot \cos(\varphi)$$
2. Calculate the Imaginary Part ($b$): $$b = r \cdot \sin(\varphi)$$
3. Combine into Rectangular Form: $$z = a + bi = r(\cos\varphi + i\sin\varphi)$$

How to Convert $a + bi$ Form to Polar Coordinates

Given a complex number $z = a + bi$:

• Modulus / Magnitude ($r$ or $|z|$): The Euclidean distance from the origin to the point $(a, b)$: $$r = |z| = \sqrt{a^2 + b^2}$$
• Argument / Phase Angle ($\varphi$): The counterclockwise angle from the positive real axis: $$\varphi = \text{atan2}(b, a) = \begin{cases} \arctan(b/a) & \text{if } a > 0 \\ \arctan(b/a) + \pi & \text{if } a < 0 \text{ and } b \ge 0 \\ \arctan(b/a) - \pi & \text{if } a < 0 \text{ and } b < 0 \\ +\pi/2 & \text{if } a = 0 \text{ and } b > 0 \\ -\pi/2 & \text{if } a = 0 \text{ and } b < 0 \\ \text{undefined} & \text{if } a = 0 \text{ and } b = 0 \end{cases}$$

Additional Complex Number Operations

This calculator also provides key derived complex quantities:

• Complex Conjugate ($\bar{z}$): Reflects the number across the real axis: $\bar{z} = a - bi$.
• Reciprocal ($1/z$): $\frac{1}{z} = \frac{a - bi}{a^2 + b^2} = \frac{\bar{z}}{|z|^2}$ (for $z \neq 0$).
• Square ($z^2$): $z^2 = (a^2 - b^2) + 2abi$.
• Exponential Form: Expressed via Euler's formula as $r \cdot e^{i\varphi}$.

Explore related tools: check out our Expanded Form Calculator, calculate with the ABI Calculator, or use the AGI.