Reviewed by: David Chen, CFA
Welcome to the **Complex Number Graphing Calculator**! This tool allows you to instantly convert between the Rectangular Form ($x+iy$) and the Polar Form ($r\angle\theta$) of any complex number, which is essential for engineering, physics, and advanced mathematics.
Complex Number Conversion Calculator
Input *exactly two* known values (either Real and Imaginary, or Magnitude and Angle) to calculate the remaining two.
Complex Number Graphing Calculator Formula
The relationship between the rectangular form ($x+iy$) and the polar form ($r e^{i\theta}$) of a complex number is defined by Euler’s formula and the Pythagorean theorem.
Rectangular to Polar Conversion:
Magnitude: $r = \sqrt{x^2 + y^2}$
Angle: $\theta = \operatorname{atan2}(y, x)$ (in radians)
$\theta_{deg} = \theta \times \frac{180}{\pi}$
Polar to Rectangular Conversion:
Real Part: $x = r \cos(\theta)$
Imaginary Part: $y = r \sin(\theta)$
Formula Sources:
Variables
- Real Part ($x$): The horizontal coordinate on the complex plane.
- Imaginary Part ($y$): The vertical coordinate on the complex plane.
- Magnitude ($r$): The distance from the origin (0, 0) to the point $(x, y)$. Also called the modulus.
- Phase Angle ($\theta$): The angle, in degrees, between the positive x-axis and the line segment connecting the origin to the point $(x, y)$.
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What is a Complex Number Graphing Calculator?
A complex number graphing calculator is a tool designed to visualize and manipulate complex numbers, which are numbers of the form $a+bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit ($\sqrt{-1}$). Graphing these numbers involves plotting them on a two-dimensional complex plane, where the horizontal axis represents the real part ($a$) and the vertical axis represents the imaginary part ($b$).
The primary function of this calculator is to facilitate conversion between the two main representations: the Rectangular (Cartesian) form ($x+iy$) and the Polar (Exponential) form ($r e^{i\theta}$). This conversion is vital in alternating current (AC) circuit analysis, quantum mechanics, and signal processing because multiplication and division are far simpler in polar form, while addition and subtraction are simpler in rectangular form.
By providing either the Cartesian coordinates $(x, y)$ or the Polar coordinates $(r, \theta)$, the calculator determines the equivalent form, instantly giving engineers and students the necessary data points to solve complex problems and plot their results.
How to Calculate Complex Number Conversion (Example)
Let’s convert the rectangular complex number $z = 3 + 4i$ to its polar form.
- Identify Inputs: The Real Part ($x$) is 3, and the Imaginary Part ($y$) is 4.
- Calculate Magnitude ($r$): Apply the formula $r = \sqrt{x^2 + y^2}$. $r = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$.
- Calculate Phase Angle ($\theta$ in Radians): Use $\theta = \operatorname{atan2}(y, x)$. $\theta = \operatorname{atan2}(4, 3) \approx 0.927$ radians.
- Convert Angle to Degrees: Multiply by $\frac{180}{\pi}$. $\theta_{deg} = 0.927 \times \frac{180}{\pi} \approx 53.13$ degrees.
- Final Result: The polar form is $z = 5 \angle 53.13^\circ$ (or $5 e^{i0.927}$).
Frequently Asked Questions (FAQ)
What is the difference between rectangular and polar forms?
Rectangular form ($x+iy$) expresses a complex number by its horizontal (Real) and vertical (Imaginary) components. Polar form ($r\angle\theta$) expresses it by its distance from the origin (Magnitude, $r$) and its direction (Angle, $\theta$).
Why is the angle usually measured in degrees for these calculators?
While standard mathematical formulas often use radians, engineering and physics applications (especially AC analysis) commonly use degrees for easier human interpretation, particularly in phase diagrams.
Can I input all four values at once?
The calculator requires *exactly two* inputs. If you input all four, the calculator will check if your inputs are mathematically consistent. If they are not (due to rounding or errors), it will show an inconsistency warning.
What is the purpose of the $\operatorname{atan2}$ function?
The $\operatorname{atan2}(y, x)$ function is a variation of $\arctan(y/x)$ that correctly determines the quadrant of the angle, ensuring the resulting phase angle is between $-180^\circ$ and $+180^\circ$ (or $0^\circ$ and $360^\circ$), which $\arctan(y/x)$ cannot do alone.