The Polar Form of Complex Number Calculator is a tool designed to convert complex numbers from their standard rectangular form (a + bi) to the polar form (r∠θ) and vice versa. This conversion is crucial in various fields such as engineering, physics, and mathematics, where complex numbers are frequently utilized to represent electrical circuits, waveforms, and geometric transformations.
Formula of Polar Form of Complex Number Calculator
Magnitude (r):
r = sqrt(a^2 + b^2)
Where:
- a is the real part of the complex number.
- b is the imaginary part of the complex number.
- sqrt() denotes the square root function.
Argument (θ):
θ = arctan(b / a)
Where:
- arctan() denotes the arctangent function.
- a is the real part of the complex number.
- b is the imaginary part of the complex number.
General Terms Table
To assist users in better understanding and utilizing the calculator, here is a table of general terms:
Rectangular Form (a + bi) | Polar Form (r ∠ θ) |
---|---|
1 + 0i | 1 ∠ 0° |
0 + 1i | 1 ∠ 90° |
-1 + 0i | 1 ∠ 180° |
0 – 1i | 1 ∠ 270° |
1 + 1i | √2 ∠ 45° |
1 – 1i | √2 ∠ 135° |
-1 + 1i | √2 ∠ 225° |
-1 – 1i | √2 ∠ 315° |
√3 + i | 2 ∠ 30° |
√3 – i | 2 ∠ 150° |
-√3 + i | 2 ∠ 210° |
-√3 – i | 2 ∠ 330° |
Note: This table includes conversions for basic and frequently used complex numbers. Additional conversions can be easily calculated using the formulas provided in the article.
Example of Polar Form of Complex Number Calculator
Let’s consider a complex number with \(a = 3\) and \(b = 4\).
Magnitude:
\[ r = \sqrt{3^2 + 4^2} = 5 \]
Argument:
\[ θ = \arctan\left(\frac{4}{3}\right) \approx 0.93 \, \text{radians} \]
So, the polar form of the complex number is \(5∠0.93\) radians.
Most Common FAQs
A: The calculator simplifies the conversion between rectangular and polar forms of complex numbers, aiding in various mathematical and engineering applications.
A: Yes, polar forms are extensively used in electrical engineering for representing alternating current (AC) circuits, among other applications.
A: The calculator provides precise results based on the mathematical formulas for magnitude and argument of complex numbers.