Master complex arithmetic with our Graphing Calculator for Complex Numbers. Perform addition, subtraction, multiplication, and division while visualizing the vectors on the complex plane.
Graphing Calculator Complex Numbers
Graphing Calculator Complex Numbers Formula
Multiplication: (a+bi)(c+di) = (ac-bd) + (ad+bc)i
Magnitude: |z| = √(a² + b²)
Formula Source: Wolfram MathWorld – Complex Numbers
Variables:
- Real Part (a/c): The horizontal coordinate on the complex plane.
- Imaginary Part (b/d): The vertical coordinate, multiplied by the unit imaginary number $i$.
- Magnitude (r): The distance from the origin $(0,0)$ to the point.
- Argument (θ): The angle formed with the positive real axis.
Related Calculators
- Quadratic Equation Solver (Complex Roots)
- Polar to Rectangular Converter
- Vector Addition Calculator
- Matrix Arithmetic Module
What is Graphing Calculator Complex Numbers?
A complex number graphing calculator is a specialized tool that allows mathematicians and students to perform algebraic operations on numbers containing both a real and an imaginary component. Unlike standard calculators, this tool treats numbers as vectors in a two-dimensional space known as the Argand plane.
Visualizing complex numbers is crucial for fields like electrical engineering and physics. When you add two complex numbers, the result follows the parallelogram law of vector addition, which this calculator illustrates graphically.
How to Calculate Complex Numbers (Example)
Suppose you want to multiply $z_1 = 3 + 2i$ and $z_2 = 1 + 4i$:
- Identify variables: $a=3, b=2, c=1, d=4$.
- Apply the multiplication formula: $(ac – bd) + (ad + bc)i$.
- Calculate real part: $(3 \times 1) – (2 \times 4) = 3 – 8 = -5$.
- Calculate imaginary part: $(3 \times 4) + (2 \times 1) = 12 + 2 = 14$.
- Final Result: $-5 + 14i$.
Frequently Asked Questions (FAQ)
What is the imaginary unit $i$?
The unit $i$ is defined as the square root of $-1$.
Can I divide by zero in complex numbers?
No, division by $0 + 0i$ is undefined as it results in a singularity.
How is the phase angle calculated?
The phase (argument) is usually calculated using the `atan2(y, x)` function to find the angle in radians or degrees.
What is a conjugate?
The conjugate of $a + bi$ is $a – bi$. It is used extensively in complex division.