Complex numbers provide a solution to finding the square root of a negative number. They are essential for modeling and solving problems in a variety of fields, including engineering, physics, statistics, chemistry, and mathematics.
Complex numbers are written in the form \(a+bi\), where \(a\) is the real part, and \(b\) is the imaginary part. The imaginary number 𝑖 is defined as the square root of \(-1\) \(\left(\sqrt{-1}\right)\), therefore \(i^2=-1\).
Complex Mode
In the Settings menu of the Scientific, Graphing, and 3D calculators, click Graph Settings and then toggle on Complex Mode . This adds the imaginary number \(i\) to the keypad and switches all angles to radians. In complex mode, you’ll also have access to functions for finding the real and imaginary parts, conjugate, argument, and modulus of a complex value.
Functions for Complex Numbers
| Function | Try Typing... | This function plots or finds the... |
|---|---|---|
| real | real\(\mathbf{(2+4i)}\) | real part of a complex number. |
| imaginary | imag\(\mathbf{(2+4i)}\) | imaginary part of a complex number (multiple of \(i\)). |
| conjugate | conj\(\mathbf{(2+4i)}\) | complex conjugate by changing the sign of the imaginary part of a complex number. This is often used when dividing by complex numbers. |
| argument | arg\(\mathbf{(2+4i)}\) | angle inclined from the positive real axis to the line from the origin to the complex value. In Desmos, this angle is calculated in radians within the interval \((-\pi,\pi]\). |
| modulus, or absolute value | \(\mathbf{|2+4i|}\) | the distance between the origin and the complex value’s corresponding point. |
Computing with Complex Numbers
With complex mode on, try adding two complex numbers in the Scientific Calculator. This adds the real and imaginary parts separately which results in another complex number. For example,
\((2 + 4i)+(1 + 5i)=(2 + 1)+(4+5)i=3+9i\)
You can also use the calculator to easily subtract, multiply, and divide complex numbers.
Multiplying any complex number by its conjugate results in a real number. Explore why using the distributive property.
\((2+4i)\)conj\((2+4i)=(2+4i)(2−4i)\)
\(=4+8i−8i−16i^2\)
\(=4+16=20\)
Graphing Complex Numbers
The Graphing Calculator plots complex numbers as points in the plane by converting them to the point where the \(x\)-coordinate represents the real part and the \(y\)-coordinate represents the imaginary part of the complex number. You open the Options menu to label complex numbers, make them draggable, and style them.
You can use the Graphing Calculator to explore what happens when you add two complex numbers. Notice that the resulting sum forms a parallelogram with four vertices: the origin, the two complex numbers, and their sum. Use the example graph to explore a geometric representation of complex number addition.
Multiplying two complex numbers multiplies their magnitudes and adds their angles. To find the length (or magnitude) of a complex number, use the modulus operator | |. To determine the angle, use the arg function.
Note: While working in complex mode, all angles are displayed in radians.
Learn More
- Scientific Calculator
- Getting Started: Desmos Graphing Calculator
- Graphing and Connecting Coordinate Points
Please reach out to us with any questions or suggestions at support@desmos.com.