## Getting Started

Complex numbers provide a solution to finding the square root of a negative number, and 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 \(i^2=-1\). Start by computing with complex numbers in the scientific calculator, then explore plotting them in the graphing calculator!

## Complex Mode

In the Settings menu of the Scientific, Graphing, and 3D calculators, you can 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. Look for real(), imag(), conj(), arg() and | | (for modulus).

## Functions for Complex Numbers

Function | Try typing... | This function plots or finds the... |
---|---|---|

real | real\((2+4i)\) | real part of a complex number. |

imaginary | imag\((2+4i)\) | imaginary part of a complex number (multiple of i). |

conjugate | conj\((2+4i)\) | 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\((2+4i)\) | 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 | |2+4i| | the distance between the origin and the complex value’s corresponding point. |

## Computing with Complex Numbers

While in Complex Mode, 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

In the Graphing Calculator, complex numbers are plotted as points in the plane by converting them to the point \((\)real\(( )\),imag\(( ))\), where the x-coordinate represents the real part and the y-coordinate represents the imaginary part of the complex number. You can label complex numbers, make them draggable, and style them in the * Options Menu*.

## 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.