# Operations on complex numbers

## • Explore how the new number, i, behaves under certain operations (N-CN.A.2).  • Perform operations with complex numbers and draw conclusions about patterns that emerge (N-CN.A.2).

With a new type of number in their world, students can begin exploring the properties of these numbers and how to add, subtract, multiply and divide them using the properties of operations to guide the work. They explore patterns that emerge when $i$ is raised to positive integer powers, and build on this work by investigating similar patterns with other complex numbers.

1 Computations with Complex Numbers

WHAT: Students calculate the value of various numerical expressions involving complex numbers, including expressions where attending to structure provides more efficient ways of doing the calculation (MP.7).

WHY: The purpose of this task is to give students practice performing operations with complex numbers using the properties of operations and the fact that $i^2 = −1$. It also leads students to see that they can use methods like factoring out a common factor with complex numbers just as they did with real numbers, reinforcing the idea that the complex numbers are simply an extension of the number system they are familiar with.

2 Powers of a complex number

WHAT: In this task students find powers of 1 + i and discover a pattern that enables them to calculate $(1 + i)^{100}$.

WHY: The purpose of this task is to foster students’ interest in complex numbers by giving an example of one remarkable property of the new system: the powers of a number cycle around the complex plane, sometimes in a regular pattern. This activity brings together everything that we expect students to know about operations on complex numbers and so it is placed here to bring everything from the last two sections together.