## Calculating Distance in the Complex Plane

#### Aligned To Common Core Standard:

**High School** - HSN-CN.B.6

How to Calculate Distance in the Complex Plane?
In the myriad of number systems that we come across in mathematics, there is one type that students might find challenging to deal with - complex numbers.
Complex numbers are a combination of real and imaginary numbers.
The set of real numbers consists of all rational and irrational numbers. The imaginary part is the one that we write with iota. These are square roots of negative numbers, and iota represents -1 in square root.
When graphing these values, you need to understand the plane you are placing them on. It is similar to the cartesian coordinate plane, but the x-axis is known as the real axis while the y-axis is known as the imaginary axis. Calculating the distance between on this plane is not difficult.
Finding the distance between two points on a complex plane is easy and requires the use of a simple formula;
Distance=√((difference between real part)^{2} + (difference between imaginary part)^{2})
Consider the following complex numbers; 0 + 0i and a + bi. In this case, you have to calculate the distance between origin and the placed coordinate. The modulus of the complex number is the distance between the origin and that value;
|a+bi| = √(a^{2} + b^{2} )
For the following two coordinates in the plane; a + bi and s + ti, is given by;
Distance=√((s-a)^{2} + (t- b)^{2} )
This selection of worksheets shows students how to determine distance on a complex plane which gives us another method to better understand these types of values.

### Printable Worksheets And Lessons

- Distance Between Points
Step-by-step Lesson- Graph two complex numbers and then find
the distance between those points.

- Guided Lesson -
Plenty of places to plop your graph down on.

- Guided Lesson Explanation
- These can take a good bit of time so make sure that you understand
that before planning a lesson.

- Practice Worksheet
- Once again, you'll need extra graph paper for this one. I just
didn't have any space for them.

- Matching Worksheet - Match the difference in distance between two complex numbers. Obviously you will need more graph paper.

#### Homework Sheets

Make sure to graph what you are given, first.

- Homework 1 - We graph the points and connect them with a line; as you see in the practice problem.
- Homework 2 - The x-axis represents the real number and the y-axis represents the imaginary number.
- Homework 3 - Draw lines to make a right triangle. This will be what you build everything off of.

#### Practice Worksheets

Make sure to go over how the negative symbol affects the outcome.

- Practice 1 - What is the distance between (25 - 125i) and (100 - 25i) on the complex plane?
- Practice 2 - Use Pythagorean Theorem to find the length of two points, imaginary number and real number. Calculate distance in the complex plane.
- Practice 3 - Start by plotting the points.

#### Math Skill Quizzes

Don't let the (+ -) symbols confuse you.

### Where in the World will You Ever Use This Skill?

We have used a similar concept to find the distance between two points on a standard coordinate system. We did this to model differences in position on a map or diagram of real-world situation. For example: Sarah is at school. Using a map, how many blocks is she from her home? Once we applied this skill to this type of problem, students start to see value in it. The same can be said when your values include complex numbers. A complex plane mirrors a standard coordinate system. You can position complex numbers on these grids in much the same way. The complex numbers include imaginary numbers, and this is where everyone gets kind of lost. If we can get students to understand that the imaginary number is just as real as a real number, this becomes much easier. Engineers and scientists use this technique model and understand systems much like the road map we discussed earlier. It is just used for many different applications like modelling the movement of particles in atoms or movement of electricity in a device.