High School: Number and Quantity
The Complex Number System HSN-CN.B.6
6. Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints.
The modulus of a complex number a + bi is defined as . This little (or sometimes big) number can be useful in a variety of things having to do with complex numbers.
For starters, what about the distance between two points? In geometry, the distance formula is pretty much our Holy Grail for everything. With complex numbers, it's no different, except instead of x1 and y2, we have a1 and b2. Using these generic values, we get .
But why plug individual a and b values when we can just subtract the two complex numbers directly? If we subtract a1 + b1i from a2 + b2i, we'll get (a2 – a1) + (b2 – b2)i as our answer. Taking the modulus of that will give us .
Whoa. That means the distance between two imaginary points is the same as the modulus of the difference between the complex numbers. Pretty nifty.
Not clicking? Find the distance between 5 + 2i and 6 + 4i to help get your students' imaginary ball rolling.
Rather than plug four numbers into the distance formula, we can subtract them and reduce them to two. Solving 5 + 2i – (6 + 4i) will give us -1 – 2i.
Now, let's find that modulus. If a = -1 and b = -2, the modulus gives us a distance of . Not too shabby.
Finding the midpoint of a segment joining complex numbers is even easier. It's just like finding the midpoint of a segment joining real points. All we need to do is average the a values and average the b values.
So if we want to find the midpoint of the segment joining 3 + 8i and 5 – 2i, we would find the average of the a values:
...and find the average of the b values:
The midpoint of our line segment is 4 + 3i. Who said math was hard?