Subject: Math Question on Magnitude and Complex Number

I am a tenth grade student in New York State. Recently, we started studying how to graph complex numbers. Our math teacher said to use what would normally be the x-axis as the real-axis and to use the y-axis as the imaginary-axis. However, when he started talking about how to calculate magnitude, that's when I became confused. For instance...

Let t = 7 -3i
Let u = -4 - i
Vector sum of t + u = 3 -4i

If I were to find the find the magnitude (which he defined as distance) then I would have to use the Pythagorean Theorum.

a2 + b2 = c2
32 + (-4)2 = c2
25 = c2
5 = c

(Obviously I would have to reject -5 since I want the absolute value)

Here's my dilemma. Shouldn't "b" be "-4i"? According to the label on the vertical axis, the value is -4i not -4. So why can you use -4?

Thank you,
Jeremy

Magnitute here is a real positive number - as are all distances.

SO for THIS calculation you are looking at the point (3,4) in a standard coordinate system. The "i" is a place holder, keeping the coordinates apart, just like the ( , ) in my notiation.

So there are TWO representations happening, and each is used in its own proper way in its own calculations.

It is analogous to having the same number in two different bases (say binary and decimal). We do NOT mix information from one representation in doing calcuations in the other!