imaginary number division

You actually need to multiply by the conjugate on top and bottom; in other words, multiply by (1 - 4i)/(1 - 4i). This is a fancy way to multiply by 1, since any quantity divided by itself = 1. This preserves the value of the original expression, but what's even better is that after multiplying, the denominator will become real. This is the whole point of the procedure. The product of complex conjugates is always real. That is to say, products like (2 + 3i)(2 - 3i) and (5 - 7i)(5 + 7i) will always turn out to be real numbers. Thus:

3i ÷ (1 + 4i) = 

3i(1 - 4i) ÷ (1 + 4i)(1 - 4i) = 

(3i + 12) ÷ (1 + 16) = 

(3i + 12)/17

In principle, this process is very much the same when rationalizing a denominator that's a radical expression. Here's an example of when this might be useful:

sec(15º) = 1/cos(15º) = 4 ÷ (√6 + √2) = 

4(√6 - √2) ÷ (√6 + √2)(√6 - √2) = 

4(√6 - √2) ÷ (6 - 2) = 

4(√6 - √2)/4 = 

√6 - √2

Did you know that sec(15º) simplifies so well? Can you find csc(15º)?

For reference, csc(15º) = 1/sin(15º) and sin(15º) = (√6 - √2)/4