Imaginary And Complex Numbers

Introduction

So just trying to keep this stuff in my head. 42 years ago I was onto all this so let's see how we go. From memory is was a pretty easy ride. Just remember   $i=\sqrt{-1}$   $i^2=-1$   $i^3=1$   $i^4=1$

Simplifying

So he gave examples and how he solved it an I guess most of the was just substitution and recognizing you can split exponent by adding e.g. $x^4=x^3.x^1$
So
  $i^7=i^4.i^3$ = $1.(-i)$ = i
  $i^{33}=i^{32}.i^1$ = $(1^4{)}^8.i$ = $i^8.i$ = i
So I guess, get it down to a value you know and you are good to go

Solving Absolute Value of a Complex Number

So another remember the formula jobby. Standard form of a complex number is a+bi where a is the real number and b is the complete number. And the formula to working out the absolute value of a complex number is
  | a + bi | = $i=\sqrt{a^2+b^2}$
So an example
  | 5 + 12i | = $i=\sqrt{5^2+12^2}$ = $i=\sqrt{25+144}$ = $i=\sqrt{169}$ = 13