Elementary Abstract Algebra- Examples and Applications, 2019a

2.1 THE ORIGIN OF COMPLEX NUMBERS 27
You also learned how to solve them either by hand, or using the SQRT button
on a simple calculator. The solutions to these equations are
• x = ±2
• x = ±6
• x = ±2.64575131106459 ...
But what about equations like:
x 2 = −1
Your simple calculator can’t help you with that one! 1 If you try to take
the square root of -1, the calculator will choke out ERR OR or some similar
message of distress. But why does it do this? Doesn’t −1 haveasquare
root?
In fact, we can prove mathematically that −1 does not have a real square
root. As proofs will play a very important part in this course, we’ll spend
some extra time and care explaining this first proof.
Proposition 2.1.1. −1 has no real square root.
Proof. We give two proofs of this proposition. The first one explains all
the details, while the second proof is more streamlined. It is the streamlined
proof that you should try to imitate when you write up proofs for homework
exercises.
Long drawn-out proof of Proposition 2.1.1 with all the gory details:
We will use a common proof technique called proof by contradiction.
Here’s how it goes:
First we suppose that there exists a real number a such that a 2 =
−1. Now we know that any real number is either positive, or zero, or
negative−there are no other possibilities. So we consider each of these three
cases: a>0, or a =0, or a0thena 2 = a · a =(positive)·(positive)=a
positive number (that is, a 2 > 0). But this couldn’t possibly be true,
because we have already supposed that a 2 = −1: there’s no way that
a 2 > 0anda 2 = −1 can both be true at the same time!
1 It’s true that the fancier graphing calculators can handle it, but that’s beside the
point.