# Complex Number Question (Easy)

• David Donald
In summary, the conversation was about verifying that i^2 = -1 using the equation (a+bi)(c+di) = (ac-bd)(ad+bc)i. The initial attempt was to choose coefficients to make (i)(i) equal to -1, but this was incorrect. The correct way to verify this is to use distributivity and the fact that i^2 = -1. However, the OP cannot use the definition of i as the root of -1 to prove this.

## Homework Statement

Verify that i2=-1
using

## The Attempt at a Solution

I tried choosing coefficients so that it would be (i)(i) = (0 - 1)+(0+0)i = -1
so then I get i^2 = -1

But I was told that this was wrong and to try again...

Can anyone explain what I did was wrong or if
theirs a smarter way to verify?

David Donald said:

## Homework Statement

Verify that i2=-1
using

## The Attempt at a Solution

I tried choosing coefficients so that it would be (i)(i) = (0 - 1)+(0+0)i = -1
so then I get i^2 = -1

But I was told that this was wrong and to try again...

Can anyone explain what I did was wrong or if
theirs a smarter way to verify?

We define i as the root of -1. So $$i^2 = -1$$ is true, no need to verify this.

The following statement is false: $$(a+bi)(c+di) = (ac-bd)(ad+bc)i$$
This would mean that the product of 2 complex numbers is an imaginary number. This is false. For example, $$i * i = i^2 = -1$$ is a real number.

What is $$(a+bi)(c+di)$$ equal to, using distributivity and $$i^2 = -1$$

Last edited by a moderator:
David Donald said:

## Homework Statement

Verify that i2=-1
using
Are you sure that you have written the equation above correctly? As already noted by Math_QED, this is false.

It's supposed to be:

And, assuming this holds for all ##a, b, c, d## show that ##i^2 = -1##.

That's what I assume the exercise to be.

Math_QED said:
We define i as the root of -1. So ##i^2 = -1## is true, no need to verify this.
But the whole point of this exercise is to prove this. In other words, the OP can't use this definition.

All,
Let's sit back and see what the OP says...