What's wrong with this proof? Please help!?

Question

Sqrt(-4) * Sqrt(-9) =
2i * 3i =
6 (-1) =
-6
But Sqrt(-4) * Sqrt(-9) =
Sqrt((-4) * (-9)) =
Sqrt(36) =
6
Therefore
-6 = 6?
Don't say that there is no such thing as Sqrt(-1)

Answer 1

Square root of 36 has two values one is +6 and the other is -6.

By two ways you have found the two values.

The correct steps are :

Sqrt(-4) * Sqrt(-9)

= (+ or-) 2i * (+ or-) 3i

1) (+2i ) x (+ 3i) = (- 6)

2) (+2i ) x (- 3i) = (+6)

3) (-2i ) x (+3i) = (+6)

4) (-2i ) x (-3i) = (-6)


In the second method after Sqrt (36) write (+ or-) 6.

You have omitted one answer.

Answer 2

The problem with this proof is that you assume that there is only one square root of 6. You use this assumption when you say "Therefore -6=6". But we know for a fact that the square root of a number isn't unique. That is why we say things like Sqrt(36)=+/-6.

The square root is what is called a multi-valued function. In that given a value it returns a set of values rather than just one. This is probably why the other respondents said that you can't use Sqrt(-4)*Sqrt(-9)=Sqrt(36). Really Sqrt(-4) = +/-2i and Sqrt(-9)=+/-3i. In this context it doesn't really make sense to multiply the two solutions sets together.

Answer 3

The fallacy lies in this step:

sqrt(-4) * sqrt(-9) = sqrt(36).

There is NO rule that guarantees

sqrt(ab) = sqrt(a) * sqrt(b), except in the case where both a and b are positive.

Along similar lines:

sqrt(-1) * sqrt (-1) = i * i = i^2 = -1
BUT
sqrt(-1) * sqrt(-1) = sqrt([-1] * [-1]) = sqrt(1) = 1

Again, it's false to assume sqrt(ab) = sqrt(a) * sqrt(b)

Answer 4

Good question.

The first approach is right and the second is wrong because the rule √a √b = √(a b) can be only applied to real numbers.When you multiply two imaginary numbers, you have to times real factors and also imaginary factors. Therefore √-4 √ -9 = √36 i^2 = -6.

Answer 5

You can not use the following rule for complex numbers:

sqrt(ab) = sqrt(a) sqrt(b)

It does not always work for them!!!