Sqrt(-1) Help

[Ted_Grendy]

Hi all

I was hoping someone could help clear up my confusion about square roots and negative numbers.

If I had the following:-

Sqrt(4) * Sqrt(4) = Sqrt(4 * 4) = Sqrt(16) = 4

Now if I followed the same steps but replaced 4 with -1 I get the following:-

Sqrt(-1) * Sqrt(-1) = Sqrt(-1 * -1) = Sqrt(1) = 1

But I thought the answer should -1 not positive 1, What am I doing wrong?

Can anyone help?

Thank you.

 

[Steven G]

Sqrt(a*b) = sqrt(a) * sqrt(b) IF both a and b are NOT negative.

 

[pka]

If I had the following: Sqrt(4) * Sqrt(4) = Sqrt(4 * 4) = Sqrt(16) = 4
Now if I followed the same steps but replaced 4 with -1 I get the following:-
Sqrt(-1) * Sqrt(-1) = Sqrt(-1 * -1) = Sqrt(1) = 1
But I thought the answer should -1 not positive 1, What am I doing wrong?

The answer is really simple \(\displaystyle \sqrt4\) exist in the real numbers but \(\displaystyle \sqrt{-1}\) does not exist in real numbers.
In fact, in the real numbers the \(\displaystyle \bf{\sqrt{x}}\) exists if and only if \(\displaystyle \bf{x\ge 0}.\)

 

[Dr.Peterson]

Hi all

I was hoping someone could help clear up my confusion about square roots and negative numbers.

If I had the following:-

Sqrt(4) * Sqrt(4) = Sqrt(4 * 4) = Sqrt(16) = 4

Now if I followed the same steps but replaced 4 with -1 I get the following:-

Sqrt(-1) * Sqrt(-1) = Sqrt(-1 * -1) = Sqrt(1) = 1

But I thought the answer should -1 not positive 1, What am I doing wrong?

Can anyone help?

Thank you.

Any number actually has two square roots. In this sense, sqrt(-1) * sqrt(-1) = ±i * ±i = ±1, while sqrt(-1 * -1) = sqrt(1) = ±1. There is no problem!

When all numbers involved are real (that is, no negative numbers under radicals), we can define the principal square root (in order to make sqrt a function), and it will be true that sqrt(ab) = sqrt(a)sqrt(b). But when anything is nonreal, that is no longer valid; no definition of the principal root can make it always true. (A principal root can still be defined, but it doesn't retain that property.)

We just learn to set aside this rule; the best practice is to immediately rewrite any square root of a negative number explicitly in terms of i before doing any other simplification, for safety.