Use complete sentences to describe why √-1 ≠ -√1

Well let's say that to compare these two numbers, we have to start with the definition first.

Definition

[tex] \displaystyle \large{ {y}^{2} = x} \\ \displaystyle \large{ y = \pm \sqrt{x} }[/tex]

Looks like we can use any x-values right? Nope.

The value of x only applies to any positive real numbers for one reason.

As we know, any numbers time itself will result in positive. No matter the negative or positive.

Definition II

[tex] \displaystyle \large{ {a}^{2} = a \times a = |b| }[/tex]

Where b is the result from a×a. Let's see an example.

Examples

[tex] \displaystyle \large{ {2}^{2} = 2 \times 2 = 4} \\ \displaystyle \large{ {( - 2)}^{2} = ( - 2) \times ( - 2) = | - 4| = 4}[/tex]

So basically, their counterpart or opposite still gives same value.

Then you may have a question, where does √-1 come from?

It comes from this equation:

[tex] \displaystyle \large{ {y}^{2} = - 1}[/tex]

When we solve the quadratic equation in this like form, we square both sides to get rid of the square.

[tex] \displaystyle \large{ \sqrt{ {y}^{2} } = \sqrt{ - 1} }[/tex]

Then where does plus-minus come from? It comes from one of Absolute Value propety.

Absolute Value Property I

[tex] \displaystyle \large{ \sqrt{ {x}^{2} } = |x| }[/tex]

Solving absolute value always gives the plus-minus. Therefore...

[tex] \displaystyle \large{ y = \pm \sqrt{ - 1} }[/tex]

Then we have the square root of -1 in negative and positive. But something is not right.

As I said, any numbers time itself of numbers squared will only result in positive. So how does the equation of y^2 = -1 make sense? Simple, it doesn't.

Because why would any numbers squared result in negative? Therefore, √-1 does not exist in a real number system.

Then we have another number which is -√1. This one is simple.

It is one of the solution from the equation y^2 = 1.

[tex] \displaystyle \large{ {y}^{2} = 1} \\ \displaystyle \large{ \sqrt{ {y}^{2} } = \sqrt{1} } \\ \displaystyle \large{ y = \pm \sqrt{1} }[/tex]

We ignore the +√1 but focus on -√1 instead. Of course, we know that numbers squared itself will result in positive. Since 1 is positive then we can say that these solutions exist in real number.

Conclusion

So what is the different? The different between two numbers is that √-1 does not exist in a real number system since any squared numbers only result in positive while -√1 is one of the solution from y^2 = 1 and exists in a real number system.

Answer:

Therefore [tex]\sqrt{1} \neq -\sqrt{1}[/tex]

Step-by-step explanation:

[tex]\sqrt{-1} \\[/tex] cannot be calculated as no two identical numbers that are multiplid could give -1. However - [tex]\sqrt{1}[/tex] means -1 x [tex]\sqrt{1}[/tex] = -1 x 1 = -1 . Therefore [tex]\sqrt{1} \neq -\sqrt{1}[/tex]

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