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]

At 6:00 AM, the temperature was 2 degrees Celsius. At 11:00 AM, the temperature was -11 degrees Celsius. By how many degrees Celsius did the temperature change?

A.
-13 degrees
B.
13 degrees
C.
9 degrees
D.
-9 degrees

Which fraction is equivalent to 6/-8? Please help! It will be appreciated

five less than 6 times a number plus seven

Please put details! I need it!

14. Books are on sale for 25% off the original price at Half-Price Books. Write an
algebraic expression for the final price of a book that costs d dollars.

PLEASE SOLVE ALL OF THE QUESTIONS IN FULL

find the exact coordinates of X on segment JK that is 1/3 the distance from j to k. J (-1,4)​ K(5,-3)

Find the number that makes the ratio equivalent to 1:4.

Is 3^2+3^3 equal to 3^5? Yes, expressions are
No are not
Explain:

I beg you, HELP MEEEE!!!!! One spinner is divided into four equal sections numbered 1 to 4. A second spinner is divided into five equal sections numbered 5 to 9. How many total outcomes are there if both spinners are spun?

13
9
20
36

what is the absolute value of 15

Please help me! I need part A and part B. Please help me! PLEASE!!

Please help. Not to sure how to do this

how is 1 half to the power of 4 =to how is 1 half to the power7

I WILL MARK AS BRAINLIEST! Solve for X and Y

need help with these two exit ticket questions

will try to post the other question when the first is finished

Help plz. Name the vector and write it in component form.

Helpppp pleeeassseee

Question3. Answer needed quickly with explanation

The diameter of a circle is 0.82 inches. What is the approximate area of the circle?

Which lines, if any, can you conclude are parallel given that m<1 + m<2= 8 ? Justify your conclusion with a theorem or postulate.

a. g || h, by the Converse of the Same-Side Interior Angles Theorem
b. g || h, by the Converse of the Alternate Interior Angles Theorem
c. j || k , by the Converse of the Alternate Interior Angles Theorem
d. j || k, by the Converse of the Same-Side Interior Angles Theorem

The first picture is the graph (already answered) and the second is the next

Natalie is making punch the recipe calls for 2 cups of water for every 5 scoops of powder mix. What will be the ratio be if she needs to double it? Triple it?

I WILL MARK BRAINLIEST! Solve The Following Questions

The upper-left coordinates on a rectangle are (-4,4) left parenthesis, minus, 4, comma, 4, right parenthesis, and the upper-right coordinates are (4,4)left parenthesis, 4, comma, 4, right parenthesis. The rectangle has a perimeter of 20units.
Draw the rectangle on the coordinate plane below.

Find the unit rate.
754 students on 26 buses
students per bus

I WILL MARK AS BRAINLIEST! Solve for Y

A rancher’s herd of 250 sheep grazes over a 40-acre pasture. He would like to find out how many sheep are grazing on each acre of the pasture at any given time, so he has some images of the pasture taken by the state department of agriculture’s aerial photography division. Here are three samples of the images.
How do the sample statistics compare to the population mean and standard deviation?

se necesitan 4 trabajadores y 9 horas para pintar un cerco si el trabajo debe completarse en 6 horas Cuántos trabajadores se necesita para pintar el cerco

Help me Guess just help please