Why can't square roots of negative numbers exist? Imaginary?

Question

Ok, I'm in Algebra 2, the year has just begun, and my class is starting out with a look into the topic of real numbers. Real numbers can be defined as ANY value, positive or negative, that exists on a number line, including rational or irrational numbers.

However, while doing my latest assignment, I tried to do a square root of a negative number on my calculator, and an error message came up. Curiously, I did some research on this, and found that negative square roots create something called imaginary numbers.

So, my question is, why do square roots of negative numbers NOT make a number that appears on a number line? And how can there be a type of "number" that can't be shown on a number line? If it can't be shown on a number line, than what is it, exactly?

I'd appreciate an explanation, but try not to make it too complicated please (something you think I'll understand)! I'm sure I'll learn this later in the year, but I'm too curious to wait.

Answer 1

A square root of a number is a number that when multiplied by itself produces the square.

sqrt (x^2) = x

However, the square root of a negative number is impossible.

Note: when multiplying two negatives, you get a Positive result.

This creates the imaginary number i

i = sqrt (-1)

Using this, it is possible to simplify imaginary numbers

sqrt(-49) = sqrt (49) * i = 7i

Answer 2

okay think about it. its kinda complicated. so if you take the sq. rt. of 4 its two. thats because 2*2 = 4. pretty simple. but if i want the sq. rt. of -4 if you multiply 2 negative numbers together you get a positive, and if you multiply a negative and a positive they aren't the same number. it cant show up on a number line because it doesn't exist or is imaginary.

Answer 3

The simple answer is this:  two negatives always make a positive.  The only way you can obtain a negative number is to multiply a positive number by a negative number.

For instance, sqrt(25) = 5, because 5 * 5 = 25.

sqrt(100) = 10, since 10 * 10 = 100

However, how would you get sqrt(-25)?  What two same numbers will give you -25?  It's not going to be 5 * 5, because 5 * 5 = 25.   It's not going to be (-5) * (-5), because (-5) * (-5) = 25.  

Taking the square root of a negative number is asking what number times itself equals that number.  Properties of negatives and multiplying make it impossible for the square root of a negative number to be a real number.