does the square root of a negative number exist?

Question

does the square root of a negative number exist?

Answer 1

Yes. But such a square root can't be a real number; it is instead a complex number (or imaginary number).



The square root of a negative number certainly can't be a real number, because any real number multiplied by itself is nonnegative.



To remedy the fact that negative numbers don't have square roots, a new number is introduced, called "i", that satisfies i^2 = -1 (by definition). So i is a square root of -1.



Numbers which are allowed to include both real numbers and this new number i are called "complex numbers."



It turns out that, with this number i, you can also represent square roots of any other negative number. Say you have the number -n, where n is positive. Then



(i√n) * (i√n) = i^2 * n = -n.



So i√n is a square root of -n.

Answer 2

By definition - the square root of a negative number does not exist - it is what is called an "imaginary number" or "complex" number.



We originally had positive integers to count, then we gained a concept of fractions, and zero, then decimals, then negative numbers, then irrational numbers (surds), then certain transcendental numbers (pi or e) - and to make our number system complete - Leonard Euler invented the idea that we can represent root(-1) with an imaginary number called "i".



Using this, we can find roots and numbers that don't exist - spanning into another dimension of the number line.



In short, the square root can be found - though it has no physical or geometric properties in the real world (though some may argue there are - they're not direct) - but they do not exist.



E.g. root(-16) = root(-1) * root(16) = 4i



Hope this helped.







If you haven't studied complex numbers yet - then don't worry about it, there is no answer as far as you need to know.

Answer 3

This Site Might Help You.



RE:

does the square root of a negative number exist?

Answer 4

The funny thing is, negative numbers don't exist. The reason for this is the two rules for squaring a number. We know that the square root of a number is the number multiplied by itself twice that equals that number. Now, a positive number multiplied by a positive number equals a positive number, and a negative number multiplied by a negative number equals a positive number, and 0 times 0 equals 0. Therefore, since positive, negative numbers, and the number zero are all of the kinds of numbers that make up all real numbers, square roots of numbers can't exist. That's why when we use them we make exceptions to the rule. Oh, and you can divide by zero, actually. The answer to dividing any real number except zero by zero is infinity. The reason for this is because zero multiplied by any number will never equal anything but zero. So, you can add zero to itself for infinity, and never reach any number but zero. The reason they call a real number other than zero divided by zero undefined is because those numbers divided by zero don't equal a number, therefore since in equations you define things by numbers, it's "undefined." There you go. I'm glad I could help. :)

Answer 5

yes, they are called imaginary numbers.



No, we cannot actually find the square root of a negative number... since no number squared is a negative number. But, yes, we have invented an abstract artifact, so to speak, that allows us to describe such roots. The constant, i, is arbitrarily assigned to describe the phenomenon, but it holds no real numeric meaning. We simply have real multiples of 'i', which are equally meaningless, but artificially create a unique number line of imaginary values... when we square it, then we have a real workable number again.



Do square roots actually exist? Difficult to interpret how you mean that. No, you cannot have a quantity of objects that can be counted in the imaginary numbers. But yes, real, physical phenomenon can have imaginary values... technology, particularly complex physics and electronics, wouldnt even be possible without the advent of imaginary numbers. They have practical, tangible use.



Consider this... are the 'real' counting numbers real? The numbers you have been taught since kindergarten are no more real... they are abstractions of the mind. No 'number' physically exists, be it imaginary or real.



Thus, the quotation below.

Answer 6

It does exists, but not generally. Negative square roots are known as imaginary numbers. Normally while we solve problems you usually donot accept negative roots.

Answer 7

For the best answers, search on this site https://shorturl.im/avsSU



If you were right that it cannot exist, then when other people cheerfully claim that sqrt(-7) = i * sqrt(7), and so on, they would eventually trap themselves in some inconsistency or contradiction. In several hundred years it hasn't happened yet, so I'm with them, not with you. But so far, every attempt to define the result of division by zero has led to an inconsistency or contradiction somewhere down the line, so it stays undefined.

Answer 8

tricky question. it doesn't really exist, but we have a term for it and it's actually a branch of maths known as complex numbers.



the gist of what it is is simply sqrt(-1) = i

thus we can represent sqrt of negative numbers and work with them

eg. sqrt(-1) * sqrt(-1) = i * i = 1

eg. sqrt(-4) = 2 sqrt(-1) = 2i

Answer 9

yes but it's not real. So technically it's "no". But if you wan't to write "yes", the numbers are called imaginary or complex numbers. So you can write i * i = -1 and solve everything else according to that.

Answer 10

Take for an example √-9, then



= √(-1 x 9)

= √-1 x √9

= √i² x 3

= i3 ANSWER.

Answer 11

yes but no ton the number line. they are called imaginary or complex numbers. when graphing the do not touch the x-axis

Answer 12

no thats not possible