What is the sqare root of negative 1?

Answer 1

Your question is difficult to answer in a way that will seem satisfying. Below is my answer to your question as well as a long winded, boring explanation of why you won't like my answer.

I wasn't satisfied with the answer that "i" is the principal square root of negative one when I was in algebra. The natural follow-up question to ask at that point, in my opinion, is "Ok. What is 'i'?"

Unfortunately, the answer to that question is "one of the two elements of the complex numbers which has a square equal to negative one."

People won't like this explanation. The reason? It seems circular. But that's only because for all of the simple questions about square roots, we have a very familiar name or notation for the answer.

For example, if you were to ask an average algebra student to tell you the square root of 4, they would probably reply with the answer "two" (which isn't the only square root of 4...-2 is also a square root of 4. We call 2 the "principal square root of 4"). People are satisfied with this answer because the idea of "twoness" is an everyday idea. We know what people mean when they say "two" or write "2."

Further, it's easy to verify that 2 (or -2) is a square root of 4 by the definition of a square root of a real number. Since two squared equals four, it follows from the definition that two is a square root of four.

But, really, the principal square root of four is the positive real number which has a square equal to four. We just have a familiar name for that number. We call it "two." We are very comfortable using the number two in calculations and we find ourselves writing (or typing) the numeral for two practically every day.

There was no such familiar name for the quantity equal to the principal square root of -1. It's not a quantity normal people use on a regular basis, and its formal name, "the imaginary unit," alludes to the original thoughts on the practicality of this number. It was given the notation "i" at some point. But, as I said above, this symbol represents one of the two elements of the complex numbers which has a square equal to -1.

We did the same thing for values like the square root of 2, except we didn't even bother coming up with a new notation for it. We simply left it as a radical symbol with a "2" underneath it. The square root of negative one probably only got a new symbol because of the frequency of its use in solutions to equations and physical applications. It shows up everywhere. Rather than using the bulky radical symbol with "-1" underneath, it was streamlined to an "i".

Here's a less mathematical analogy to help explain why I think your question shouldn't be answered with a symbol. Suppose a 3 year old asked you what a "booger" is. You couldn't just point at the word "booger" and hope that they understand. You see, "booger" is just the notation (in this case, a string of letters commonly referred to as a "word") we use to describe a very specific thing. To explain what a booger is, you would use its definition, not just allude to its name. If "booger" had a more familiar symbol (say it had an everyday synonym that the little guy knew), then you might describe it that way (in the same manner that we can call the principal square root of four "two"). But without that, you can only describe it by its definition.

In my opinion, it's the same thing with "i." When someone tells you the square root of -1 is "i," they aren't really telling you anything at all UNLESS you already know what "i" represents. And if you did, you probably wouldn't ask the question.

That symbol "i" is, one last time, one of the two elements of the complex numbers whose square is -1. And it is the notation we use to describe one of the answers to your question.

By the way, -1 has two square roots. The additive inverse of i, denoted "-i", is also a square root of -1.

There are many more tangential conversations one could take from your rather innocent quesiton. My apologies for going so far down this one.

Answer 2

There is no such thing as the 'square root of negative 1'. There is no such thing as the square root of a 'negative number'. Complex analysis is really built on the properties of trigonometric functions/series and the accepted fact that i^2 is assumed to be -1. This is what makes 'complex analysis' possible. The number i need not be sqrt(-1). You could define i as follows:

i = sqrt(-a)/a where a is some positive
real number.

Answer 3

The square of any positive or negative number is positive, and the square of 0 is 0. Therefore, no negative number can have a real square root. However, it is possible to work in a larger number system, called the complex numbers, in which negative numbers have square roots. This is done by introducing a new number, called the imaginary unit, which is defined to be a square root of -1. It is usually denoted by i (sometimes j).

Answer 4

Square root of negative 1 is called an "imaginary unit", abbreviated as i. The word "imaginary" is somewhat misleading and is only used because the first mathematicians (Rene Descartes was among them) to discover imaginary numbers didn't believe they actually existed.

However, imaginary numbers, or more generally complex numbers, are just as valid as the real numbers. One way to understand this is by realizing that numbers themselves are abstractions, and the abstractions can be valid even when they are not recognized in a given context. For example, fractions such as ⅔ and ⅛ are meaningless to a person counting stones, but essential to a person comparing the sizes of different collections of stones. Similarly, negative numbers such as − 3 and − 5 are meaningless when keeping score in a soccer game, but essential when keeping track of monetary debits and credits.

Imaginary numbers follow the same pattern. For most human tasks, real numbers (or even rational numbers) offer an adequate description of data, and imaginary numbers have no meaning; however, in many areas of science and mathematics, imaginary numbers (and complex numbers in general) are essential for describing reality. Imaginary numbers have essential concrete applications in a variety of sciences and related areas such as signal processing, control theory, electromagnetism, quantum mechanics, and cartography.

Answer 5

Many people have answered it almost correctly. However there are two square roots of -1 rather than one. These are +i and -i that is + iota and - iota. Iota (represented by curled "i") is an imaginary number. Although it is an imaginary number it has significance in science and mathematics.

Answer 6

In the Field of the Real numbers the square root of -1 doesnt exist.

In the field of the Complex numbers the square roots of -1
is i + k*2*PI, k =...., -1,0,1,.... "they" usually only consider k=0.

Answer 7

This is a comple number i(real part is 0 imaginary part is 1)

Answer 8

It is an imaginary number (i or j) used in many areas of physics, electrical engineering, and in complex variable mathematics. A fundamental relationship is

e^(pi^i) = -1

Another is

e^{ix} = cos x + isin x

Answer 9

the square root of -1 is "i"

Answer 10

i
-1 is a complex #