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Often written in mathematics as simply "i", it's the "simplest" imaginary number. Not to be confused with i-hat, which is assumed to be the unit vector on the x-axis of a ((0,0,1),(0,1,0),(1,0,0)) space.

1*1=1
(-1)*(-1)=1 (Both are technically the square root of one, but when people say the square root it should be assumed they mean the positive square root.)
There is no n^2=-1. So they call it an imaginary number.
Some simple equations based on the fact that i is the square root of negative one.

i^1 = i

i^2 = -1

i^3 = -i

i^4 = +1

and this pattern repeats forever

It should also be known that "i" is its own inverse reciprical. Now thats pretty cool.
The square root of a positive integer may be rational or irrational, but is always real. The square root of a negative integer is always imaginary, and is plus-or-minus a real multiple of i.

The square root of a multiple of i, however, can always be represented as a complex number -- the sum of some real number and some imaginary number -- and so can all other roots of all complex numbers. It can be said, then, that the complex numbers are as "complicated" as numbers can possibly be... at least, until we invent some new mathematical symbol to make it even worse.

The imaginary and complex numbers may seem like mathematical novelties with no real value whatsoever, but they do turn out to be useful in certain electrical engineering problems, as well as theoretical physics. (Tachyons, for instance, are particles that are supposed to travel faster than light, and therefore have imaginary mass.)

The square root of -1 is i to mathematicians, but to physicists, I is the abbreviation for current, so the square root of -1 is j.

This seemingly minor fact can cause problems in interdisciplinary arguments between mathematicians and physicists, as I well know.

This is also the name of an album by Illbient NYC-based We. Features some good tracks to space out to, as well as some to just groove to. I believe they call it Square Root of Minus One.. but it's all the same.. the album has the mathematical symbol on it.

This is the answer to a common question about why we need to define the square root of -1 at all.

Well the answer rests on the fact that you want all algebraic equations to be solvable. If you consider the set of all real numbers, equations such as

x 2 + 3 = 5

can be solved. The solution above is of course sqrt(2).

However if you want all algebraic equations to be solvable the real number system is insufficient. For example the equation:

x 2 + 1 = 0

does not have a real solution. It turns out that the only addition you need to make to the real number system is to introduce this funny quantity called i and say that numbers of the form a + i*b are to be called complex numbers. With this addition every algebraic equation is solvable.

When people say that the square root of -1 is i they are only partly correct. Just as 4 has the two square roots 2 and -2, -1 has the two square roots i and -i. Both these numbers give -1 when squared and are therefore square roots of -1. The difference between any complex number (a+bi) and its complex conjugate (a-bi) is therefore just that we use the alternate square root of -1.

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