What does the imaginary number equal in math?

Answer 1

The following may seem far-fetched if you are not accustomed to imaginary or complex numbers, so before I continue, let me assure you that complex numbers have many practical applications, including electricity, quantum mechanics, art, and several other more.

The imaginary number is neither a positive nor a negative number. Imagine two perpendicular axes of numbers. The directions are arbitrary, but the way it is commonly drawn, from left to right you have the real numbers - the numbers you are probably most familiar with, which include positive and negative numbers. Positive at the right, negative at the left. The number line which you may have seen already.

From top to bottom is another line, that crosses the origin - the line of the imaginary numbers. One unit up is +i, two units up is +2i, one unit down (from the origin, or zero) is -i, two units down is -2i, etc. The "imaginary unit", then, is called "i", although in electricity the letter "j" is used instead (to avoid confusion with the unit for current).

A combination of a real number and an imaginary number is called a complex number - for example, 2 + 3i. Adding and subtracting complex numbers is fairly straightforward. Just add the corresponding terms. To multiply complex numbers, multiply them as you normally multiply binomials - then use the definition i2 = -1.

It so happens that when complex numbers are used, not only do negative numbers have a square root, but any root - square root or otherwise - has a solution. In a way, this makes the complex numbers more "complete" than the real numbers.

Of course, common sense should be used. Just as negative or fractional numbers don't make sense for some real-life problems, complex numbers don't make sense for some real-life problems, either. So if, for example, the quadratic formula gives you a complex solution (or a negative solution, for that matter), analyze the original problem to see whether the specific solutions found make sense, given the problem statement.