Imaginary Numbers If you can imagine them, they exist. What kind of number can we imagine that is not a Real number?!? Any number which cannot possible exist could be called 'imaginary'. Imaginary numbers are actually ones like this ... Think about this number. It is certainly a number ... we just wrote it. But what kind of number could it be? What is the square root of negative twenty five? A square root is a value that will multiply by itself to give the original number. For instance: The square root of 25 must be 5, because 5 x 5 = 25 So ... what can the answer to the 'square root of negative 25' possibly be? If this number exists, then its value must multiply by itself to give -25. Can  - 5 be the answer? No. -5 x -5 = +25 Can +5 be the answer? No. +5 x +5 = +25 Apparently the square root of -25 has no answer! You can write the square root of a negative number, but you can't work it out. It has no value. Numbers like this are called imaginary numbers, and they always involve the square root of a negative number. It's a little like unicorns. They don't exist. But we can read stories about them, draw them, talk about them, and use them to teach little kids about how to behave. Similarly, we can write imaginary numbers, examine their properties, and use them in problems. The problem with imaginary numbers really just arises from the negative under the root sign. In fact, since every negative number can be thought of as a positive number multiplied by -1 (ie: -74 = 74 x -1), every imaginary number can be simplified like this: where i is the label given to the troublesome 'square root of -1'. Every imaginary number, then, is just some multiple of  i. Here's another example: Complex numbers are ones which have a Real and an Imaginary part. A typical complex numer might be: which is the same as 3 plus the square root of - 4 Since it has an imaginary part, a complex number has no value either.

What possible value can an complex or pure imaginary numbers have? Since they have no value, they can't be used for measuring things, and can't be the result of any measurement you might have made. Nevertheless, it is important to understand how complex numbers work, because some problems can be more easily solved by using imaginary solutions to guide us to the real ones. Rules about what sort of solutions that an equation can have, for Math 30 students, make more sense if you allow for the possibility that some of the answers might be imaginary. And some equations in engineering mathematics can only be solved using methods involving Complex numbers, even though the final answers aren't imaginary. The Complex number system is like a tool, to help mathematicians understand how numbers work. BACK