Imaginary numbers:

What does lowercase j actually mean in the expressions of impedance such as 4+ j7 and 45 - j83?

Mathematicians use the lowercase letters i to represent j. This imaginary number is square root of -1. It is number that, when multiplied by itself, gives -1. So i= j, and j x j = -1.

The whole set of imaginary numbers derives from the single unit. The square of an imaginary number is negative-always. No real number contains this property. Whether the real number is positive, zero, or negative, its square can never be negative.

The notion of j came about simply as some mathematicians wondered what the square root of -1 would behave like, if there were such type of things present. So the mathematicians assumed the existence of this animal, and found that it had some properties. Eventually, number i was given a place among the realm of numbers. Mathematically, it is as real as the real numbers. But original term imaginary stuck, so that this number carries a mysterious aura with it.

It is not important, in this context, to debate reality of abstract, but to reassure you that imaginary numbers are not special particularly, and are not intended or reserved for a few eccentric geniuses. Imaginary numbers are as real as real ones. And just as unreal, in that neither kind are concrete; you can hold neither type of number in your hand, nor throw them in the wastebasket.

The imaginary number j can be multiplied by any real number, getting infinitude of imaginary numbers forming the imaginary number line. This is a duplicate of real number line you learned about in school. It should be at a right angle to the real number line when you are thinking of real and imaginary numbers at same time.

 
Figure--   The imaginary number line.