√-64, the answer can't be -8 as -8 x -8 = 64. The answer can't be 8 as 8 x 8 = 64 and so on. So how can we evaluate such a number? Well the answer is to split √-64 into to parts, remember that we can write this number as

√-1 x 64 which is √-1 √64, now we can evaluate the √64 which is 8, but we can't evaluate √-1, so this value of √-1 is given the letter j. (In mathematics √-1 is given the letter i but in electrical engineering we use i to represent current).

Now lets have a look at some powers of j: Well j squared is j x j and we know that as the operations of square and the square root are opposite we simply remove the square operation and we have what is left which in this case is -1. That might sound a bit complicated so I'll use another example which you can work out, take the number √81, now if we square this number we get √81 x √81 which is √81 squared. Now if you work this out you will find that your answer will be what ever number you had in the square root sign which was 81, I hope this is clear.

So back to powers of j then, j x j = √-1 x √-1 = -1 = j

j x j x j would be j squared x j which is -1 x √-1 which is √1. What about j x j x j x j, well this is exactly the same as writing j squared x j squared which is -1 x -1 = 1, so from a completely irrational number of √-1 we gen get a rational number. So if we were asked to calculate j to the power of nine that is exactly the same as saying j to the four by j to the four by j to the one which is 1 x 1 x √-1 = √-1 = j.

Now lets consider an expression such as 2 + √-81 , well again in our expression we are faced with this un - solveable √-81, but we can seperate this out as we did before we can change our expression to; 2 + √-1 x 81 which is the same as writing; 2 + √-1 x √81 , and we can solve √81 so our expression becomes 2 + (√-1) x 9 and as we give √-1 the letter j our expression becomes 2 + j9 also notice that we do have a number in our expression which isn't indeterminate (the number 2) we call these numbers

**real numbers**and we call the number 9 an

**imaginary number**as it is associated with the letter j, sometimes called the j operator.

The next thread will describe complex numbers in greater detail.

Nirvana.