Up until now, you've been told that you can't take the square root of a negative number. That's because you had no numbers which were negative after you'd squared them so you couldn't "go backwards" by taking the square root. Every number was positive after you squared it. So you couldn't very well square-root a negative and expect to come up with anything sensible. Now, however, you can take the square root of a negative number, but it involves using a new number to do it.
This new number was invented discovered?
Complex Number -- from Wolfram MathWorld
At that time, nobody believed that any "real world" use would be found for this new number, other than easing the computations involved in solving certain equations, so the new number was viewed as being a pretend number invented for convenience sake. But then, when you think about it, aren't all numbers inventions?
It's not like numbers grow on trees! They live in our heads.
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We made them all up! Why not invent a new one, as long as it works okay with what we already have? Anyway, this new number was called " i ", standing for "imaginary", because "everybody knew" that i wasn't "real".
That's why you couldn't take the square root of a negative number before: you only had "real" numbers; that is, numbers without the " i " in them. The imaginary is defined to be:. But this doesn't make any sense!
And i already squares to —1. So it's not reasonable that i would also square to 1. This points out an important detail: When dealing with imaginaries, you gain something the ability to deal with negatives inside square roots , but you also lose something some of the flexibility and convenient rules you used to have when dealing with square roots.
We see, then, that the factor i 2 changes the sign of a product. Problem 1. Evaluate the following. To see the answer, pass your mouse over the colored area. To cover the answer again, click "Refresh" "Reload".
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Do the problem yourself first! Problem 2. Express each of the following in terms of i. Let us begin with i 0 , which is 1. Any number with exponent 0 is 1.
Each power of i can be obtained from the previous power by multiplying it by i. We have:. And we are back at 1 -- the cycle of powers will repeat. Any power of i will be either.
Problem 3. Evaluate each power of i. Complex numbers follow the same rules as real numbers.
For example, to multiply. Again, the factor i 2 changes the sign of the term.
Problem 6. The difference of two squares. Again, the components are real. Problem 7.