Exams soon (just a reminder for those of you who haven't seen many stressed out teens lately) and today, I have been revising imaginary numbers! Because in maths, not only do you have nice, real tangible numbers like 1 and 2 (1 orange, 2 shoes) we also have made up numbers that don't really exist - have you ever seen √-1 of anything?
How is this any different to √1? Is the answer just -1? NO. Any number squared has to be positive (because positive x positive = positive and negative x negative = positive) Since square rooting is just the opposite of squaring, to find √-1, we just need to find a number which is negative when squared... which we've just shown does not exist. Whoops. But mathematicans are not happy with just accepting that it doesn't exist because that means they have a whole lot of numbers that they aren't allowed to play with. So they call √-1 'i' and just carry on trying to do maths with it anyway (By this, this also explains awesomely nerdy t-shirts like the mental_floss one: √-1 <3 Maths, which is obviously completely true...)
But to make life more complicated, these numbers don't just not exist. I know I just said they were imaginary but they can magically become real. If you square a root, it cancels out the square root part (opposite functions remember?) so you're just left with the bit under the root:
√-1 x √-1 = -1
Which is very real. It's a bit like having two imaginary friends together and ending up with a real person at the end. Weird, huh?
And sometimes, like when solving quadratic equations, you get things which have actual real numbers, and imaginary numbers like this: 1+ √-1. This is called a complex number (quite understandably). You have to treat them as two seperate bits because you can't suddenly add between the boundary of existance.
As for their uses... er I'm not really sure (I'm not exactly great at this maths thing!). Apparently, they are used in electronics, but really, in exams, they come up in quadratics and roots where the descriminant (The b² - 4ac bit of the quadratic equation) is negative (because we have to square root it, and if it is negative then it's not real). And they can be used to find the root of negative logs! Which may or may not have any use whatsoever.
So there you have it. √-1. The only time where an imaginary answer will get you the mark.
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