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Transcript of Imaginary Numbers
is less than zero. i² = −1 It’s a mathematical abstraction, and the equations work out. Don't worry you'll understand eventually...
We can solve equations like this all day long:
X^2=9. It's 3 and -3. But, what if somebody put this?
X^2=-9. Wait... What? You want the square root of a number
that is less than 0? Whhhaaatttt?!? It seems crazy, just like negatives, zero, and irrationals must have seemed crazy at first. There’s no “real” meaning to this question, right? Wrong! "So-called “imaginary numbers” are as normal as every other number (or just as fake): they’re a tool to describe the world. In the same spirit of assuming -1, .3, and 0 “exist”, let’s assume some number i exists where: i^2=-1 That is, you multiply i by itself to get -1. What happens now? Well you get mad... like that. ------------------> New, brain-twisting concepts are hard and they don’t make sense immediately. But negatives are just as wack, and they can still be as useful. (1) X^2=-1
-We can’t multiply by a positive twice, because the result stays positive.
-We can’t multiply by a negative twice, because the result will FLIP back to positive on the second multiplication.
But what about… a ROTATION! It sounds crazy, but if we imagine x being a “rotation of 90 degrees”, then applying x twice will be a 180 degree rotation, or a flip from 1 to -1! If we multiply by -i twice, we turn 1 into -i, and -i into -1. So there’s really two square roots of -1: i and -i.This is pretty cool. We have some sort of answer, but what does it mean?-i is a “new imaginary dimension” to measure a number-i (or -i) is what numbers “become” when rotated-Multiplying i is a rotation by 90 degrees counter-clockwise-Multiplying by -i is a rotation of 90 degrees -clockwise-Two rotations in either direction is -1: it brings us back into the “regular” dimensions of positive and negative numbers. And if we think about it more, we could rotate twice in the other direction (clockwise) to turn 1 into -1. This is “negative” rotation or a multiplication by -i: The powers of "i" are:
i^0= 1( any number to the power of 0 is 1)
i^1= i(anything to the 1st power it is the number it's self)
i^3=-i (i^2 =-1xi=-i)
i^4= i x i^3= i x -i = (-1)(i)(i)
(-1) / = 1 The cycle then repeats it's self.
i^0 = i^4 An imaginary number bi can be added to a real number a to form a complex number of the form
a + bi Complex Numbers -21= -21 + 0(i) or -21 7i = 0 + 7i = 0 + 7i A complex number is a real number plus an imaginary number. a + bi where a is the real part and bi is the imaginary part. converting to complex numbers These complex numbers are useful to represent two dimensional variables. The key point to remember is that imaginary numbers are often used to represent a second physical dimension. Example:
The size of a photograph (2 dimensions, one for length, one for width). We could use a complex number to describe the photograph where the real part would quantify one dimension, and the imaginary part would quantify the other.