-x^2 = 4 - 5
-x^2 = -1
x^2 = 1
x = +1 or -1 He tells his brother that the skiers will cross the green laser at positions -1 and +1, since they satisfy the equation of motion. He then tries to work out when they cross the red laser: -x^2 + 5 = 6
-x^2 = 6 - 5
-x^2 = 1
x^2 = -1 Then Ed gets stuck, since there's no positive or negative number he can think of which would fit as x here. Thus he declares to De "There are no solutions, so the skiers never get above my laser.". De is not satisfied with this. He says "That's cheating. I want you to finish working it out.". Ed knows how stubborn his brother is, since he carried on wearing fake tan even after going on "Snog, Marry, Avoid" the previous year, so he invents a number to satisft this equation: "The skiers cross my laser whenever they get to a position which I will call N. They will also cross the laser if they get to a position -N. Thus, +N and -N are the solutions to this equation." De is still not satisfied, so he asks his brother what N is. The doctor then explains imaginary numbers to the younger man: "N is a quantity that is not a positive number, or a negative number, however it is a very useful quantity, since it lets us solve the equation for my laser. You can't find it no matter how closely you look between the numbers, or how far along you go. This means that its value is orthogonal to the real numbers, since the two are independent, in the same way that you can't change an x coordinate by moving in the y direction. In fact, we could draw a diagram using the real numbers as an x axis and these imaginary ones as the y axis." "I'm confused." said De, "Will you please draw me such a diagram?" "I can't be arsed," Ed replied, "so here's one I found on Wikipedia." Using this diagram, Ed could explain to his brother how a Real number (Re) sits on the horizontal line, how an Imaginary number (Im) sits on the vertical line, and how the quadrants are filled with numbers that have an imaginary part and a real part, and are called Complex numbers. De seemed to understand, but had some questions: "OK, I'll accept what you just said, but then where does our quantity N fit in this diagram? Also, there are some undefined variables; what is the value of the "i" that's written on the diagram?" "The "i" represents the Imaginary unit." Ed explained, "It's equivalent to the number 1 for natural numbers, and -1 for negative numbers. Every Imaginary number is made out of some quantity of i." "So what is the value of i?" "The imaginary unit is defined by the equation i^2 = -1. No more, no less." "But that's the equation for the skiers crossing your laser! That means that N = i and the skiers will cross if they ever get to a position i, which they can never do because imaginary space is orthogonal to physical space, so no force can ever push them off the Real lines for x, y and z." "Well done" said an impressed Ed "but you're forgetting how to do square roots! There are always two square roots of a number, just like we had +N and -N. That means that i^2 = -1 and (-i)^2 = -1. So tell me, does N = i or does N = -i?" De thought about this for a minute before declaring "There's no way of telling! If N = i then N^2 = -1 since i^2 = -1, but also (-N)^2 = -1 since the minus signs cancel so -N = i. But also, (-i)^2 = -1 so N = -i and, because the minus signs cancel, i^2 = -1, so N = i. So N = +i or -i and -N = +i or -i, they're equal!" "Don't be too hasty to call them equal!" cautioned Ed "If N = -N then i = -i and I could do the following: i = -i This is what you are claiming
i + i = -i + i We can add i to each side to preserve the equality
2i = 0 Adding i to i gives 2i, whilst adding i to -i gives zero
i = 0 We can divide both sides by 2 and preserve the equality
i*i = 0*i We can multiply each side by i and preserve the equality
i*i = 0*0 We showed that i=0 so use that on the right hand side
i*i = 0 We know 0*0=0