and also ELI16 what a an imaginary number is and how it works in real life
In: Mathematics
ELI5:
Start with a line 100 meters long. Write zero at the beginning and then mark off each meter up to 100. This is a short section of a number line such as you see in mathematics, with all “real” numbers (as named by Descartes) from negative infinity to positive infinity laid out on it.
Now, imagine you have a field that is 100 meters by 100 meters, with this line starting at a corner labelled zero running along the southern side. Add another identical line starting from the same zero corner at right angles along the east side, so that now by using two numbers from 1 to 100 you can denote any location on the field. You just say “22 East by 57 North”.
Or, 22 + 57i, 22 “real” number, 57 “imaginary” number. Together, they were named “complex” numbers.
Line one is laid out in real numbers. Line two is laid out imaginary numbers. Just as there is a “real” number number line, there is at right angles to it, starting with the same zero, a line of “imaginary” numbers from imaginary negative infinity to imaginary positive infinity. When you use both, you get a complex number.
Replace “East” and “North” in our example with “real” and “imaginary”, and now you can mark out any point on an infinite plain, which cannot be done with just “real” numbers. But the result is as real as 22 East by 57 North.
ELI6
Long ago, Descartes was working with square roots, and he ran into a difficulty. When you multiply a positive number by a positive number you get a positive number, and when you multiply two negatives you get a negative number.
This works fine if you want the square root of a positive number like 1. You can two roots, 1 * 1 = 1, and (-1) * (-1) = 1, giving you 1 and -1. But what about the square root of -1? It can’t be positive or negative, since either way you end up with a positive number.
The positive and negative numbers along a number line were “real” numbers to Descartes, because he could see them on a number line. But he couldn’t quite figure out where the square root of negative one was because it was not on a number line. So, Descartes labelled the square root of -1 “imaginary”.
However, imaginary numbers were not a mere abstraction. They actually had real world impact.
So, start with a standard number line with only real numbers. When you multiply two numbers, imagine that their sign + or – as directions on a circle, with positive numbers being zero degrees (they continue in the positive direction) and negative numbers as 180°. When you multiply two numbers, add the number of degrees, remembering that 360° is the same as 0°.
If you follow this rule, a positive 0° and a negative 180° multiplied together end up in the negative direction, (0° + 180° = 180°) as do a negative multiplied by a positive. Similarly, a positive 0° and a positive positive 0° end up positive 0°, while a negative 180° and a negative negative 180° get you 360° (0°), or positive.
But what about i, the imaginary number. How does it change?
If you go to the right of zero on a number line, you are going in the 0° direction, or positive. If you go to the left of zero on a number line, you are going in the 180° direction, or negative. But where is the imaginary number?
i is 90°, at right angles to the number line. If you multiply i * i = -1, you are adding 90° to 90°, getting 1 at 180°, or -1. 1i is the “imaginary” square root of -1.
i is 1 at 90°.
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