and also ELI16 what a an imaginary number is and how it works in real life
In: Mathematics
Basically mathematicians were trying to figure things out but at some point in calculation they encountered a part where that had √-1 which can’t be solved since any number squared ends up being positive so they just went like “well let’s imagine there was a number that when squared equals -1” and they called that number “i” so “i = √-1” and whenever they found √-1 on their work they just substituted it for i, but then how do you solve the equations and apply the in real life if you have to use a number that doesn’t exist? Well luckily later in the equations there was a point when they had to square i so they end up with just -1 which is a regular number so the problem solves itself.
On a normal numbering you have 1 2 3 4… in one direction and -1 -2 -3 -4… in the other direction.
The rules with multiplying negative numbers are as follows:
– negative x negative = positive
– positive x negative = negative
– positive x negative = positive
Now this makes a lot of sense with not much effort to picture if you’re using that numberline. Multiplying by a negative makes you switch direction on the line.
With this logic we can say that if you square a negative, you will always get a positive number. Because negative x negative is always positive. But this brings up a point:
Squaring a number multiplies it by itself. Therefore I must always be doing either negative x negative or positive x positive. So any negative number squared becomes positive. If that is the case: WHAT HAPPENS WHEN WE LOOK FOR THE SQUARE ROOT OF A NEGATIVE?
There is no rule of nature that says there can’t be a square root of a negative number, but by the rules of our mathematical model, one cannot exist. So mathematicians invented one. Put another numbering over than first one, perpendicular and crossing at 0. We have now invented an imaginary numberline. Except instead of going 1,2,3,4… we go i1,i2,i3,i4. Or -i1,-i2,-i3,-i4…
Square root your negative i numbers and we get j1,j2,j3,j4…
What you might notice if you’ve been drawing this or visualising it is we now have effectively another dimension to maths. Further than this wasn’t covered by my brief course on further maths but I recommend looking into it further if you’re interested. Iirc, Veritasium has a decent video on it.
One thing that I didn’t see in a ***very quick skim*** of top-level comments was the concept of extending the x-y plane into the third dimension. Other answers that I saw have done a pretty good job of getting to the point of rotation and periodicity, but another application for the imaginary (complex, really) set is for 3-dimensional physical modelling of the solution to a function.
Whenever you go to Wolfram Alpha and get a solution to a function, and the graph extends into the third dimension, that’s because of non-real solutions to the function – in this case, specifically complex solutions.
I wonder if this thread has been had before? Maybe search for it?
Imagine you have a puzzle, and some pieces are missing. Real numbers are like the pieces we can easily find and use. But sometimes, we need a special piece that doesn’t seem to fit anywhere at first. This special piece is the imaginary number “i.”
When we say we “make up” numbers like “i,” it’s because we need them to solve certain puzzles (math problems) that real numbers alone can’t solve. For example, if you have a problem where you need to find the square root of -1, real numbers don’t have an answer for that. But if we use “i,” we can say the answer is “i”.
Even though it sounds like we’re making things up, using imaginary numbers helps us solve real-world problems in fields like engineering, physics, and computer science. So, it’s like adding a new piece to our puzzle collection to complete more challenging and interesting puzzles.
Lots of great answers!
I’d like to offer a different approach. “What is a number anyways?” Seriously.
How do we know that when I write “2” you and I both know what this means?
– Because of rules.
If two objects follow the same rules exactly – then they are the same. Easy enough, so we think…
Answer me this: “what comes after 2?”
If you’re dealing with only natural numbers, whole numbers or integers – the answer is “3”. Perfect.
If you’re dealing with irrational numbers (like 2.1, 2.01, sqrt(2), pi, 7, etc) then the answer is …. There is no answer. Uh oh!!
The concept of “what number comes after a number” is totally invalid with irrationals. It’s not 3, it’s not 2.5, it’s not 2.0000000001, and there no such number of “2.000-infinitely-many-0s-then-1” because infinite implies no end so there can never be a 1.
Therefore “2” in irrational numbers is a totally different object from the one in integers. Woh man.
So this is a long post to illustrate that numbers are objects that follow rules. And if we want to solve some problems in the world, we have found that allowing sqrt(-1) = i to be a very handy thing indeed!
In short – it simplifies anything that oscillates or rotates. Which is a lot of what goes on in machines and electromagnetic waves.
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°.
Draw a dotted line, set any dot in the middle as 0, the dots to the left and right are negative and positive numbers. Draw a vertical line through any dot and that’s the imaginary number line.
Common notation is use of i for imaginary. I.e. 1i 2i 3i is progression through the imaginary number line.
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