and also ELI16 what a an imaginary number is and how it works in real life
In: Mathematics
Just to offer another perspective here, if you’re interested in real-life applications of imaginary numbers, a good example is Electricity… specifically how Alternating Current works.
The characteristics of various loads on our electrical grid mean that almost all AC power has both an Active and Reactive component. Active power is what you’re used to seeing, such as for purely resistive loads like lights or running your toaster. But plenty of loads also have capacitive and inductive components, such as motors, transformers, electronics, etc. This means that the alternating current passing through these kinds of loads leads to a mismatch between the current and voltage waveforms, resulting in the need for what we call Reactive power.
Some people like to call it “imaginary” power because the math that allows you to easily calculate these reactive characteristics heavily involves imaginary numbers (though in electrical engineering we use the letter “j” instead of “i” and I’m not entirely sure why). But of course, there’s nothing imaginary about it. Reactive power is just as “real” as Active power, it just serves a completely different function. It also disappears completely when we’re talking about DC circuits instead.
I just want to add with all the excellent responses here is to not get too hung up on the terminology. Real is a defined sub set of numbers. Just like Imaginary is a defined subset of numbers.
Imaginary numbers are still real numbers ( lower case r) they are just not in the set of Real numbers. ( upper case r) also, Imaginary numbers are used all the time in electronics. They are real.
They could just be called Fancy numbers and Less Fancy numbers. They are all numbers.
So you’re running around a circular racetrack. If you looked at it from above, you could track your movement in terms of how far north/south you are from the center of the circle and how far east/west you are. This would end up with two numbers that are constantly changing as you run around the track.
You could describe the same movement by tracking how far from the center you are and how far around the track you are from the start/finish line. This way of doing things you have only one number changing (how far around the track you are) and the other stays the same (how far from the middle you are). If you go to a different track that is a larger or smaller circle, you just change the number that corresponds to distance from the center and continue tracking how far around the circle you are.
So you’re using two numbers to describe a thing in motion. A person running around a track in this case. It turns out there’s tons of things in the world that you can describe like a runner around a track. To do that you need two numbers to describe one thing. For historical reasons we say one of these numbers is real while the other is imaginary. In the end, they are both describing real things that happen to move or exist in cycles. Things that after a certain time get back to where they started.
It helps to think of numbers as vectors, having two values… That is, they have a magnitude and a direction. You can think of them as arrows. Magnitude has no sign. Positive numbers point 👉 Negative numbers point 👈 180 degrees off. When you add vectors, you put them tip to tail. so 5👉 (5) plus 3👈 (-3) ends up at 2👉 (2). When you multiply vectors, you multiply the magnitudes and add the directions. this is not intuitive! But it explains why multiplying a negative by a negative gives a positive — you add 180 degrees and 180 degrees and get 360 degrees which is 0 degrees. Or you can think of it as rotating the grid to he other direction. Imaginary numbers point ☝️. That’s it, just 90 degrees. All the other rules apply… In particular, multiplying an imaginary number by another imaginary number has you add their directions, 90 degrees and 90 degrees to get 180 degrees — that is, a negative number. Once you understand vectors visually, all those weird rules like a negative squared is positive, or i squared is -1 — they just make sense because multiplication has that rotation step they never talk about. The best part is these concepts extend right into complex numbers, linear algebra, etc.
Some problems are complex enough that a point can’t be represented by just one number, it needs two. You can imagine it like a point on a graph. For these “2D numbers” to be useful, you need to define how to add and multiply them in a way just like regular numbers. The way the rules shake out, (0, 1) * (0, 1) = (-1, 0). The y axis is called the “imaginary numbers” (using the letter ‘i’) and the x axis behaves just like the regular real numbers, so you can write it as i * i = -1.
Basically, the moment you start talking about imaginary numbers, you’re actually talking about these “2D numbers” (complex numbers), which is a different system than the regular 1D numbers (real numbers) we usually work with.
They’re really good for problems involving waves and transformations. For example, multiplying by (0, 1) rotates your complex number by 90 degrees!
ELI5:
Imagine you have a number line to count things on. Regular numbers, like 1, 2, 3, and so on, all live on this number line. But what if you wanted to count something that wasn’t there?
Imaginary numbers are like special numbers that help us describe things that don’t exist on the regular number line. They’re kind of like pretending there’s a whole new part of the number line, just for imaginary things!
The most important imaginary number is called “i” and it’s defined as the square root of -1. That means if you multiply “i” by itself, you get -1. It might seem strange, but imaginary numbers are very useful in math, especially for things like electricity and engineering!
ELI16:
Regular numbers are like points on a number line, covering positive and negative values. But what if there was a kind of number that, when multiplied by itself, gave you a negative number? That’s where imaginary numbers come in.
They’re numbers that include the unit imaginary number, “i”, defined as the square root of -1. Since no real number squared equals a negative, imaginary numbers seemed impossible at first, hence the name.
But here’s the cool part: despite seeming imaginary, mathematicians discovered they were useful for solving equations that regular numbers couldn’t. These numbers extended the number system, creating complex numbers. These are numbers of the form “a + bi” where a and b are real numbers.
Even though imaginary numbers might have seemed strange at first, they’ve become a powerful tool in math, science, and engineering.
So you’ve got the “natural numbers”. They go 0, 1, 2, 3 etc. People seem generally happy with those.
Then we discover some rules which apply to those numbers. 1+2=3. 6-4=2. Lots of other rules.
But what is 4-5? That question doesn’t have an answer in the “natural numbers”. But what mathematicians did was they said “Let’s pretend there is a number that answers that question”.
We call this made up number “negative 1”. What we discovered is that most of the rules of the “natural numbers” apply to these “negative numbers” – by pretending this number exists, we find that maths still works!
Then we came to a different problem – what is the square root of -1? Again mathematicians imagined a new number, which they called “i”. And again, they found that most of the rules still apply. Maths still works by pretending this number exists as well.
There are lots of usages of this number, but the key usage of it is it lets us deal with the square roots of negative numbers when they pop up. If it didn’t exist, then any square roots of negative numbers would break our equations. By “pretending” an answer exists, we can continue working through them, and end up with sensible solutions anyway. One such example is the cubic formula, where by continuing to work through the maths as though it makes sense, we can find sensible solutions.
5: If I show you a picture of a ball in the air, you know its position but not its velocity. So you don’t know anything about its past or future state. A real number is like that picture—it tells you something concrete, but there can be additional hidden information.
16: As others have said, starting with positive numbers leads to roots problems with negatives, negatives produce rationals, rationals produce irrationals, and reals produce complex. But *it stops there*. Roots starting from complex numbers can only ever be complex. That makes them even more fundamental than the reals in a critical sense.
i have an explanation for complex numbers. for this to understand, you need to know that complex numbers are of the form a+ib. so they are real part “a” and imaginary part “b”. so it kind of two dimensional where each dimension is like real numbers. so real numbers are like bridge compared complex numbers which are full land surface. while travelling in bridge of it breaks in the middle then there is no way to cross it but if are travelling in a land road and it breaks in the middle, you can still go to other side by going outside the road. hope this makes some sense to the question why complex numbers are useful
Think of the number line of -infinity to 0 to infinity, as a line, you go forward or backwards along one axis. Imaginary numbers allow you to go left or right on the number graph.
As for why they’re needed, I’d look at someone else’s post
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