I think your premise is wrong, but that could just be a language problem on my part. As to your actual question. When something has no solution 'no solution' is the solution. Try to devide any number by zero. The calculator will no just make up a number. When you do it manually the solution looks something like this: L={}

Sure, in the same way that, when you don't know about fractions, you can say 3 divided by 2 isn't possible, because there's no integer that, when multiplied by 2, gives you 3.

But you don't do that, because inventing the idea of a number 3/2 is pretty useful, and lets you do many calculations that you couldn't do without it. In particular, you get ideas like "3/2 is the same kind of no-solution-is-possible as 6/4 is".

Imaginary numbers are the same kind of thing. They aren't "impossible", and "imaginary" is just a name (you could call them the "Bob numbers" if you wanted to). They are a perfectly well-defined algebraic object. They're only imaginary in the sense that they don't fit onto the number line, and human intuition about numbers tends to come from ideas like length that happen to be real-valued.

But in practice, imaginary numbers show up all the time in descriptions of the world around us. In particular, they're critically important in quantum mechanics, where the fact that the states of particles aren't real-valued is essential to the theory (many quantum-mechanical phenomena actively require this). They also turn out to produce very compact representations of things like periodic behavior, as with a spring or a pendulum. And relativity tells us that the relationship between space and time is the same kind of thing as the relationship between the number "3" and the number "3i".

You could do all of this with objects that are just pairs of real numbers (a,b), defined in such a way that (a,b) "times" (c,d) = (ad + bc, ac - bd) and (a,b) "plus" (c,d) = (a + c, b + d). But the object you've invented has exactly the same properties as imaginary numbers do, so why add the extra complexity?

If you put 10 coins into your bank account and the balance ends up being +3 coins, how much balance did you have to start with?

Technically speaking, yes, you can say that "no solution is possible" since there is no such thing as an anti-coin (or even "half a coin," depending on how you define it) but life becomes immensely easier if you accept the concept of "negative coins" to represent debt. That way you can easily have a balance of -7 on your account and offset that by just adding money to the account.

A similar thing happens with imaginary numbers. Instead of representing things which flip between two states (positive and negative) they rotate between four states, and like negative numbers, that enables a lot of extremely important math.

>I get that squaring a negative leads to a solution that’s impossible, but why do you have to make an impossible number into a number? Can’t you just say, “no solution is possible”?

I'm not sure what you mean by "impossible". Could you elaborate on your premise?

Subtracting a number from a smaller number is impossible within the realm of N (Natural Numbers). So why do we need Integers if they are "impossible"?
Well, sometimes we need to subtract a number from a smaller number.

Imaginary number exit because we define them to exit. Maths is an internally consistent system that is based on a few unprovable axioms..

So we define the imaginary unit i as i^2 = -1 and the rest of the usage follows from that definition

I meant square rooted not squared

You can say "there is no solution." But that's boring.

Mathematicians don't like being told they cannot do something. Instead they try to come up with new rules or new definitions to do whatever it is there isn't already a rule for. And those rules can be anything, but generally we look for consistency (those new rules should complement or add on to existing rules), usefulness (the new rules should help us do something we couldn't do before), and interesting consequences (things that make us go "ooh, that's neat").

And the more useful, interesting and neat those rules are, the more likely they are to be used by other mathematicians, and adopted as standard.

With complex numbers, we take all our usual rules for numbers and throw in one more; there exists some number(s) i such that i^2 = -1. It is consistent with all our existing rules, and turns out to be really useful in a bunch of areas of maths and science (and leads to some really interesting results).

i isn't impossible. It doesn't appear on a standard number line, but that's not a huge problem. The number line is an interesting and useful tool, but not the end point of numbers. Interestingly the first mathematical paper to use something like a modern number line was published about the same time Newton was publishing his Principia Mathematica; there weren't number lines when Newton was learning maths. Number lines are fairly modern.

Some classical Greek mathematicians had a very different way of looking at numbers, seeing them more as a lose collection of concepts, with fractions being connections between the different concepts (so 1/2 was a connection between 1 and 2). This did cause them problems, though, when it came to irrational numbers...

Because the maths is consistent, so it's interesting to mathematicians to explore it. And like so many "purely academic" things in maths, it turns out that it's no such thing; it has practical applications in things such electronics.

Why do negative numbers even need to exist?

Because the math is usefull to describe real world situations.

For imaginary numbers, for instance to describe both amplitude and phase of waves.

Aren't all but maybe the natural numbers "impossible"? You cannot have -5 sheep. You can even less have 5/3 trees, even less so -pi hamsters. Why is imaginary numbers suddenly the issue?

Well, it simply isn't. All those extension of the numbers made sense, do not cause contradictions, and most importantly, turn out to be useful in everyday life, engineering (electricians, for example use them) and physics (all over the place). Notwithstanding the immense effect on mathematics itself.

>For imaginary numbers, for instance to describe both amplitude and phase of waves.

They really drop that one on you out of left field in higher level math/science/engineering classes. One day, they are just like 'remember that imaginary number bullshit from junior high school? Here is what that is actually for.'

I think this video explains it much better than any short reddit reply could, but basically mathematicians historically had a similar opinion compared to yours, until they actually discovered places where this number is relevant and useful (quantum physics).

You could say "no solution is possible", and for certain starting assumptions that's true, and you can solve certain problems with that. But you could also use the starting assumptions that let you make a coherent answer involving imaginary numbers, and that lets you solve all the old problems and a lot of new ones as well.

I didn't learn about imaginary numbers until several decades after leaving college.

To be fair, Euhler's will mess with people's heads, and it's honestly the answer to this question but it sure isn't ELI5 :).

Excellent recommendation.

When you take calculus and differential equations (or Calc IV) Imaginary numbers (and eulers number as well) turn up there are are used in what are called "transforms".

The idea of a transform is: I have this disgustingly difficult calculus problem that is beyond my ability to do by hand. But...... I can transform it into an algebra problem, solve the algebra problem, then transform it back and get an answer that is useful. Yes, these transforms can get very messy on their own because you have to transform the entire algebra equation and take it with you.

The easiest analogy I can think of (and this is imperfect): You have a problem were you need to travel across the US continent, by foot. Walking 3000 miles kinda sucks though. So instead you arrange a plane ticket, fly in the plane, then get off and end up and the same place as if you had walked. There are some different tricky issues with the plane flight (like getting through airport security), but overall it is faster.

In this case, the plane is the transform. And if you have never seen a plane or used one, you look at it and say "what is that big aluminum can, I don't see any use for it"

Many things you learn as impossible in math are possible, just not possible with the current level of understanding. Calculus isn't an end. It is the start of everything. Compared to what comes after Calculus, learning Calculus is the equivalent of learning to count to 10.

Told, you can't divide things by zero. Did this for multiple years in engineering and calculus. We divide by zero all the time.

Told there is no square root of a negative, there is. We called it imaginary, its not imaginary. it exists. We could have called them extradimensional numbers, because that's what they are. Numbers on a different dimensional line than the normal real number dimensional lines. In early school, there is no need to understand this yes as its not something you use in day to day life. Its physics math tool offering a new dimension just like, x, y, z. Now you got xi, yi, zi. That's 3 new dimensions on top of the existing 3 that can used for more detailed information about a point.

Told there is nothing bigger than infinity. There is. 2*infinity is bigger in that is grows faster. Power sets which are infinite sets of infinite sets are bigger as well. You find there is an entire new number line just for the infinite numbers, often called the non-ordinals.

There are lots of solutions for those things learned in the past that had no solutions. The math just wasn't known yet.

The best way this was explained to me was instead of using a number line, imagine numbers more as a graph where you have an x and a y axis. The c axis is all the regular numbers we normally use, and the y are numbers that we don’t yet have names for and hence called them “imaginary”. But they both still intersect at 0, and the “distance” between them would be the same for regular numbers, so it’s handy to invent a nomenclature for turning the axis 90 degrees, so i.

People always fixate on the "square root of -1" thing, but that's not what imaginary numbers are for.

Essentially, numbers give us forward. Negative Numbers give us backwards, and imaginary numbers give us "sideways"

Imaginary numbers give us numbers in 2 dimensions. We don't just look at imaginary numbers on their own, but as a pair. We have a "real" part and an "imaginary" part. Something like 3+4i .

Now, we can convert real numbers to imaginary numbers by multiplying by i. We're basically rotating them. What happens if I rotate something twice? We end up reversing it.

Another way of saying we rotate twice is we multiply by i twice. i x i. Or i². We get -1. Rotate again (multiply by i again), and we get i³ = -i. Rotate again and we're back to where we started. i⁴ = 1.

Like I said earlier, there's an imaginary part and a real part. So 3 + 4i. Rotate by multiplying by i we get 3i + 4 i² = 3i - 4.

"Imaginary" and "real" numbers are so-called because at one time, it was thought that real numbers exist and imaginary ones don't. We now know that this is not true, but the labels "real" and "imaginary" stuck around, which is unfortunate because they are misleading.

Imaginary and real numbers are both real in the sense that you can do maths with them. And they're both imaginary in the sense that they're concepts, not physical things.

It's also worth noting that imaginary numbers are not just a contrivance for taking square roots of negative numbers, they are a core part of mathematics with uses far too numerous to list.

They’re really useful with electrical circuits (which, if you use a phone, car, or anything else powered by electricity, is something you should care about)

Because they are useful for describing certain things. For example, they are heavily used in electrical engineering and signal processing, because you want to work in two dimensions, and imaginary numbers are a convenient way to do that.

But the question is itself nonsensical. Nothing man-made needs to do exist. Negative numbers don't need to exist, you could just say there's no solution. Fractions don't need to exist, you could just say there's no solution. Natural numbers don't need to exist, you could just not count things. Language doesn't need to exist, we could just not speak, plenty of animals get on with their lives just fine without a language.

People really get hung up on the word "imaginary", and it drives them crazy, but it's just a name. Real numbers aren't any more or less real than imaginary numbers. Natural numbers aren't more or less natural than integers. Rational numbers don't have a mental capacity, and irrational numbers don't run around making dumb decisions. A turing machine isn't actually a machine with cogwheels inside. They are all made up, and they are all just names we gave them.

>why do you have to make an impossible number into a number?

From a mathematician's point of view, because we can. And it leads to a new number system that has a very rich structure that we can ask interesting questions about. (Unlike trying to make some other impossible numbers into numbers, like division by 0.)

1. Why do we need natural numbers?

Because we need to count things.

2. Why do we need whole numbers?

We need number where we can show that someone owes something.

3. Why do we need fractions?

Because we need a number to show how many parts od whole thing is taken/eaten.

4. Why so we irrational numbers?

So we can calculate area of round things.

5. Why do we need real numbers?

See 3rd and 4th answer.

6. Why do we need lateral (imaginary) numbers?

Because we need it for our alternate current calculations.

If you like video idea than watch all the parts

https://youtube.com/watch?v=T647CGsuOVU

This isn't a great comparison, the problem with dividing by zero is it could be literally anything, it is ambiguous. To have zero in the denominator is to say "I haven't provided you the necessary information to extract a value from this ratio, therefore the value could be anything." Like if you were to say 1/2. I am saying "I have divided something exactly once evenly to produce two equal parts, and I have handed you 1 of those parts." 0/2 is saying "I divided something exactly once evenly to create two equal parts and I have given you zero of them." 1/0 is saying "Something was divided into equal parts but I don't know how many times and here is one of them." <-- that isn't enough information to create a rational number.

The square root of negative 1 is saying "in a real sense negative square roots can't exist, but if they did exist we can manipulate them in this way." It is similar to anything in a computer that is 'virtual', it literally doesn't exist but if we imagine they do we can still do interesting thing.

Would something like '2.5 people per square mile' be considered an imaginary number? Since you can't have .5 of a person (well, you can, but that involves investigations and awkward questions XD)...

It is also REALLY important in Electrical Engineering, especially along with Fourier Transform for example

No, "imaginary" has a specific technical meaning in mathematics. It doesn't just mean "constructed for some purpose". It means a member of a particular set of numbers that solve equations of the form x^2 = a, for some negative number a. For example, 2i is an imaginary number that solves the equation x^2 = -4.

Ah, okay. Thanks :)

Two positive numbers can't multiply together to create a negative number, so it should be impossible.

But... if you 'imagine' that there IS some number that is multiplied by itself to get a negative number, you can actually use that 'imaginary' number to get useful results. All we do is define that number explicitly, so you say i*i= -1.

From this you get:

i^1 = i

i^2 = -1

i^3 = -i

i^4 = 1

A 'Complex number' is the sum of imaginary number and a real number. (for example 3i - 1) The complex plain is a coordinate system that plots complex numbers using the imaginary component for one axis and the real component for the other axis.

If you're using a regular x/y coordinate system, you can easily move the point on the X axis by adding or subtracting from the x value, and you can move the point on the Y axis by adding or subtracting from the y value, but what if you want to do something like rotating the coordinate around the origin? It's not super straight forward to find an operation that will turn 3x-y to -3y-x that will work for all coefficients. Sure, you could just add -4x -2y, but that only works for those specific coordinates.

But if you're using a complex plain, you unlock another operation! You can rotate the coordinate around (0,0) simply by multiplying it by some power of i!

Multiplying by 'i' rotates the coordinate 90 degrees clockwise, multiplying by i^x rotates the coordinate by 90*x degrees.

i^4 = i*i*i*i = (-1)(-1) = 1 because you've completed the 360 degree rotation returning the coordinate to where it started.

So if you're using 3i-1 instead of 3x-y, you can easily rotate it.

(3i-1) * i

3(i*i) - i

3(-1) -i

-3-i

And this operation will work regardless of what your starting coordinates are. Say you want to rotate -9i -5 by 270 degrees, you just multiply by i^3.

(-9i -5) * i^3

(-9i * i^3) + (-5 * i^3)

(-9i^4) + (-5i^3)

(-9 1) + (-5-i)

5i-9

We learned about them in like 9th grade. Times change. The mathematics being taught in high school now was university level a decade or two ago.

What did you take then? I can't imagine an engineering degree that wouldn't use them, especially if you have any classes on anything electrical

We didn't learn imaginary numbers until Algebra 1. Not a lot of us took that in Jr hi. Well, I did, but the excellent teacher we had went to teach at the high school, so we got the world's worst Algebra teacher. I'd ask her a question and be more confused afterward, so I learned to figure things out on my own. I felt bad for the kids who couldn't do that and got no help from our very nice but clueless teacher.

He understood, the thing is that the square root of a negative number is only impossible if you exclude "imaginary" numbers as a solution.

Very much like saying 3-6 is impossible if you don't think negative numbers are possible.

Just like banks might need negative numbers to say you're in debt, imaginary numbers are useful for other things, engineers use it to describe electricity in AC circuits for example

So you can say that imaginary numbers aren't possible, but that excludes real solutions that help you solve problems, very much like excluding fractional numbers would stop you from sharing an apple with a friend since you can either have an entire apple or none at all

I can't imagine an engineering degree I could have done.

Background: I went to high school in the late 1970s. We had geometry (Euclid style, not Descartes) and an algebra class for the students who were going to university.

Took an accelerated trigonometry class during summer bridge, and then failed Calculus I three times my freshman year.

That's when I shifted my academic goals from the natural sciences to history. I still retain my youthful enthusiasm for the sciences, which is why I learned about complex numbers in the first place.

Most of my friends view my ongoing efforts to understand mathematics as a charming eccentricity. As my eldest brother put it, paraphrasing Oscar Wilde, 'all logarithms are quite useless.'

Imaginary numbers become useful when working with a lot of advanced mathematics, and make equations work nicely.

A simple example though, is square rooting a negative number as you mentioned, but there's a little more to it. square_root(-1) = i, but that doesn't mean "no solution possible," it means that it's not possible at that moment. For example, if I plug that result into the function "f(x)=2+x^(2)" it now becomes a valid answer, because I am squaring the "impossible," answer of square_root(-1), cancelling out the square root

Ah I see, then it's totally understandable, unlike fractions and negatives they don't really serve a purpose for day to day.

Education changes from place to place, I see a lot of people apparently had calculus in highschool while I only started it in college (this was only a few years ago), I did however learn about imaginary numbers in high school

Math, like every logic system, must have axioms. These are statements that are true only because they are stated to be true, not because they can be shown to be true.

The imaginary unit is axiomatic. There is nothing special about axioms. I think people are just thrown off by the unrelated associations people have with the term "imaginary" and how/when it is taught in school.

In this case, imaginary numbers don't just lead to interesting structures and algebras, they are exceedingly useful in physics and might even be required.

Imagine getting to a solution is a pathway and you walk along it to find a solution, you find a tree in the path there is no way to get to the other side you could give up and say it is not possible to get to the solution but that is not true you could leave the path and walk around or climb over.

Imaginary numbers give you a way to leave the real numbers and find a solution. The solution is not always imaginary just because you used them to get it.

Imaginary is a bad name because they are actually numbers and the real numbers are just a subset of them.

We use imaginary numbers a lot in engineering, it would be impossible for modern control systems and power networks to work without them.

A lot of the comments answer why we have them but an interesting thing is the way they came up with them (from what I remember). Basically, math always evolves, and mathematicians were at a crossroads trying to figure out problems. They needed a number that when it was squared resulted in a negative number. Previously thought impossible because squares can only be positive numbers. What’s the answer? Create a new “imaginary number” that solves your problem, and it leads to many different applications and uses in modern mathematics. Always a complicated topic to think about though!

TIL that N stands for natural numbers.

You can owe the mob 5 sheep. Which means that if you want to have 2 sheep, you'd better buy 2 sheep so you can pay your debts before you start farming wool.

And you can have 1 2/3 pizzas, because you bought two and then ate a third of one.

Sure, you can't have -pi hamsters, but if you have a circular sheep pwn, then you'll need to have pi times twice the radius of fencing, otherwise your sheep will escape.

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For basics i is basically the square root of negative 1, this lets it do a couple useful things cause when dealing with control systems you can help avoid real zeros by tweaking the controls to create zeros at imaginary points ( which for a quadratic is always in a A +- root(B) form). So instead of stalling out it will occilate. For a simplified case, imaginary numbers let us make a cruise control that won’t just stop working on a highway.

It's not what it's "actually for", it's one application among many, including their original algebraic origins.

Not really true. Imaginary numbers were put on a sound footing by Gauss in the 1800s. Mathematicians were very comfortable with them by the time of quantum physics. Generally mathematicians are not concerned about whether there are physical applications.

Thanks, that's a great video

Imagine you tie a golf ball to a fan and lie beneath it and mapped its position. The golf ball would make a circle, and we could graph its movement using the x-y coordinates.

Now imagine you’re looking at the golf ball from the side. It’s still making the same movements, but it’s difficult to graph it’s movement because it looks like it’s only moving along the x axis.

In this example, imaginary numbers are the missing coordinate plane. They help us graph the behavior even though part of the movement doesn’t necessarily make sense/isn’t visible from our perspective.

If you didn't know, Z, Q and R are also used. Z are for the integers (-inf to +inf) while N are the natural numbers (0 to +inf). Q is for the rationnal numbers, all the numbers that can be written as a quotient a/b where b isn't equal to 0 (you also have Q' for the irrationnals like pi). Finally R are for the reals. It's important to note that each categorie can contain some of the other categories. For example, all natural numbers are integers, and all integers are real.

EDIT : As commented below, apparently 0 isn't included in N, but in N* (Natural numbers including 0) or W (Whole numbers). Didn't learn it that way, but maybe some people did ?

His original post wasn't talking about square roots. I replied to his og version.

This is probably the best explanation - it definitely helps to be visualizing the number line for this. I really enjoy the 3blue1brown YouTube series on this concept

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> integrating by parts

I guess it's a calculus class ? Good luck with that, it was dreadful for me; I had to do the class 3 times before passing the class lol

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Tbf I started really understanding calculus once I was in Uni; I'm now studying engineering and I can integrate in my sleep lol

The simplest answer is they exist to provide a round about way to get to a legitimate real solution.

In the same way negative numbers exist in scenerios that should be impossible.

For example i have 5 apples and on a day I lose 7 I gain 3 5-7+3

Now its impossible to lose 7 apples if you only had 5. And yes you can re-arrange the order. Or you can just go into negative and get a legitimate solution.

The math didn't care that it might have been lost 2 found 2 lose 3 find 1 lose 2.

TLDR: The name imaginary numbers is bad and people would find it less futile if it was called Bob numbers

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But they DON"T exist?

This is my basic understanding. One small use for them....
Maths functions are to describe relationships between things.
You can graph these relationships.
If the relationship is exponential, you can incorporate squared items.
But if the result is always -ve, it can be better to write the function with an imaginary number in it.
Then you can also manipulate the function without losing the property of a -ve square.

> You can owe the mob 5 sheep. Which means that if you want to have 2 sheep, you'd better buy 2 sheep so you can pay your debts before you start farming wool.

Your math is wrong. If you owe the mob 5 sheep and you want to have two sheep, you're going to need to buy 8 sheep (5 for the mob, 1 for the "big" and 2 for you).

> Essentially, numbers give us forward. Negative Numbers give us backwards, and imaginary numbers give us "sideways"

So what gives us a vertical? SQRT(-i)?

> while N are the natural numbers (0 to +inf)

I was taught that Natural numbers were 1,2,3,..., and Whole numbers were 0,1,2,3...

Looking online I can see that W is used for the Whole numbers, but I never saw the W symbol used in all my math classes.

Personally I learned that 0 was a natural number and looking online it is said that 0 was not included in the original definition, but was later added. To avoid mistake, N* is often used to indicate that it contains 0. At the end, it's not really that important but it's good to know !

Big? You mean vig?

This is where my answer is a bit messy. Imaginary numbers can be thought of that way, but using them for 2d space is an application.

We do have quaternions. This gives us 3 imaginary numbers, we call i, j and k, where squaring any of them gives -1, and ij = k, jk = i and ki =j. Weirdly, we lose commutivity here. ij = k, but ji = -k. Reversing the order gives a negative result.

As a control systems engineer......what?

Why not?

But seriously. A better question is actually: "Why do we need to work exclusively in a real number?"

I invented a new kind of mathematical object. I call it a "Weird Pair", or WP for short. It's represented as a pair of real numbers [a, b]. Two WPs are equal if each number is equal. Let's define a few operations we can do to a weird pair:
[a, b] + [c, d] = [a + c, b + d]
[a, b] * [c, d] = [a * c - b * d, a * d + b * c]

Hmm. Interesting. What is this useful for? Here is one of the possible uses. Let's interpret the Weird Pair as a regular pair. Looks like we can represent a point in a R^(2) space. The addition looks like a translation formula. But what does the multiplication do? Notice that the formula of rotating a point around the point of origin is (x cos θ - y sin θ, x sin θ + y cos θ), so, if the second WP is [r cos θ, r sin θ], we have invented a way to describe scaling and rotation.

Now, I wonder what is the sample operation of WP.
[0, 1] * [0, 1] = [0*0 - 1*1, 0*1 + 1*0] = [-1, 0].
Hmm, interesting. Also, what if I only care about the first element:
[a, 0] * [b, 0] = [a*b - 0*0, a*0 + b*0] = [ab, 0]. [a, 0] + [b, 0] = [a + b, 0].
So, I can treat a WP with the second element zero as a regular number. Wait a minute.

Whoops, I accidentally reinvented an imaginary number.

But you still do not have -5 sheep, just a debt of 5. That is conceptually not exactly the same. Sure, you can now define(!) negative numbers as debts, and that's okay. This would be one formal way to extend the natural numbers to the integers. Similarly one can extend further and further if careful.*

But exact numbers, irrational ones in particular, are already esoteric in real life. No fence ever will have exactly length pi. No diagonal of a square with side 1 truly has length sqrt(2), however precise you drew it. And maybe that third of a pizza was actually slightly less or more (but that one can be done, if we get down to counting atoms).

So what we do is to accept that those numbers mostly exist conceptually and abstractly. But if pi and sqrt(2) are fine, why not i? We artificially added the circumference of a circle and a solution of x² = 2, why not also x² = -1? And as mathematicians realized this is maybe where we can stop: every (non-constant) polynomial equation has already a solution in the complex numbers (they are algebraically closed), and every limit (such as pi as an infinite sum) that should exist actually exists (they are complete).

It also has applications in real life, and you don't need to go to quantum mechanics for that: The laws of electricity for DC extend neatly into those of AC. But only if you treat capacitors and coils as resistors of imaginary(!) "resistance". Hence like pi being the best way to deal with a fence of arbitrary precision, i works really well to deal with currents.

There are more abstract reasons in mathematics as well. As a simple example (as going into the true applications would go way beyond ELI5 or ELI18):

What is sin(0°) + sin(1°) + sin(2°) + sin(3°) + ... + sin(179°) + sin(180°) ?

Complex arithmetic tells you that sin(x) = ( e^ix - e^-ix ) / 2i, with x in radians. Using that and the geometric series

1 + a + a² + a³ + ... + a^n = ( 1-a^n+1 ) / (1-a)

will lead to the result; details are left to the reader ;-) .

tl;dr: they just work and make life easier, so why not use them?

*: The more common one is to work with pairs (a,b) of naturals, which we treat as if it were the number a-b: we consider (a,b) equal to (c,d) if and only if a+d = b+c (note how this only involves natural numbers now), similar to how "a/b" and "c/d" are the same if and only if a·d = b·c. And so on...

Didn't the YouTuber 3Blue1Brown suggested rotation numbers as a more user friendly name ?

This comment took a sharp turn from "Explain Like I'm 5" to "Explain Like I'm 5i+30".

Orthogonal numbers would be nice term for it.

Imaginary numbers are 2D numbers.

To give a brief example, you could point to a map and say walk 5 meters east, and 5 meters south. This would also mean 5 -5i on a Cartesian graph. Also this is used in wave calculations as waves can be denoted as vectors and plotted on a graph.

The term imaginary is stupid. They should be called lateral or perpendicular numbers

Hexagon numbers are the bestagon numbers

Imaginary number is a bad name for them... and intentionally so. It was meant to be derogatory. Mathematics has a history of people not liking new developments because people think of math as a very logical and objective thing and those developments can fly in the face of what they believe.

There's an apocryphal story that someone in the Cult of Pythagoras proved that the square root of two is irrational and they were so outraged by the idea that a number could be irrational that they took him to sea in a boat and returned without him. They believed that all numbers could be expressed as a ratio of two integers and an irrational number by definition is one that can't be expressed as a ratio.

The Ancient Greeks also didn't believe in the idea of zero or negative numbers and both were very controversial in the Western world for many centuries afterwards. In math today, the standard form for polynomials to put all the coefficients on one side and set it to zero because it's really useful. For example, Ax^2 + Bx + C = 0 for the quadratic polynomial where B and C are allowed to be zero and A/B/C are all allowed to be negative. But up until I think the 1500s, Western mathematicians didn't have a standard form but a family of forms specifically to avoid zeroes and negative numbers. Ax^2 + Bx = C, Ax^2 = Bx + C, Ax^2 = C, Ax^2 = Bx, etc.

Imaginary numbers first saw real use in the cubic equation because the people who found it realized that in some cases it involved taking the square root of a negative number, which people believed to be nonsensical. However, the cubic equation worked because the imaginary numbers cancelled each other out. Thus, they were called imaginary because people didn't think they were "real" and were only a mathematical trick that happened to work out and not something that's meaningful.

To get a feel for what it was probably like when irrational numbers, zero, negative numbers, and imaginary numbers were first introduced, look at the comments whenever 1+2+3+4+5+6+... = -1/12 comes up.

It's interesting how "useful day to day" can change so much in context. Logarithms are the basis for how slide rules work so in the time before personal computers when logarithms were more useful then than today.

And while most people don't actively use logarithms in their day to day life, they are incredibly important because human perception is generally logarithmic, not linear. The decibel scale is logarithmic because of this. There's even some evidence to suggest that logarithmic thinking might even be more natural. https://news.mit.edu/2012/thinking-logarithmically-1005

So a lot of the time with math names like "imaginary" just leads to confusion. In general these are numbers we can use in math to calculate things, but that you cannot physically hold in your hand.

Forget imaginary and just think of negative. You cannot hold in your hands "negative 5 apples". You can however represent it with a note and poof you now have credit and debt, all because you are considering a number that cannot physically exist (negative of something). Imaginary numbers are just another leap in this sense going even deeper into math.

The point ultimately is that they let you calculate things that might otherwise be impossible to calculate by filling in gaps.

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“In practice, it Happens all the time in the world around us, in theory”.

You may be correct I’m not smart enough but that’s sort of how you explained it.

Electrical engineer here, which is 90% dealing with imaginary numbers. On that comment on using pairs of real numbers instead of imaginary numbers:

My uncle had (and gave me) a very old book on electrical engineering that described all of the equations and theory without ever mentioning imaginary numbers - they used vectors instead, so they had to tap into vector calculations for even the most basic operations.

Needless to say I never really read it as it was borderline impossible to follow once you knew how easy it would be to do the same with imaginary numbers instead...

edit. typos...

Excellent description. I’d maybe save quantum mechanics to last though, because many people have the same feeling of doubt about qm as op has about imaginary numbers. Let’s just say that periodic phenomena are much easier to describe using complex numbers. To give a sense of why that is you can say it’s because these need to shift energy back and forth between two different forms of energy, which is reflected in the two different kinds of numbers that make up a complex number - the real part and the imaginary part. A pendulum, for example, converts potential energy to kinetic energy and back again. Quantum mechanics is just one example of periodic phenomena.

The same issue with grammatical gender. Instead of “masculine, feminine and neuter nouns” we could call them “Type A, B and C” and avoid many pointless discussions.

Turns out imaginary numbers are just a way to name coordinates, as in seeing in 2D.

What you used to know as A(-2;4) is now a=-2+4i

I think the real challenge with imaginary numbers is that most people don't experience them in real life. They are just as abstract and useful as fractions or negatives, but the application of complex numbers is so much more niche that they can feel made up

Imaginary numbers get like that. Since well most there uses are in higher maths.

It’s been awhile for me, not a control engineer but did my studies in mechanical engineering. So can’t speak for what you use in your day to day, but from what I remember controls (the math behind it) is tied with laplus transformation and the whole 1/t. Basically from those classes for feedback controls the zeros in laplus will say how it behaves in real life. And one way to avoid real zeros is forcing imaginary ones.

It’s not a bad name honestly, because “real numbers” also means something important in math and it excludes imaginary numbers

If we called them Bob numbers, you’d either have to change the name of “real numbers” to “non-Bob numbers” which isn’t great, or you would feel a lot less natural to exclude them.

It’s the same as rational vs irrational numbers. It’s not that pi or the square root of 2 is overly emotional, it’s just a convenient binary for mathematicians to use.

Honestly, "complex" numbers is so much more accurate. As they're formed of a real and "imaginary" part (even if the real part is 0), they are, as stated, more complex than real numbers.

Laplace*, he was a French mathematician

Another commenter mentioned that you can substitute calculations in electronic circuits that would normally use imaginary numbers instead with vectors. But it's a lot more work doing that, whereas imaginary numbers make calculations easier to do.

It seems like imaginary numbers provide a way of rotating between magnitudes in a sine/cosine way

It's this related to quaternions?

>Generally mathematicians are not concerned about whether there are physical applications.

They're not anti-applications, it's more that they spend time in abstract-land for a while investigating something before they look for applications of what they just created. It's their creative process.

You're mistaking something having a definition by being axiomatic. Imaginary unit is a construct that's a consequence of first defining natural numbers using Peano's axioms and then (in layman's terms) further "creating" other, more complex structures based on your previous results.

I agree with your other statement - imaginary numbers are just an ordinary mathematical object and there's nothing special about them. I consider real numbers to be the really, really weird ones (transcendental numbers anyone?). Imaginary numbers are just a simple extension of that weirdness. And they are also very useful, so that's a big plus.

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Ohhh Laplace transformations. I graduated electrical engineering, we used it a ton in school.

I haven't used it since class.

Given what I’ve heard of most engineers, because we have software designed to solve all that stuff most of the stuff in class we don’t use ourselves lol. Though understanding the theory behind it still helps. Specially when something goes wrong you know what to look for.

It’s fine to call it axiomatic. You can get into that Hilbert style formalism all you want. That’s all just backwards justification of math.

There is a reason nobody picked it up and continued that work when he abandoned it.

> Big? You mean vig?

"Vigorish (also known as juice, under-juice, the cut, the take, the margin, the house edge or simply the vig)..."

I guess I do. I must have mis-heard it.

Yeah, definitely, what I meant was that at first it wasn't seriously considered, root of a negative was just end of the road, and then later it was seen as a useful curiosity, something to help in intermediate steps but living purely in the theoretical/mathematical space, but only with Schrödinger they realised that it was part of nature and could be an answer in itself, not just an intermediary.

Nice, didn't know that, thanks!