
How to explain infinity to kids - gbacon
https://blog.plover.com/math/infinity-for-kids.html
======
scandox
But before you do that it's worth just asking them what they think comes after
everything else and see what they say. Because kids often have really
interesting ideas on those kinds of topics and once you tell them something,
then their ideas get pushed out.

~~~
jackgolding
This is one of my mottos for life - you can only be naive once so it can be
beneficial to let your imagination run wild based on your unique background
before you seek out best practices.

------
romwell
Picking the first infinite ordinal as the infinity to explain to kids might
not be the best choice for every kid, though. Oridnals are tricky. I am not
too comfortable with them myself, and I say that as an adult with a degree in
math!

The thing is, this kind of infinity just doesn't come up that often when
dealing with other objects in math. Even though, as Tom Lehrer sang, one can
count up to infinity - or somewhere in that vicinity - and that's mathematics,
the question then is - so what?

The other kinds of infinity - cardinals, for example - are encountered early
on, and there are _things you can do with them_.

The first time I've seen the notion of infinity was in a Russian children's
book. There, they made a bijection between all the (infinite number) of points
in a small segment and a larger segment - and even with all the points of an
(infinite) line! I didn't really _get_ it then; while I could find nothing
wrong with the argument, it certainly looked like bullshit that a _short
segment_ could have as many points as a _long one_.

But some things don't have to make perfect sense right away.

The next time my understanding of infinity really improved (ignoring the
notation for "x growing without bound" of calculus) was in the first year of
college, with Cantor's diagonal argument. And I think that's when the picture
from the book I read in kindergarten made sense, at last.

The bijection in that picture would have been boring if one could always make
it. But with the diagonal argument, one sees that's not the case. That's what
makes these infinities _interesting_ and _fun_ , to me.

So, I might be biased in that, but I think that the _cardinals_ are the most
_playful_ type of infinity. And really the kind you can explain to kids.

One night I've had a long tea on a rooftop of a Brooklyn apartment building
with a friend who is an artist, and by the sunrise, she understood Cantor's
diagonal argument - and enjoyed it.

It is quite regrettable that this is the kind of knowledge that's only
generally shown to math majors in college. This is reason #712889 why we need
to change the way we teach and talk about mathematics.

~~~
gowld
Get a sheet of graph paper. Imagine that it goes on forever in 2 of the 4
directions. Name the intersection points in order -- left to right, top to
bottom.

What's the name of the first point in the first row? 0

What's the name of the first point in the second row? \omega

How is that more complicated or less interesting than cardinals?

~~~
empath75
I guess I’m mixing up ordinals and cardinals but it seems odd that if you
order the points differently ((0,0), (1,0), (0,1), (2,0), (1,1), (0,2),
etc...)you never get to omega and cover all the same points.

~~~
cokernel
Yup, you never get to \omega that way. Cardinality is a coarser notion than
"ordinality". So your example shows that Card(\omega * \omega) = Card(\omega)
= \aleph_0, even though \omega * \omega and \omega have different order types.

EDIT: Just wanted to add that an order isomorphism has two requirements:

(1) it needs to be a bijection (so order-isomorphic objects have the same
cardinality); and

(2) it needs to preserve all inequalities (so a strict inequality among items
in one object turns into a strict inequality in the same direction among the
corresponding items in the other object).

------
sandov
As a kid, I asked my mom if all numbers have a name, she said yes, and then I
asked the obvious follow-up question: If there are infinite numbers, and all
numbers have a name, then there must exist a number called door, another one
named airplane, because at some point you would run out of _numbery_ names, I
don't remember what she said, but she didn't answer the question.

That question was unresolved for me until recently, when I revisited the
question and thought that at some point you could start naming them a, aa,
aaa, aaaa, etc instead of _seventen hexadecillions five hundred and sixty
octillions_ ... or something like that.

I would love to remember all the other questions I had.

~~~
jpulgarin
If you're referring to real numbers, then virtually no real numbers have a
name. If all real numbers had a name then you could order them alphabetically
and put them in one-to-one correspondence with the natural numbers, which we
know is impossible:
[https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument](https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument)

~~~
wwweston
Does this apply if names are infinitely long?

~~~
shoo
[edit: for clarity i'm not going to use the term "name"]

the proof crucially relies upon each real number being represented as an
infinitely long (perhaps ending in an infinite sequence of repeated digits for
less interesting numbers) sequence of digits. i.e. the representation that
Cantor uses for reals is infinite sequences -- effectively representing each
real number infinitely long (in the countable sense) strings.

Cantor represents each real number as an infinite sequence of digits. apart
from technical details`+` you can think of this as the base 10 or base 2
expansion of the number.

for simplicity, you need only consider counting the real numbers between 0 and
1. there are plenty enough of those. each such number can be addressed in base
2 as some infinite sequence of 1s and 0s after the binary point. Cantor's
argument shows that if you try to enumerate all such numbers (i.e. count them)
by their series expansion, then you can generate a new number that doesnt
apppear in the enumerated list, which is a proof by contradiction, refuting
the assumption that you could enumerate them all in the first place.

`+` :

technical details include that some numbers have non-unique representations as
infinite sequences of digits.

For example, in base 10, 1 can be represented as 1.000... (where the 0s keep
going) or 0.999... (where the 9s keep going).

if this irritates you, let z = 0.999...

then 10 * z = 9.999... then 10 * z - z = 9 then 9 * z = 9 so z = 1

------
Izmaki
I don't understand this explanation, and I'm an adult with an interest in
mathematics and a degree in software development (although not mathematical
per se, they usually go hand-in-hand). I like how "w" is the smallest number
you cannot count to - if I understand it right it's almost 0 but not quite -
but I don't understand the bonus questions and answers.

If this is the best "explain it like I'm 5" explanation of infinity, I believe
I can think of a few examples that give a better idea of it. Heck, even the
concept of "never ending" seems simpler to me. "Never ending" \+ 1 is still
never ending.

~~~
TuringTest
I believe that approach comes naturally to us software developers, as we tend
to think about mathematics in procedural terms (which is how programs run in
the execution environment) rather than in string rewriting (which is how
mathematical proofs are made). I also consider infinity as a process that
never finds its ending condition, my mental model is the "infinite loop".

"ω" is not almost 0, that would be epsilon "ε". Omega ω is a number higher
than any other natural number, i.e. "to the right" of the infinite line of
numbers.

Mathematicians "will" this number into existence, so to say, starting from a
contradiction. It's the same that they do with irrational numbers ( _" imagine
there's this number "i" that, when multiplied by itself, it gives you -1"_).
With infinity, it's like: _" you know this process that never ends? Well,
imagine that it finishes, and let's call the result "ω"._

Once they have this new number defined, they do lots of mathematical
operations with it, trying to find its properties. What they never remember
again after that is that the number dit _not_ appear as the result of
following the initial process to completion; they had to assume that it
existed independently from the process.

* BTW, this is also why they have different kinds of infinities. They are using different never-ending processes in their respective definitions, and using the same name for all of them.

~~~
enedil
Notice that mathematicians don't take real numbers for granted too - we need
to assume their existence first. Looking at this, complex numbers stop being
that arbitrary, just another extension of reals.

Also, it's nitpicking, but i is imaginary, not irrational. Also, complex
numbers were accepted by mathematicians before negative numbers (it's
something that boggles minds of some people).

------
mhneu
_A professor of mine once said to me that all teaching was a process of lying,
and then of replacing the lies with successively better approximations of the
truth._

This is the opposite of what Feynman thought. He said, and I agree, something
like "the hard part about teaching is making concepts simple without saying
things that are false." The whole challenge of teaching is NOT lying, but
instead saying simple things that are still true.

That said, I see the appeal of choosing omega.

~~~
n4r9
Feynman might have been sympathetic to the idea that all of science is a
process of "replacing the lies with successively better approximations of the
truth". Certainly Asimov was:

[https://chem.tufts.edu/answersinscience/relativityofwrong.ht...](https://chem.tufts.edu/answersinscience/relativityofwrong.htm)

------
duncancarroll
I like the concept, but if I may editorialize, I feel the phrasing needs work.
"The smallest number you can't count to" is a negative statement, which makes
it confusing right off the bat.

What do you mean, a number I can't count to? If I'm 8 years old this is like
throwing a null pointer exception in my brain.

~~~
kylec
They might also take it literally. Like, they might think that infinity is
less than a million because they know they couldn’t physically count to a
million.

~~~
marcelluspye
>couldn’t physically count to a million.

This is only because they/you haven't tried.

Source: counted to a million once.

~~~
lwansbrough
How long did that take you? Seems like it take at least a week.

~~~
kirubakaran
A million seconds is over 11 days straight without stopping for sleep, food,
or anything! I very much doubt that they actually did it.

~~~
cyborgx7
If you count one a second, you're counting very slowly. Also, you can take
breaks and pick it back up, making a note where you left off. I did it this
way once, growing up.

~~~
udkyo
_If you count one a second, you 're counting very slowly._

That stops being true long before you reach the finish line. Try timing
yourself counting from 147,895 to 147,900 and see how long it takes.

~~~
cyborgx7
fair enough

------
Boulth
> A professor of mine once said to me that all teaching was a process of
> lying, and then of replacing the lies with successively better
> approximations of the truth.

Well put! I'm in the process of explaining my 3yo daughter different molecules
(there are some cheap kits on Aliexpress) and cells and life on a micro scale
and this description of moving from simple but inaccurate models to more
complex and accurate is something that I've also noticed in my explanations.

~~~
mncharity
>> all teaching was a process of lying

> Well put!

Shudder. Consider a thought experiment - a military briefing. A captain
briefing generals. One must necessarily simplify. But imagine a briefing that
is grossly incomplete, assortedly incorrect, very misleading, written without
understanding and without mentioning and characterizing that lack, and
pervasively incompetently bogus, and that captain later on the carpet before
the generals, defending the briefing with "well, all briefing is a process of
lying". Shudder. I consider the "teaching is lying" meme to be _vile_. I've
heard it most often associated with pre-college chemistry. Chemistry education
research describes pre-college chemistry education content as "incoherent",
leaving both students and teachers deeply steeped in misconceptions. Oh well.

You mentioned microscale, so I write just to note the top "How to remember
sizes" section of my slowwwly-loading wasn't-intended-to-be-public page
[http://www.clarifyscience.info/part/Atoms](http://www.clarifyscience.info/part/Atoms)
. It might help you provide a framework for your kid to think about small
things. FWIW.

~~~
oska
> Shudder. I consider the "teaching is lying" meme to be _vile_.

Couldn't agree more. And when I read this in the article I also thought of
high school chemistry even before I read your comment. I was put off chemistry
in high school precisely because of its incoherence.

The best thing to do, as always, is to be honest. Tell your students that what
you are teaching them is a simplification; a model that is useful at this
level. Many will be satisfied with this answer and not want to go to a deeper
level. But some children, those with a greater need for coherence and for
things to make deep sense, will be reassured to know it's only a model and
either wait for the deeper model to be taught or want to start exploring the
deeper model at that time. Which is fine.

I found the approach taken by the author of this article quite patronising and
rather distasteful.

~~~
vilhelm_s
The approach in this article is to give a "technically unimpeachable"
definition. It doesn't involve any lies.

~~~
oska
My comment, and I believe the one I replied to, is directed at the first
paragraph in this piece, which is about the idea of teaching as 'a process of
lying' and one which the author agrees with 'in principle'.

I also do not like his suggested definition. It comes across as smartarse-ish
and I think most children would feel the same way.

------
peteretep
If we're going for useful lies to tide kids over, I quite like the explanation
in the Postgres docs:

> infinity (date, timestamp) later than all other time stamps

I think you could tell a kid who wasn't quite ready for Aleph numbers that:

> infinity is a useful made-up number that's bigger than all other numbers

Which is useful when kids first hear about it, I guess.

I think my first "practical" introduction to infinity came with "Space is
infinitely big", and both mjd's explanation and my own fails at that point.

~~~
dahart
> infinity is a useful made-up number that's bigger than all other numbers

I love the emphasis on this being a choice, and I wish this kind of thinking
was taught more in math. So many math explanations act like these things are
immutable facts of the universe, rather than human constructions. We lose some
of the history and character of math when we teach it as law rather than
invention.

Zero is also a useful made-up number, and it didn’t always exist as a concept.
We understand 0 so deeply now, it’s not longer possible to imagine a world
without that idea. Another fun one, 0^0 gets defined (chosen) to be 1
sometimes, because it makes other formulas work out, not because it’s the
correct or only answer.

~~~
KennyCason
I wish I could upvote this more. I encounter this very often in discourse. I
extend this to logic and philosophy as well. Without maintaining both the
rigor and the creativity of reasoning, only a hollow shell remains.

------
antidesitter
_A different bright kid might ask about ω−1, which opens a different but
fruitful line of discussion: ω is not a successor ordinal, it is a limit
ordinal._

ω - 1 makes sense in the surreal numbers
([https://en.wikipedia.org/wiki/Surreal_number](https://en.wikipedia.org/wiki/Surreal_number)).
The surreal numbers are the "greatest" ordered field, in a sense. They contain
all ordinal numbers, which in turn contain all cardinal numbers. The
cardinality of a set is just the least ordinal it can be put into a one-to-one
correspondence with (this is called the von Neumann cardinal assignment).

------
tzs
Or you could raise them as ultrafinitists [1]!

[1]
[https://en.wikipedia.org/wiki/Ultrafinitism](https://en.wikipedia.org/wiki/Ultrafinitism)

------
injb
From the article: “Imagine taking all the numbers that you could reach by
counting,” I said. “Then add one more, after all of them. That is infinity.”

How is 'adding one more' not the same thing as counting?

------
peter30744
I am not a mathematician, but I know that at least in standard mathematics it
is fairly ill-advised to treat infinity implicitly as a number. Doing so can
result in various contradictions (two different seemingly valid solutions to a
problem). It should be thought of as a property of a process, i.e. for any
number x you can change the process so that it results in a number greater
than x. One example of infinity might be: "Think of a number which you are
allowed to change after each number I suggest. If for any number I suggest you
change it for a bigger one, then what you think of is infinity."

Chapter 15: Paradoxes of probability theory in Jaynes's "Probability Theory:
Logic of Science" is a great reading on the topic (you can find a pdf easily
on google). It starts with a quote from Gauss:

"I protest against the use of infinite magnitude as something accomplished,
which is never permissible in mathematics. Infinity is merely a figure of
speech, the true meaning being a limit." \-- C. F. Gauss

Anyway, there are plenty of theories in mathematics which use infinity
implicitly, but one should perhaps be cautious.

~~~
enedil
Now you're conflating two different notions. You're (and your quotes) are
talking about the first bullet point from the article. In turn, when taking
about the last one, the one chosen by article author, from my experience, I'd
say that it's beneficial not to even consider it in time, i.e. don't think
about it as process, don't think of it as something completed. There's no
completion, since there was no process to begin with. The object just is, and
we can manipulate it according to the rules.

------
SublimeWarior
Many mathematically-trained people fumble the infinity question because
infinity is just a term or notation for something that is non-ambiguously
defined. People tend to view infinity as a sort of weird, maybe even mystic,
concept. Certainly the intuition behind it is useful for reasoning, but
mathematically-speaking it's just a term for another formal definition.

Consider limits. One can say that a function converges to a certain value v
when its input approaches infinity. This seems all subjective and mystic and
non mathematically trained people would come up with all sorts of
interpretation of that. But this just means that no matter how small a number
ε you chose, then there exists a number x such that for any input greater than
x, the function's output will be contained within v-ε and v+ε. That's it. It's
as simple as that. There is no mystery to it, nor judgment. "Infinity" is just
a name that is involved in such formal property.

------
JeanMarcS
Teaching mathematics is lying to kids all along, at least in France :

\- No, you can’t do 2-3. If you have 2 apples, you can’t give 3

\- Well, in fact you can, it’s negative numbers. But if you have 5 apples you
can’t split them equally between 2 persons.

\- Well, in fact you can. They got 2.5 each. But you cannot solo 2 apples in 3
equally !!

\- Well in fact you can. That’s 2/3 each.

\- etc...

~~~
likeclockwork
> \- No, you can’t do 2-3. If you have 2 apples, you can’t give 3

Well, how soon is too soon to teach children about debt?

~~~
JeanMarcS
Apparently the 18th century :-D !

[https://betterexplained.com/articles/a-visual-intuitive-
guid...](https://betterexplained.com/articles/a-visual-intuitive-guide-to-
imaginary-numbers/)

------
sattoshi
I loved how a math teacher I knew talked about infinity: it is a garden where
they grow very large numbers.

It communicates that you can't work with infinity like with a number, because
it is a group of numbers.

It swallows all the operands as infinity does: 2 groups = still a group; half
a group is still a group; a group plus one is still a group, etc.

Then you have groups of groups, or infinity powers which work the same way. A
group of groups is clearly greater than just a group.

At least, I think this is what he meant. He never really expanded on it beyond
the garden analogy.

------
dnate
> Instead we can decisively say that there is another number after infinity,
> which is called “infinity plus one”.

I see a problem with saying that. One of my earliest troubles when dealing
with infinity in algebra was understanding, that you cannot add or subtract
real numbers from infinity to make it something else.

e.g. inifnity - infinity is not 0. Suddenly saying that infinity + 1 \neq
inifinity would just make it more confusing.

The article is a nice mind exercise but IMO not really helpful in explaining
infinity to a child.

~~~
navane
“Imagine taking all the numbers that you could reach by counting,” I said.
“Then add one more, after all of them. That is infinity.”

Disclaimer: I'm an adult that doesn't understand infinity. If you "add one
more" you're still counting.

~~~
romwell
You are correct. _Counting_ here changes its meaning.

The correct form should be "all the numbers reachable by counting _in finite
time_ ".

E.g. you can count up to 100 in a minute. Up to a million in a month[1]. Up to
a billion in quite a long, but finite time. You take all such numbers, and you
say that a number named Omega comes _just after_ all of them (just like
million and one comes _just after_ all numbers that are less than or equal to
ine million).

Omega is your first _infinte ordinal_ \- or, simply, infinity.

And now you're counting in a new way.

[1][https://www.mathsisfun.com/activity/count-
billion.html](https://www.mathsisfun.com/activity/count-billion.html)

~~~
navane
What I hear you say is that you can count to finite numbers in finite time,
and infinite numbers in infinite time. That doesn't help me in understanding
infinity.

~~~
enedil
The key point here is that you shouldn't be thinking of counting as a process
that happens in time. How is infinity (here, omega) defined? We just say that
it comes after any other natural number. So how to think about it? You
shouldn't tell yourself:

\- ok, take number n, it's smaller than omega, so what about n+1, n+2, etc. ?

You should be telling yourself \- ok, take number n. It's smaller than omega.
Now check any other number m. If this property holds for any given number,
then omega is greater than any given number, and thus infinite.

------
Sniffnoy
Nitpicking: This article writes 2⋅ω where what's meant is ω⋅2.

------
User23
John Conway's On Numbers and Games has a great overview of infinities. And
since it's about games it'll be great for teaching kids too!

~~~
OscarCunningham
I think Conway and Guy's "The Book of Numbers" might be a better source. It's
more focused on infinities as different types of number, and certainly at a
more suitable level. ONAG is quite a technical book; children don't really
need to know that the surreals are the universally embedding real-closed
field.

------
Pete_D
Another approach is taken in the comic "Life on the Infinite Farm" (pdf:
[https://www.math.brown.edu/~res/farm.pdf](https://www.math.brown.edu/~res/farm.pdf)).
I'm not sure it's entirely appropriate for children though.

------
swfsql
(related - and I enjoyed the vibe he put into it):
[https://www.youtube.com/watch?v=SrU9YDoXE88](https://www.youtube.com/watch?v=SrU9YDoXE88)
\- Vsauce's "How to Count Past Infinity"

------
mchahn
I got excited because I thought that I might finally understand infinity.
Unfortunately I am still not there as I am not there with quantum mechanics. I
have read many books on both with equal non-success.

------
avip
Just give them two mirrors and then play.

------
DomreiRoam
A very nice book in French is "Le Chat au pays des Nombres" by Ivar Ekeland.
The book tell a story that happens in the infinite hotel on the numbers
planet. This hotel managed by the Hilbert is always full. It is a very good
approach of infinity for children around 6.

[1] [https://www.babelio.com/livres/Ekeland-Le-Chat-au-pays-
des-N...](https://www.babelio.com/livres/Ekeland-Le-Chat-au-pays-des-
Nombres/434839)

------
secabeen
When my daughter was learning fractions, it was fun to use the infinity to
discuss the real numbers between 0 to 1 by squeezing 1/<whatever number she
named> into them.

~~~
gowld
You don't even need the reals for that. Rationals is plenty.

------
YouAreGreat
> It's the smallest number you can't count to

Funny. If explaining infinity to a kid had been my problem, I'd have tried
hard to _avoid_ calling infinity a number.

------
mbrumlow
THis is one of the best things I have read on HN lately!

I can't wait to use his answer to what infinity is to my kid when he gets old
enough to ask such a question.

------
enturn
I like to think about the length of the outside of a circle. It's only when
you use a line with a beginning and end in the shape of a circle that you can
measure it. The idea can be extended to 3D with a spiral staircase that has no
beginning and end. The steps of the staircase cannot be measured until you
give it limits.

------
tomclive
This is interesting to read as I've been struggling with the best approach for
my son who's coming up for five.

He has a pretty solid understanding of basic addition, subtraction and
multiplication.

He was asking about infinity and although he knows it's very big he was still
asking about infinity plus one. Maybe it's not too soon to explain the
concept.

------
SideburnsOfDoom
What came to my mind was "a number is where to stop counting up. e.g. I'll
walk for five blocks. I'll eat ten jelly-beans. I'll throw the ball twenty
times. Infinity mean never ever stopping."

but that's much the same as "It's the smallest number you can't count to."

------
nurettin
So we use a symbol and when they ask what it is, we use recursion to
demonstrate the idea.

------
skate22
I like the way I was taught better:

Imagine you are 2 feet from the wall, and with every step you move forward 50%
of the remaining distance. How many steps does it take to the wall.

And the answer of course is that you never get to the wall, no matter how many
steps you take

~~~
throwawaymath
That ceases to be a paradox once you distinguish between _boundedness_ versus
_completeness_ for a set of infinite steps.

There are infinitely many steps on the interval [0,1]. But if you add 1 "step"
at any point of the interval greater than 0, you've still "passed" it.

------
abhchand
most of these comments are picking apart the explanation, but honestly this is
a great way to introduce kids to the concept and have them "arrive" at the
concept of infinity through their own logic.

------
mirimir
> It's the smallest number you can't count to.

I don't remember whether I was ever taught that. But I do remember the line
from Marilyn Manson's "Posthuman".

> God is a number you cannot count to.

------
axilmar
What an awful way to explain infinity to kids and adults. How about something
simpler? like 'infinity is when something never ends'?

~~~
enedil
Your explanation is worse, since it doesn't allow to distinguish different
infinities.

~~~
axilmar
It doesn't have to explain different infinities. When we talk about infinity,
we need to realize the concept of 'never ending', and not anything else.

------
asdffdsa321
I thought infinity was most always described in terms of limits. I have
trouble thinking about it as a number

------
bronlund
I'm glad you guys didn't try to explain this to me when I was a kid. Infinity
is that which never ends.

------
magerleagues
Wait, why can't you count to it?

------
raincom
> A professor of mine once said to me that all teaching was a process of
> lying, and then of replacing the lies with successively better
> approximations of the truth.

In philosophy of sciences, it is called "idealization" and "concretization".
Many philosophers from Poznan even name this philosophy of science as "Marxist
philosophy of science", since Marx uses the technique of idealization and
concretization in his work "Das Kapital".

------
nicodjimenez
\infty = \lim _ { 0 \to \infty } x

that's how

------
Aeolun
I don’t know if this would make any sense to kids, it certainly doesn’t to me.

If infinity + 1 exists, then that number would be infinity. It’d be more
accurate to say that Infinity + 1 = NaN

------
hutzlibu
Why so complicated?

Just ask them to do i++ and never stop. That is also infinity.

(or in simpler words, imagine you have a table than you put one apple to it,
then one again, amd again and again and never stop. And yes, they would also
soon understand, that the table needs to be infinitly big)

edit: of course it is about the sum of the process. they should imagine the
pile of things or the number if you never stop adding.

~~~
kovrik
That's an infinite process (which is an awkward way to describe anything), but
not infinity. You'll get more and more natural numbers, but will never
actually get infinity.

I love ops explanation because it gives more natural picture: there are
_other_ numbers beyond naturals. Then you tell kids there are also negative
numbers (integers) and rational numbers and real and complex and so on. I
believe that way it is much easier to grasp that very abstract (but
fundamental) idea of 'number'. That it is not just 1,2,3...

Because otherwise, we get people thinking "Complex numbers do not exist, that
is just a silly thing used by mathematicians that has nothing to do with the
real world, it is useless."

~~~
dragonwriter
I think the problem with people thinking complex numbers don't exist actually
stems from the imaginary component and the same problem of imaginary numbers,
which is rooted not in failing to teach about other kinds of numbers but in
the name. Especially given the name of the real numbers.

The idea that real numbers are real, imaginary numbers are not real, and
complex numbers that have a real number part and an imaginary number part are
also not real is a natural consequence of unfortunate naming choices.

~~~
kovrik
Agree. Scientists love confusing and weird names.

For me, back in school days (or was it university?) it was a revelation when I
came across quaternions. It suddenly clicked. I finally understood that there
was nothing special about complex numbers (despite their special names and
weird look). It was just an extension and a very intuitive one!

I finally understood that real numbers are the same thing: tuples. They just
happen to have exactly one element, hence we omit parens and everything else
and just write that element (number)! Complex numbers have 2 elements (real
and imaginary). Quaternions - four. And so on.

~~~
throwawaymath
_> I finally understood that real numbers are the same thing: tuples. They
just happen to have exactly one element, hence we omit parens and everything
else and just write that element (number)! Complex numbers have 2 elements
(real and imaginary). Quaternions - four. And so on._

Yes, but if you go too far with thinking this way it becomes easy to confuse
complexes, quaternions, octonions, etc with 2-,4-, and 8-dimensional vectors.
The additive and multiplicative behavior of complexes, etc changes in
increasingly pathological ways as you increment the dimension. This is not the
case with vectors.

In many cases you can safely replace R^2 with C. But there are specific cases
where you cannot, because treating the complexes as just a pair of real
numbers doesn't work the same way as if it was just a vector. Differentiable
functions come to mind because they behave differently in R^2 than C, and when
you're working with rings (instead of fields and vector spaces) they are also
different. In a lot of places the isomorphism between R^2 and C is actually a
happy accident rather than a definition.

