Imaginary Numbers Are Just Regular Numbers

tauting is ugly, share and people will absorb, that simple😉

Hey, thank you for the clarification...But this still doesn't clear up the mess I am having in my mind...this doesn't make sense....why rotation ? Are we saying that real numbers and lateral (imaginary) numbers are functions of each other ? Acting like dependent variables ? What exactly does it mean when I find +or - i as the solution of my quadratic equation ? Does it mean that I actually have a solution, but in different dimension? Eh ? What ?

I guess both Platonic and non-Platonic people have a point. Numbers exist in nature though. Take the harmonic series, if you’re interested in acoustics. But sometimes not all numbers exist in nature.

Wow. Finally I see an intition I can just easily grasp. Thanks!

congratulations. Very well explained !

In my opinion imaginary numbers are just a mathematical crap because it breaks fundamental rules such as that sqrt(x) has a meaningful result only for positive x. Most people use them because it is taught that they are needed to understand fancy Hollywood style theories (relativity, quantum etc) but no one really understands what is the tangible result. And of course none of those theories are proven, but many books are written and sold, papers are published etc.

Can anyone explain me the meaning of the equation 5x1 = 5x5 = 25 ?

Destiny Bombay.

So the obese drug dealing pimp dj khaled pretended that my dick banging connie was his? Pathetic. Anyways I got pics from when i was there.

simple yet taught me more than 3Blue1Brown videos

they are still imaginary and not real though, the 90 degree rotation does not really represent something real like 4 times 4 people= 16.

What a great video. Onya.

anyway i can just see one thing which is actually imaginary is you =my girlfriend (ik u are married maybe..cool!) but all i think is kissing you so hard ❤

Great video. I completely agree about 'imaginary' being a bad name.
As implied in your video, we might as well call all numbers imaginary apart from the natural numbers. Fractions (I.e. rational numbers other than the integers) were invented by humans to give us ratios; irrational numbers numbers invented by us to solve equations; and negative numbers invented to make sense of debt/oweing/any kind of 'shortfall'.
Would be better to call real numbers 'X Numbers' and Imaginary numbers 'Y Numbers'

Can you make videos about the different schools of thought in math as you introduced in the video?

They drowned the discoverer in the Mediterranean! - And we think our generation's nerds make big deals over things and fight too much on internet... That's cute....

Imaginary Numbers Are Just Regular Numbers - exactly they're all not real.
Actully not heard of the 'Platonist' view of numbers. Really? People take that seriously?

I had an idea of what imaginary numbers are (highschool curriculum and a video from Veritasium), and now you've just made it more clear. 🎉🔥

I never imagined nunbers to be two dimensional, though I knew this concept. Awesome!

Also, I wonder in which dimension do 10-adic numbers or p-adic numbers exist 🤔

That's a Brilliant way to learn imaginary numbers intuitively ! ✅💯

When you really think about it, negative numbers are just as "imaginary" as imaginary numbers.

Men with beards are just men without beards, with beards.

You can never multiply any negative number by either another positive or negative number as how you have been educated. This nonsensical human notion is one of the causes of the nonsensical notion of complex numbers
Here is a very easy way to understand that you have wasted many years on learning such nonsensical mathematical notions of mathematics at universities:
If you know anything about hardware architecture in computer science, you will see that all such mathematical operations, such as multiplication and division operations are actually done by the ALU, in addition and subtraction operations like this:
5 x 5 = 5 + 5 + 5 + 5 +5 = 25. The number of the number 5 here is 5, which can also be expressed as:
1 2 3 4 5 = 5 times of 5 = 25
Now putting the negative sign before " 5 times" will give you the following nonsensical mathematical expression:
- 5 times of 5 = -- 5 x 5 , which means that the number of the number 5 is negative 5 , which is obviously an expression of human ignorance because it is the same as answering the question about how many times you go to school each week, and you'd say that negative 5 times

You can never ever multiply any negative number by another negative to come up with a positive number number either , as how all of you guys have been ignorantly educated , because if so, you will basely change a decreasing in quantity value into an increasing in quantity value, which is disastrous wrong in the real physical world and objective reality, outside human-created subjective reality and defective concepts of mathematics
I'll take your number 5 to invent a mathematical problem as an example for you to understand my mathematical statement:
The population in San Francisco, CA before the pandemic was 2,500,000 people. The population decreased by 5% in 2020 when a lot left SF due to the lockdown and available telework, and in 2021, it decreased again by 5 times of 2020's decreasing number of people. How many people still remained in SF in early 2022?
The number of people left in 2020 = 2,500,000 people( 5/100) = 125,000 people
125,000 people is the decreasing number of people, and thus it must have negative sign to indicate that it's the decreasing number of people to be taken away from the total population before 2020 when the pandemic started------> - 125,000 people.
Hence, the population at the end of 2020 is
2,500,000 people - 125,000 people = 2,375,000 people in 2020

In 2021, the decreasing number of people is 5 times of the decreasing number of people in 2020, which is -125, 000 people :
5 times of -125,000 people = -125,000 - 125.000 - 125,000-125,000-125,000 = -625,00 people
1 2 3 4 5 = 5 numbers of -125,000 people = 5 x - 125,000 people = - 625,000 people

The negative sign of the number of 625,0000 people here merely indicates that it is a decreasing in the number of people who have left SF, but it does not have any actual quantity value at all
Therefore the population at the end of 2021 is:

2,375,000 people - 625,000 people = 1,750,000 people in 2022

The number of -125,000 people is 5 numbers as shown above, or 5 numbers of -125,000 people is the same as 5 times of -125,000 people. Now baselessly or ignorantly putting a negative sign before the number of -125,000 people, which is 6 times, will give:

- 5 times of -125,000 people = - 5 x -125,000 people = + 625,000 people = 625,000 people,

which is not only nonsense because the decreasing number of people has now become changed from 5 number or 5 times of -125,000 people into negative 5 numbers or negative 5 times of -125,000 people, but the negative sign - before 5 times has also turned the decreasing number of people - 625,000 people into an increasing number of people + 625,000 people, with the positive sign, which actually represents an increasing in the number of people, who were to be added to the population in 2021, instead of substracting 625,000 from the 2020 population. As a result, the population in 2022 would be :
2,375,000 people + 625,000 people = 3,000,000, which is obviously completely wrong

- & + are also arbitrarily used to indicate 2 opposite directions, while all corresponding negative & positive symbolic quantity values are the same and expressed as absolute values | +/- Number| = number, where number is actual symbolic quantity value, & - and + merely indicate 2 opposite directions


That is only one small example extracted from a book of STEM which i will publish in a few years, and which covers all natural sciences, including quantum mechanics, particle physics, mathematics, chemistry, astronomy, computer science, and micro-biology

Your understanding of human-invented concepts of mathematics that there are values which are less or smaller than nought or zero or nothing is a fundamental misunderstanding of the meanings of the negative sign. There is not even anything anywhere in the universe, which is actually existent in such a a form of non-existent-being as human-invented concept of nothing or 0, and the only things which can be nothing or 0 are the non-existent within human wild imagination. Hence, there is not any actual form of matter, which can be less or small than 0 in the physical world. Such nonsensical quantity values exist only in human -invented and defective concepts of mathematics. There are a lot if flaws in human-invented concepts of mathematics, including number systems, which you guys have been taught at schools of all levels of education, including post PhD. That's why there are ignorant human mathematicians, and scientists who worship such nonsensical human notions as the empty set in linear algebra and abstract mathematics

Your understanding of numbers as invented by humans is correct. However, your idea that number sign - & + were invented, & have been used by humans to keep tract of things is not very mathematically correct because what the actually represent are a decreasing in quantity value which is to be taken away from another quantity value, and an increasing in quantity value which is to be added to another quantity value.
Additionally, your understanding of imaginary numbers is incorrect because square root or even root of all negative integers can be certainly be easily found.
If you really want to know how it can be so, you can see my explanation in the replies under some of the videos about imaginary numbers
Numbers signs are also used to indicate opposite properties of forms of matter in physics, such as electrons and cations, and electrons and protons or nuclei

Do I 'revolve a lever about a pivot' or 'rotate the lever about the pivot' ? [ The pivot' in this case being 0+0i on the complex plane and the length of the lever being |Z|, where Z=a+bi=|Z|(cos(A)+i.sin(A)) & 'angle A of lever wrt ground'=Arg{Z}=A. ]

Carl Gauss called them 'lateral numbers'.

He said the following:
"Hätte man +1, -1, √-1 nicht positiv, negative, imaginäre (oder gar ummögliche) Einheit, sondern etwa directe, inverse, laterale Einheit gennant, so hätte von einer solchen Dunklelheit kaum die Rede sein können."

English translation:
"If +1, -1, √-1 had not been called a positive, negative, imaginary (or even impossible) unit, but rather a direct, inverse, lateral unit, then there could hardly have been any talk of such obscurity."

The value 'i' can be thought of intuitively as the the result of the operation 'rotation of +90 degrees' (and translation) on the value of a real or rational number, just like rotating and translating a 2d vector
v=|v|(c,s) by applying a 'tangent & translate' operator 't' to it, so that t(v)=|v|(-s,c) is the anticlockwise rotated and translated vector of the same magnitude that is orthogonal to v, equivalent in direction to a tangent vector of a circular trajectory at the end of a radial vector, but then mapped back onto the original positional plane wrt it's origin.

The number i on the complex plane maps a complex number to a another of the same size that is 90 degrees' ahead of it, so technically it's a 'revolution about the origin' rather than being like a pure rotation of a vector which would be about the midpoint of that vector.

For 2d vector v:

Revolve(v):=
" Rotate about v midpoint + Shift v tail end to origin & head to opposite direction "

Rotate(v):= 

" Displace vector v head H by angle A, tail T by angle A such that H moves along circular

arc |v|A ~ sum{ 2Rsin(dA/2) . dv} & 

T moves along arc |v|A 

~ sum{2R sin(dA/2). (-dv)} "




Comparing position vector v=|v|(cos(A),sin(A)) relative to origin O
to complex number Z=|Z|(cos(A)+isin(A)),
where angle A=Arg(Z) and O is overlapped by 0 of a complex plane:
Zi=|Z|(icos(A)-sin(A))~t(v)=|v|(-sin(A),cos(A)) _rel O

One way to decompose the operator t is to:
[1] Double the size of v (relative origin O: v=OH), having the tail at the opposite position of the original (instead of point O it's now at O'), call this v'= (2v) _ rel O', with the head opposite O' at point H twice the distance from O' as O ,
so that : v'=O'H=O'O+OH, |O'O|=|OH|.
[2] Rotate v' about it's midpoint by 90 degrees' relative O: R(v')= -v' _rel O'' ,
H' =my(H) is new head and O''=my(O') is the new tail, m=mirror about Y axis.
[3] Shift R(v') tail at O'' to O, halfing the vector size: S(R(v'), O'')= -½v'_rel O'', O''—>O.
[4] New vector is -v_rel O =t(v).

You can carry out all these steps, or just say 'v has revolved 90 degrees about O'.

Your videos are good. I wish you would have made one for complex numbers also...

Thanks for the extraordinary explanation. I really like it!

They were so upset they did what??

This made SO MUCH more sense to me that when we were learning about imaginary numbers back in HS. That was in the 80s. And I could grasp it pretty well as a concept. But this actually made sense to me! Thanks! :)

Jade, you've actually PUT ON SCREEN,a depiction of the physical tangible property "seven" of "seven puppies", which is something that can be confirmed by observation just as much as the property "playful", and then your words contradict what you're saying, denying that "seven" is a tangible physical thing. You say that WHILE SHOWING US a demonstration of the opposite. (At the very least if the count is not a thing then it's a physical property of things just as much as a frequency is a physical property of a ray of monochromatic light even if the "frequency" can be argued not to be "a thing" with a physical reality without a light-ray to HAVE the frequency. Which I'd say is a flawed argument but not NEARLY as absurd as what YOU'RE saying.)

Go into the backyard and play with 7 puppies for an hour. For what portion of that hour did the seven puppies become six puppies? Eight puppies? If they DID become 6 puppies, what would your reaction have been? Would you have said "Nothing changed. Seven wasn't a property of this group of puppies. It was just the way I, as a human, chose to see them. I'm now re-inventing things to see them as six in number."? Or would you say "OH NO! One of the puppies has gotten lost!!"

The assertion that "7" is an imaginary number is completely wrong and contrary to science. Seven has a real and material existence in the world of physical objects, and it has been so since before humans evolved and could say "there's seven yaks". If we find a petrified nest (at least one dinosaur-nest has been found) of three dinosaur eggs and later find another nest of three dinosaur-eggs, and find a third nest that has four dinosaur-eggs, then at some point before humans evolved it was true that the first two nests and eggs possessed a PHYSICAL PROPERTY (just as they possessed mass, volume, and density) of being "just as many eggs as" each other, while the third nest, at that point, did not have that property. The integers are clearly NOT invented. But by extensions, you can figure out that zero's not an invention either (it's a PHYSICAL property of my basement and dragons that my basement contains zero dragons on Tuesdays), and nor are negative numbers (because a mound of dirt with a positive volume of dirt can be added to a hole to result in mound of zero volume and a hole of zero volume, proving that AS A PHYSICAL PROPERTY (not as a human way of constructing the world for our discourse) a hole may have a positive volume of air or water but has a negative volume of dirt (or pebbles or whatever it would be that WOULD displace the water if it were there).

Addition too is NOT something that humans invented. Imagine a sack of tennis-balls. Imagine that the number of tennis-balls in the sack is what we'd call "seven". (In other words, it really has what we think of as seven balls in that the wordless PICTURE of seven balls is what it looks like, although you can imagine a world in which language evolved differently and this number of balls is CALLED by the word "six". It'd still be what WE call "seven" even if we didn't CALL it "seven". That number was "seven" before any human had any name for ANY count of things.) Imagine two other sacks, one with three tennis-balls and one with four, of course by the same rules as for seven, it's that we SAY "that's three" but that there's balls in there that LOOK like what you're imagining when you imagine three of something, even if you have 100% aphasia because of head-trauma and words just don't have meanings anymore.)

Now pour the sack with three tennis-balls into the sack with four tennis-balls. Does the latter sack now hold "just as many" tennis-balls as the sack with seven tennis-balls? If the "maths is invented" people are correct, then the answer to the question could be either "yes" or "no" depending on how we "invent" the result of adding three and four. But that's bonkers. The fact is that it is a PHYSICAL PROPERTY of this operation that after the four-sack and the three-sack are combined, the resulting sack's tennis-balls can be put in one-to-one correspondence with the seven-sack. We cannot change the actual PHYSICAL REALITY resulting from combining the sacks to make it so that it's just as many tennis-balls as a sack containing only six, or eight. Addition is a process for describing, and working with, PHYSICAL RULES about the way MATTER behaves, and WE can't re-write those rules, and those rules existed before there were any humans around to engage in rule-making. We dd not invent, and we did discover, what addition is and how it creates results.

Subtraction and multiplication are also not invented, as you can work out for yourself, but I will say that if you lay out apples in three rows and two columns, and sack them up, then it's impossible for that sack to NOT contain "just as many" apples as a sack that contained six apples before you started. This just doesn't happen. Think of it: if addition and such are human inventions, then "conservation of mass" could NEVER have been regarded as a law of physics, because how would it be proved without this idea that the result of adding two things together is determined by PHYSICS, not by humans writing rules as we please.

Division too works by rules we don't invent, and division will lead us into a place where we are forced to deal with ("deal with", not "invent") fractions, and the rules for performing the four operations of arithmetic on fractions can also be shown by similar arguments to derive from PHYSICAL MATTER and the way cut-up objects behave in the MATERIAL WORLD. None of this is invented.

The illuminati are among us. I had never heard of this rotation idea. Jade has just illuminated imaginary numbers in my brain. 🙂 Great stuff.

Yeah but what does it even mean to multiply by a negative number ?

I like the "apartment complex" illustration!

My dear Up and Down Atom
First time I understood complex numbers so clearly

I have studied Electronics and Communication Engineering in UV but before I was in college(for class/grade 11 & 12, schooling is till 10 in my country then comes college then UV if not the college comes again) and their they taught us the angle included with an imaginary number, maybe I was lucky.

That’s exactly why I’m watching this video cuz of me questioning what I actually know about “imaginary” numbers due to wave functions lol

It's a first step in dymestification of complex numbers - Thanks

I'm no expert, so I will defer to you to correct me should I be mistaken, but the way I understood it, complex numbers don't exist in the quantum physic reality. The reason why they are being used is that they make very "complex" (pun intended) calculations with nasty sin x and cos x easier to do. The "i" part of the equation eventually cancels out or is simply ignore, keeping the "real" value as the answer to an otherwise lengthy integral calculations. I'm clueless whether the imaginary part tells us something about the nature of things at the quantum level, so maybe there is more to it. Feel free to chip in if I'm missing a core concept that makes complex number something more tangible.

A pragmatist could ask you, "how difficult to think about this problem without them" Do imaginary numbers make it easier? Or do they make the impossible possible?

Rotating systems are not used just in higher order physics. They are used in alternating current. The standard way to calculate alternating currents is to use complex numbers.

Awesome 👌

I notice that this channel only accepts nice comments, so critics are not allowed. That's the way modern mathematicians and scientists do their science, worst that in times of the Holly Inquisition.

This is a truly mind-blowing illustration of imaginary numbers! Thank you so much!