What are imaginary numbers?

Imaginary numbers are any numbers that include the imaginary number i. A mix of imaginary and real numbers gives you what’s called a complex number.
The primary reason we use imaginary numbers is to give us a way to find the root (radical) of a negative number. There’s no way to use real numbers to find the root of a negative number, which prevents us from solving even simple polynomial equations.
But with imaginary numbers, we define the square root of -1 as the imaginary number i, and that allows us to easily find the roots of imaginary numbers.
0:21 // Are imaginary numbers real numbers?
0:45 // Examples of real and imaginary numbers
1:22 // The definition of the imaginary number i
1:39 // Why do we need imaginary numbers?
1:53 // Why is the imaginary number helpful?
2:52 // What happens when you square the imaginary number?
Music by Joakim Karud: / joakimkarud
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Hi, I’m Krista! I make math courses to keep you from banging your head against the wall. ;)
Math class was always so frustrating for me. I’d go to a class, spend hours on homework, and three days later have an “Ah-ha!” moment about how the problems worked that could have slashed my homework time in half. I’d think, “WHY didn’t my teacher just tell me this in the first place?!”
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[@bobbrownjr]

Krista, your videos are excellent. I have taught at East Georgia State College for 15 years after I retired from AT&T Bell Labs after 31 years working on satellites, fiber optics, and the first generation WI-FI. I would say about the same things to my students about imaginary numbers that you said. I would then show them that when applied to communications engineering that complex roots like a+bi and a-bi that are derived from polynomial equations give rise to sinusoidal waves that allow radios, cell phones, and fiber optic systems to be possible. We would not have television, radio, cell phones, electric cars, motors, power systems and the list goes on without imaginary numbers. As you said, these "imaginary numbers" make a lot of "real things" possible. Keep up your good work.

[@kristakingmath]

Thank you so much, Robert! I'm honored! :D

[@baruchben-david4196]

I like to say that imaginary numbers require some imagination to understand.

[@byteaesx1373]

We could never thank you enough for your work Krista. I like the term "Lateral" better as Gauss suggested but is to late for that now.

[@CE113378]

Nice video! You should do a video talking about how multiplying by imaginary numbers "rotates" a number, just as multiplying by negative numbers "flips" a number (or rotates the number by 180 degrees). The concept of "rotation" may not make too much sense, until you talk about the phase difference between two quantities that are both oscillating with the same frequency. In other words, by Euler''s identity, e^(i*(omega*t + phi)) = cos(omega*t + phi) + i*sin(omega*t + phi). If a voltage and a current are both oscillating with a radial frequency omega and are phi radians out of phase with each other, and if you multiply the current by an imaginary number, then you have shifted its phase by pi/2 radians. In other words, you "rotated" it. Likewise, if your argument is a function of time (as in my example), then the number is "rotating" with time (aka - oscillating). This would kind of give an intuitive feel for what imaginary numbers actually do - in "real" life.

[@kristakingmath]

Great point! Thank you! :D

[@Mattiethekorbat]

This is the clearest description of imaginary numbers I've heard. However, at 1:53 you say why do I have to bother because I'd never have to use this. I'd rather thought you'd give an example of a real life application! Some time of scenario where it is used outside of academia (even if the average individual is not the one directly using it).

[@fayadrahman552]

A very sweet vivid description of imaginary numbers. Loved it! 😀

[@kristakingmath]

:D

[@santafucker1945]

The field of real numbers is not algebraically closed. However a non-trivial fact is that the field of complex numbers is algebraically closed. Further, the complex field has geometry which the real field lacks. This is why the complex field is important.

[@Ridgerunner2270]

Thanks for the refresher on imaginary numbers!

[@kristakingmath]

You're welcome! Glad you liked it. :D

[@alextl97]

Your videos are very helpful, thanks.

[@kristakingmath]

I'm so glad they're helping!

[@MatthewSuffidy]

I am trying to get to the bottom of it, and I think the point is that some equations, being solved algebraically suggest a number system paradox involving a sqaure root of a negative number. You get real roots when you multiply two of these paradoxical values. I guess this suggests an imaginary number dimension that is accepted today.

[@rollingrockink1]

Your videos are insane, like sensory overload of arts and crafts insane. Your ability to write, produce, direct, act, edit and distribute your videos is impressive. Keep up the creative momentum. Slow down a bit though, I'm sure your subscribers would love to hear you talk more and expound upon your knowledge in the topic. Thank you for the uploads.

[@kristakingmath]

Thanks for the feedback!

[@heha9752]

Really easy to follow! Thanks so much!

[@fuckyou60286]

@KristaKing you were very close to what I thought was a breakthrough in understanding it, in x^2 = -4. Could you simply say that the imaginary number i is just an additional operation that much be performed multiplying the number by -1? so x^2 = -4 is really just (-1 * 2^2) = -4

[@BC-2]

I once asked a well known math professor what the use of the unit circle is. He said "nothing. It does nothing." So how can imaginary numbers be of practical use?? What on this planet would require the use of something fictitious?????

[@vrendus522]

Thanks Krista.I think that imagination is part of a good mathematician's repertoire. Hamilton in England while on a walk with his fiance, had an epiphany beneath a stone bridge and said that he had seen these imaginary numbers on the buttress work of this bridge that gave him the basis of modern Algebra in England.Algebra was also developed in the Islamic and Indian empires, as well as by Leibniz in Germany, but it's funny how inspiration helps us grow in this area of endeavor. Like your earnings, you always look pretty to me and this helps brighten my math.Thanks,. Dan

[@aarondrummond4463]

Great job

[@godofgodseyes]

Can you show me imaginary number representations with apples, oranges and bananas? Hope you can think simple things deep.

[@leoguillon1062]

Great videos, maths explained the easy way. like feynman

[@jupitereye4322]

It reminds me of quantum physics in a way. You know that these sub-atomic "mechanics" result in particles that form atoms, and without it atoms wouldn't exist, but you can't really pinpoint what exactly is it.

[@sarahbracha]

I'd love to hear some examples of how imaginary numbers are used in real life applications? i.e. what professions or functions are they used in, how do they actually work "in real life" since they're not actually real numbers. I mean, if there is no such number that multiplied by itself can result in a negative number, I'm having a really hard time understanding the real life use and application of imaginary numbers. Is this just math theory/philosophy? Thank you! (I'm a math teacher and will need to teach imaginary numbers later this year, so I'd really love to know what they _actually_ are! Not just what i stands for, which I get. I want to go deeper.

[@idigmath]

i is an element of the set of complex numbers. i=(0,1) in the complex plane. The fundamental theorem of algebra shows the connection between the set of reals and it's field extension the complex numbers.

[@indian_scouser_ynwa]

excellent Krista. thank u

[@kristakingmath]

Thanks!

[@johnpeter7832]

Wow my mind just got blown. Nice narration thanks.

[@kristakingmath]

😊

[@Durfit]

Do you hold your markers between your middle and ring finger? I don't recall ever seeing that before. Very helpful video though.

[@kristakingmath]

Thanks, and yes I do. I always have, and never got rid of the habit. :)

[@avak6666]

thank you this is going to help me not fail my math test tomorrow

[@kristakingmath]

I hope the test went great, Ava! :)

[@m.d.lu.m.d9292]

you are my hero

[@kristakingmath]

:D

[@compsciag]

Could you possibly make a video on finding the equation of an osculating circle at a point on a curve? Currently taking multivariable calculus and your videos are a big help. Thank you so much for what you do!

[@kristakingmath]

I'm so glad the videos have been helping! I'll look at doing more osculating videos, here's the general one that I have about osculating planes, in case you haven't already seen it. kzbin.info/www/bejne/sITWiYyBrspmlbM

[@cis7775]

Ah, i = √(-1). Ok, so calculate i·i = √(-1)·√(-1) = √((-1)·(-1)) = √(1) = 1, mmmhh..... So the correct definition is: "i is the operator with the property i² = -1". Especially √(-1) has the two values √(-1) = i or √(-1) = -i.

[@HolahkuTaigiTWFormosanDiplomat]

Really make sense.

[@kristakingmath]

Oh good! I'm so glad it helped! :)

[@zackryderwoo3]

That was a great video! Also, your eyes are captivating.

[@naikkelas1241]

Say I want to compare |sqrt(-2)| vs |sqrt(-3)| or |sqrt(4)| vs |sqrt(-5)| which one is bigger? or what should we do if we have a set of real and complex number and we want to arrange them?any advice?

[@craighalpin9521]

PPIE Salemba sqrt(-3) > sqrt(-2) because they're on the same axis. choosing more convienent numbers sqrt(-9) > sqrt(-4) which is the same as. 3i > 2i. it could be copaired to. 3 apples > 2 apples as long as it's comparing the same type of numbers it makes sense. if you have something like 3i vs 2 that doesn't make sense unless you are talking about complex numbers in which case you would need to compare their total magnitude. ex. 3 + 4i magnitude would be 5 so 3 + 4i > 4 you get the total magnitude by pytherian formula. a^2 + b^2 = c^2

[@Darlene.nails.]

Thanks you are the oknly reason ikm going to pass my math class.

[@courtcomposer]

Well done. Thanks!

[@kristakingmath]

Thanks, Josef, I'm glad you liked it! :)

[@TriptoPal]

I feel Lucy, because I am your student.

[@tahminaktarlaskar1377]

30395 this digits always rotates in my mind what does it mean?????pls help

[@Chil137]

didnt explain anything. Just brought up i and said it solved things

[@baruchben-david4196]

Travis Britton Well, yeah, she did explain them.

[@cliffordwilliams9597]

Thanks for the contribution Travis.

[@mmajid09]

Beautiful method as beautiful you are yourself.

[@Jaimon-kw1tf]

Bruh 2:35 just do square root before to subtract 4 from both sides.

[@JohnnyYenn]

Thanks, this helped me a lot! :D

[@kristakingmath]

Yay, awesome! Thanks for letting me know!

[@darylallen2485]

why are your eyes each a different color?

[@indian_scouser_ynwa]

u have videos on complex numbers

[@Geodendronitrian]

Awesome video!

[@kristakingmath]

Thanks!

[@hamsack981]

would you please do a video on centroids? thank you for your time!😃

[@chriscozilas]

thank you !! finally i understand ;,D

[@enriquediaz6306]

What is the square root of i?

[@michaelhdesimone]

The term 'Real Number' was introduced after the term 'Imaginary Number'. Imaginary Number was used to describe a 'made-up' number. For instance, sqrt(-1) and sqrt(-4). Thus, 'Real Number' represents the opposite of imaginary. For instance, 1, -56, 1/2, and sqrt(2) are 'Real Numbers'. However, I don't associate these numbers as real life numbers. I can't draw a line on paper that is exactly sqr(2) in length. But one can still say that 'Real Numbers' explain real-world phenomenon. Which is also true of 'Imaginary Numbers'. Which is the goal of your video. Well done Krista!

[@kristakingmath]

Thanks, Michael! :D

[@OddAntSounds]

hm ok thanks but couldn't we just say this number don't exist ? if you multiply i by itself you get a negative number which makes absolutely no sense. As an analogy, dragons don't exist but fire, wings, and reptiles do. Maybe an imaginary number is just something that the human mind constructed but doesn't exist in the extrinsic world.

[@aname3504]

Pretty sure they are real life applications of it though in science and technology

[@parthibansachin5137]

Please do video with your complete presence on screen..it's more helpful to concentrate on things you are saying

[@mohamedessam7036]

perfect 👌

[@kristakingmath]

:D

[@moneyharry]

OH!

[@bruceedward3079]

The way your move your eyes and lips is beautiful.

[@AnasAsifSA]

this is basic stuff....i thought i would know how they exist and where

[@muhammadsadiq1792]

derivative of imaginary number is zero am i true?

[@theophonchana5025]

Square root of -1 = error

[@filipsperl]

Actually, one real number doesn't give you a positive value when squared, but I'll let that slip because of your beauty :)

[@lalsenarath]

if -2 * -2 = +4 (by definition), if we define -2*-2 = -4 we don't need imaginary numbers!

[@theophonchana5025]

Irrational numbers

[@lunamanueladelosriosduque6202]

the best

[@AliHassan-hb1bn]

Christie cool

[@Kwaku211]

Got a Degree in Mathematics, still don't really understand iNumbers...

[@kristakingmath]

I'd like to dig into imaginary numbers more, too. I'm hoping someday to make a Complex Analysis course! :)

[@Kwaku211]

@@kristakingmath That would be cool; will check it out!

[@delilithkerk1130]

So .............it is kind of like (the) god(s), it is imaginary but still influence much of our daily practices?

[@sneakywolfiegaming5350]

I dont know if your teaching us real cuas my teacher did not say eniting about that

[@jiezanzhang1323]

i = x

[@solom4590]

hadir boss

[@theophonchana5025]

#eulernumber #e #euler

[@godofgodseyes]

everyone repeat the same thing.

[@khizarhayat8541]

what is wrong with this number didnt find it 1/9=.1111111...... 2/9=.2222222...... 3/9=.333333....... 4/9=.4444.... upto 8/9=.888888....... and 9/9=.9999999............(this number never exist in computers,calcultors and any other device is it real number(i know square is +ve)? ) no real life use even uncertain exist or not.

[@kristakingmath]

What you're missing is that all of those numbers have one higher digit on the end. For example, 1/9=.11111111112, and 8/9=.88888888889, which means that 9/9, using your same logic, rounds up to 1.

[@khizarhayat8541]

i have to round-off the value of 9/9 . or is absolutely 1 difference of .111111111112 and .111111111111 =.000000000001 which so big difference if the .111111111112 and .111111111111 are considered equal.

[@khizarhayat8541]

is the number 9/9=.999999.....so on exists

[@FreeFromAllThings]

All numbers are imaginary

[@nutnutbc4781]

ร

[@CarlosOrtiz-ht6rn]

You're awesome! And soooo Kawaii!! 😁. Thank you for there tutorials!

[@kristakingmath]

Glad you like them!

[@texasmiller5781]

you say too much

[@Stuff_happens]

You are much too distracting :-)