What are the Types of Numbers? Real vs. Imaginary, Rational vs. Irrational

We've mentioned in passing some different ways to classify numbers, like rational, irrational, real, imaginary, integers, fractions, and more. If this is confusing, then take a look at this handy-dandy guide to the taxonomy of numbers! It turns out we can use a hierarchical scheme just like we do for biological organisms or fundamental particles, and it's really useful for understanding the properties of different types of numbers.
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[@emaq2006]

I've been struggling at every single lesson in my math class, but when professor dave explains ANYTHING its as easy as 1+1

[@A_Good_Boy.]

Really 😮🥺 not for me

[@lawsonrhea4834]

"This triangle exists, and therefore, root 2 exists" You sound like an ancient philosopher

[@plantae420]

Yes but it doesn’t proof that the square root of two wasn’t invented by people because he didn’t proof that this special kind of triangle can emerge naturally. But at least squares can emerge naturally and the ratio of the diagonal and the edge of a square is the square root of two. Sodium chloride has a cubic crystal system and the surface of a cube is made out of squares.

[@lawsonrhea4834]

@@plantae420 dude i commented this a year ago

[@solarsystem1605]

@@lawsonrhea4834 time is imaginary so Plantae saw your past post in his present

[@JB-iz8bi]

@@solarsystem1605 more like relative. not imaginary

[@A_Basic_Maths_Teacher]

Well, ancient math was grounded on logic, which is what philosophers use, as are most sciences, going as far as using geometry as proof. And mind you, the introduction of, and hence the acceptance of negative numbers and zero were much more recent than you would think (if you would just look at the centuries, you'd see that 2000-2099 is the 21st century all because the Greeks or Romans do not have the concept of zero), and as with the spread current number system that we use today. The name imaginary numbers was a comment of disdain by famous mathematicians of the time when it was first conceptualized, but now it had apparently allowed us to build technology faster by embracing the fact that "real" numbers are insufficient to describe the universe. If I'm not mistaken, all of these concepts were well within the last 1000 years, and I believe although knowledge is spreading and advancing really fast, there is just so much we can learn or teach that we end up graduating and working all the while not seeing the relevance of the math subjects nor the beautiful history, philosophy and logic behind them.

[@SquareFoil31362]

"Imaginary numbers don't exist, they can't hurt you"

[@mwzngd1679]

_i_

[@BurningShipFractal]

Quaternions

[@HafJaf]

Mfs when x^2 is a negative: 😨

[@Johnathan9743]

i = -1 -i = 1

[@crificz]

​@@Johnathan9743dumb

[@julianemery718]

This is... deliciously digestible. I'm sure this took a lot of work for you to distil this information down to what you presented here, I appreciate it very much.

[@ashleyroday3002]

This is the best explanation I’ve found thank you so much!

[@sunilvarma321]

Good explanation....... Forgot all these in busy life

[@gurmeetsharma5870]

No I have no baby 🍼

[@P4R4ND]

Who is watching this because they have an exam soon

[@cringe_lord01]

me

[@humphreyipeli8706]

Yes,please how do u know it

[@TheFatSigma777]

Me

[@PR0DBYKANI]

Tomorrow 😢

[@kervinboyles2638]

Me tomorrow

[@aveinmovil]

First, thank you sincerely, since you put me on the track of learning mathematics, something i was trying unsuscessfuly for a long time . Second, it would be more acurate (althoug not completely) to have the philosopher on the right, whom is Aristotle, pointing out the existence of irrationality in the numbers realm and the one on the right, whom is Plato, getting nuts about it. The one who is said to be annoyed was Pythagoras but Plato belived in the ideal existence of perfect forms so it is good enough. Thank you again!!!

[@alexandragaltseva3493]

This is perfect explanation!)

[@fatamabegum4436]

This is beautifully explained. Thanks 😊

[@HaczzVR]

The intro is 🔥🔥🔥

[@Jasmita-8]

Thank u for the session professor Dave!

[@Omi-to-7-Bunnies]

Thank you!!! You summed it up perfectly and not in an annoying way like some.

[@moradsarwari864]

Thanks for sharing your knowledge of maths.

[@adivasimh39]

What is decimal piont 🤔

[@rossholst5315]

It would seem for root 2 to exist, it requires lines to be constructed with 0 thickeness or width. However such lines are difficult to see. On any physical construction of a triangle, the perimeter of the triangle will be different along the outer edge compared to the inner edge. And then which one is root 2 the outer edge or the inner edge? So it would seem to me that it’s just a mathematical concept. We can draw a very accurate approximation, but true mathematical perfection only exists on the paper.

[@barshakhatun9742]

Thanks for the update

[@kusumvishwakarma9025]

I wish you sir for 1 million subscribers Sorry for bad english

[@sergeantblue6115]

Your wish came true

[@joejomahkerkula9369]

Thanks for the video❤

[@Rorablink]

You are the best professor These videos are so helpful.

[@jpruiz2811]

Hello profesor dave, I didn’t know you make Math classes too. Thank you, i learned a lot from your videos

[@mutabib]

thanks prof .

[@welcome-k7h]

you explained it so simply

[@whitefeather8387]

Professor thank you for your hard work❤

[@removeduser3006]

Thabkyou i have a test tomorrow this helps alot 👍

[@biomolecules7483]

I'm very bad at maths.

[@i_really_like_cats]

Me too

[@FLOURISH2033]

You are also bad in English 😂

[@gomza226]

Real

[@deathmeter7243]

@@FLOURISH2033They didn't make any grammar error in their comment.

[@mousumipaul7319]

Thanks sir this help me too much

[@aftabmohd8783]

Does pi exist as a fraction because of the circumference and diameter of any circle.

[@kripashankarshukla4073]

Sir please tell that what you say after ''Support me on patreon?''

[@ProfessorDaveExplains]

support me on patreon so i can keep making content! i need a little help with the cost.

[@makeyourmommaproud6500]

@@ProfessorDaveExplains Sir we need more professors like you. I can give my support by watching your videos.

[@Curtessiupostol]

finalllyyy. I finish algebra 1😊

[@vincentxie3090]

As a college student, this video has been helpful to me

[@ekobutter]

Sent here by my math teacher

[@rewakfullsworth360]

The song was amazing and explaination is more awesome.Thanks Professor

[@sebastiansantos-se1pu]

So... if it is not an imaginary number it is a real number and vice versa?

[@mathmistico]

True

[@jonykhan4395]

Can division of Two Numbers belonging to Z set produce an irrational number? how can we produce an irrational number through division without involving Two irrational Numbers as numerator and denominator ? If One day super Computers are able to find the exact decimal places of Pi , then pi would no linger be irrational number and it will become the subset of Z and can be expressed as ratio??

[@kuyacuys4065]

Its a really great help thank you so much sir you helped me finished my module

[@Antianti-ti]

great video. I was confused weather imaginary number is irrational or not, but it was solved by you. Thank you.

[@littlequeenp1368]

This is the one i find all day mi a look for one like this thank you so much❤

[@lumri2002]

Would you say that an irrational number is really a real number? After all irrational numbers such as pi and square root of 2 are formula generated values and can not be exactly pinpointed on a number line.

[@carultch]

Without other information, yes. Irrational numbers that lie on the real number line, are real numbers. There are imaginary and complex numbers that are irrational, so not all irrational numbers are real. But if a number's defining characteristic is that it is irrational, and no one bothered to specify if part of it is imaginary, most likely, it is a real number. The terms integer, rational, irrational, algebraic, and transcendental, all refer to how the number fits between previously understood classifications numbers on the real number line, unless otherwise specified that it is non-real. The terms real and imaginary, refer to which number line it resides upon, with complex referring to a sum of real and imaginary.

[@Scynthius137]

Is i ^ i rational or irrational?

[@charishma1102]

Your explanation is mind blowing sir

[@myname3305]

Thank you this is very helpful.

[@ravenarcilla1270]

This is also helpful. kzbin.info/www/bejne/opTLn6CVYp2Knas

[@ramshendage4709]

I will never forget this 😉 Professor Dave explains very well 🤗

[@jocelpearlsantos4183]

Good explanation 5/5 💖

[@nurjohar9170]

Thank you so much you were really helpful 😊

[@samadmerchant7649]

I HAVE NOW CONQUERED ALGEBRA...1. I've conquered Algebra 1. ONTO THE NEXT ADVERSARY!

[@jyothiswaroop2964]

Precise and well explained!! Thanks a ton!!

[@simonesewero9405]

Thank you :))

[@iniyatheanimator8416]

Educational for us👌

[@alishadesai1966]

Well explained just looking for a video giving info about the numbers and found ur video very helpful..

[@clivehaanyika5667]

Thank Professor

[@marcusgeneroso1920iloveyou]

nice one

[@naleanibbs3922]

Thank u smh im am struggeling with that tyyy

[@sreerage5329]

Is 7/22 rational or irrational?

[@chenpinkofficial]

Great explanation, Prof. Dave!

[@Ilovebrushingmyhorse]

I can’t wait to use this information in my life

[@James-hs1eq]

Update?

[@Ilovebrushingmyhorse]

@@James-hs1eq I still am waiting but still can’t wait to use the information in my life, it will be so great to have this knowledge in a situation where a number is irrational or imaginary and not real but will be able to identify that because I learned about these types of numbers, it will be so worth it for moments like those yknow

[@Ilovebrushingmyhorse]

@@James-hs1eq you just never know when you need to know if the number you’re looking at is real or not, and it’s really important to know if it is. Taxes? Bills? Preparation for adulthood? None of that is as important than this bro, this is like life changing information tbh

[@James-hs1eq]

@@Ilovebrushingmyhorse Sarcasm 100 lmao

[@abinayaananthanarayanan3241]

Best ever explanation I have ever seen Thank you so much

[@ReimskyToussaint]

Hey, thanks for the video. Could you detail which naming convention(s) you use to nominally classify numbers? Especially for the terms "whole numbers", "natural numbers" (I also once came upon a source claiming "counting" and "natural" numbers are different by the exclusion of zero).

[@angelmendez-rivera351]

This is a good question. There is no authoritative source that uses the symbol N or the name "natural numbers" and excludes 0, and the ISO does not even recognize the "whole numbers" nomenclature, since in most of the rest of the world, "whole numbers" refers to the set of integers Z. Mathematically, the only sets you make axiomatically are N and Z, and N includes 0, with the definition that 0 := {}, and if n is a natural number, then Union(n, {n}) is a natural number. There is no "whole numbers" nonsense, which is largely an invention of North American schools of the 20th century.

[@ramashankerkumar9233]

Love you Sir,U r the Great

[@josephreimarkco9693]

Excellent!

[@brightbloomingstars6473]

Great video! Thanks Professor.

[@YamadaDesigns]

Is there only one type of imaginary number, or are there significant subsets in there?

[@carultch]

The same subsets of real numbers also exist in the subset of imaginary numbers, once you factor out the imaginary unit. For instance, given 2.54*i, you would identify this imaginary number as a rational imaginary number, because once you factor out the imaginary unit i, you end up with 2.54, which is an ending decimal and therefore a rational number.

[@blackrasberry58]

yo my teacher is making me watch this and this is the first video that i actually understood what they were talking about so thanks

[@mariaheizels.agujar6658]

Best explanation indeed!

[@jeyanthr6284]

Lol I am in grade 12 studying applied mathematics but I came back here to understand the basis properly

[@GRasputin91]

Do irrational numbers really exist? Because even if you can't calculate the exact number, obviously circles do exist and...well I'm not sure how else to say it but circles exist and they have boundaries.

[@wadkarchandrakant4091]

Hats off to yout teaching style 🙇

[@channel12381]

❤👌Professor. Love from India!

[@yamatanoorochi3149]

I'm confused as to why 9/9 would ever give 0.999999... I get that it's following a pattern with all the other digits, by why would that override the fact that splitting a nine by itself would give us 1, plain and simple? I understood the explanation of 0.999999... and 1 not having anything else in between because the decimal extends infinitely, I'm just confused as to why 0.99999...=9/9 Well, I gues 0.9999999... would be equal to 2/2 and 3/3 snd 4/4 anyway, so maybe ny question is pointless

[@albertleinstein5378]

I LOVE YOU

[@albertleinstein5378]

1/9=0.11111111111111111111111111111111111111111111111111111111111111 2/9=0.22222222222222222222222222222222222222222222222222222222222222 3/9=0.33333333333333333333333333333333333333333333333333333333333333 4/9=0.4444444444444444444444444444444444444444444444444444444444444444 5/9=0.5555555555555555555555555555555555555555555555555555555555555555 6/9=0.66666666666666666666666666666666666666666666666666666666666666 7/9=0.777777777777777777777777777777777777777777777777777777777777777777 8/9=0.8888888888888888888888888888888888888888888888888888888888888888 9/9=0.99999999999999999999999999999999999999999999999999999999999999999

[@albertleinstein5378]

1/3=0.333333333333333333333333333333333333333333333333333 2/3=0.6666666666666666666666666666666666666666666666666666666 3/3=0.99999999999999999999999999999999999999999999999999999999999

[@albertleinstein5378]

1/7=0.14285714285 2/7=0.28571428571 3/7=0.42857142857 4/7=0.57142857142 5/7=0.71428571428 6/7=0.85714285714 7/7=0.999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999999

[@yamatanoorochi3149]

@@albertleinstein5378 sir, do you have any idea how much joy your replies brought me? Mildly pisses me off that KZbin decided to make them near impossible to access except through your profile though But anyway, NO, I LOVE *YOU*!! THANK YOU!!

[@৩৬স্বপ্নাআক্তার]

Thanks

[@OmegaCat9999]

I know I'm about 5yrs late but what about p-adic numbers?

[@renji20]

Most fire intro

[@DaNerdWiiGamer6512]

Pi is also 3.1415926535897932384626433832 and that’s all I memorized

[@orotewilderness2913]

my brain cant comprehend this. idk what hes saying anymore.........good luck on my test

[@ibrahimhafez7885]

I understand easily.... Thanks, professor!

[@roblovestar9159]

So is the square root of -2 both irrational and imaginary?

[@thedude882]

Is an imaginary number.

[@carultch]

Yes. The two classifications are not mutually exclusive. All the classifications of real numbers, also occur for imaginary numbers.

[@monettelabiche3601]

Explaination was very clear

[@pista69420]

isnt 9/9 = 1?

[@panf7738]

i am a student from mexico, professor Dave explain much better than my professor.

[@kevinbenavides92]

Thank you! 🖖

[@ArhantLone]

Great!!!!!!!!!! 👍👍👍

[@kathypan7713]

Great explaination!!!

[@countvlad8845]

I don't understand how a repeating decimal that repeats to infinity is rational and an irrational number with nonrepeating decimals to infinity is irrational. They both go to infinity. It seems arbitrary.

[@ProfessorDaveExplains]

Because they can be expressed as a ratio of integers. Irrational numbers can't.

[@countvlad8845]

@@ProfessorDaveExplains I thought pi was the ratio of the circumference to the diameter of a circle. I'm thinking the circumference and diameter could be an integer. At least in this world.

[@ProfessorDaveExplains]

Yes it is that, but it can't be expressed as the ratio of two integers.

[@countvlad8845]

@@ProfessorDaveExplains Thanks for the answers. But I don't see what it has to do with infinity and rationality. Both rational and irrational numbers' decimal places approach infinity, so in that sense, you could say that both types of numbers are the same. But why use the word rational or irrational if both are approaching infinity?

[@ProfessorDaveExplains]

I can't express it any more simply. Some repeating decimals can be expressed as the ratio of integers. Others can't. That's the difference.

[@user-yr3uj6go8i]

Just to clarify: Do I only conjugate if there's an irrational number in the denominator?

[@simularfreak9232]

4

[@bradleyplummer5778]

my face after realizing 9/9 is .99999..... is also.... one... wtf.

[@manishpal5510]

Thanks Dave..... This would help my child a lot...... Love from India

[@TerrelleCheers1]

0 > 1

[@Priceygames]

6:30 wait 9/9 is equal to 0.999999 🤔

[@riahchav151]

Best explanation I found thank u

[@sthandiwezwane8948]

Isn't it he knows a lot about science stuff... professor Dave explains

[@Drsimranmbbs]

Thank you so much♥❤😍

[@tanujapolamuri7634]

I have a doubt, what's the difference between a rational and a fraction.Both are a/b forms & b not equal to zero.

[@angelmendez-rivera351]

"Fraction" is the colloquial English expression we use to refer to expressions of the form "a/b," but in mathematical language, it means nothing. In mathematical language, if you want to talk about expressions involving division, you need to specify what the dividend and divisor are, and from which side you are dividing.

[@carultch]

Rational number is a general term, fraction is a specific term for the notation we opt to choose for it. Rational number means it is possible to express it as a fraction of two integers, while fraction means we opt to use the fractional form to represent it. For instance, 2.54 is a rational number, as are all ending decimal representations of numbers. However, I wouldn't call it a fraction, because I opt to write it in decimal form instead of fractional form. I could alternatively write it as 127/50, and then I would call it a fraction.

[@wongcheehen-un7dr]

Is e not a imaginary number?

[@ProfessorDaveExplains]

nope

[@Anna-zm3td]

I am in 9 grade and was still confused in this, but now I am not thanks to you ~

[@RavendraSingh-sz3nr]

Same

[@cubefromblender]

i just want you, or anyone else rewatching this 6 year old vide is the number 0.05(1206) [1206 is repeating], a rational number?

[@julijuly7461]

Are all irrational numbers non-terminating and non- repeating decimals?

[@ProfessorDaveExplains]

yes indeed

[@zaza8870]

Please don’t tell me I’m the only one triggered at the fact that root-4 is 2i instead of +/- 2i

[@LVP201]

But it's not... it's just 2i

[@abstractguy9]

I am too triggered by seeing √4 equals to 2, but √4 gives ±2... Similarly √-4 gives ±2i

[@angelmendez-rivera351]

@@abstractguy9 No, it does not.

[@angelmendez-rivera351]

By definition, root(z) for any number z is a single number, not two numbers. This is just how the symbol is defined. Look at any textbook and you will see that they agree with what I said.

[@sabrinasabrine2246]

tanks l am from morocco 👍

[@businessenglish8562]

What about decimal?🙏

[@aweebthatlovesmath4220]

Decimals are just rational 4.6=46/10 or 23/5

[@yasmeenabanoo]

NICE