What are the Types of Numbers? Real vs. Imaginary, Rational vs. Irrational
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- Опубліковано 11 жов 2017
- 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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"Imaginary numbers don't exist, they can't hurt you"
_i_
Quaternions
Mfs when x^2 is a negative: 😨
i = -1
-i = 1
@@Johnathan9743dumb
"This triangle exists, and therefore, root 2 exists"
You sound like an ancient philosopher
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.
@@plantae420 dude i commented this a year ago
@@lawsonrhea4834 time is imaginary so Plantae saw your past post in his present
@@solarsystem1605 more like relative. not imaginary
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.
I've been struggling at every single lesson in my math class, but when professor dave explains ANYTHING
its as easy as 1+1
Really 😮🥺 not for me
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.
Professor Dave, do you have any worksheets to accompany this? I wish I could print your chart out with little descriptors. In the meantime I will make this on my own, thank you so much.
This is the best explanation I’ve found thank you so much!
Good explanation....... Forgot all these in busy life
No I have no baby 🍼
Hello sir!
Sir I’m trying to improve my math and take out my math fear and i guess I’m just at the right channel. Thank you sir
This is beautifully explained. Thanks 😊
This is a grade-saver! Thanks!
Thanks for sharing your knowledge of maths.
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).
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.
Thank you, it was pretty useful 👌☺
Thanks for the update
Thank you so much Sir.
A save once again. Thanks professor 🥹🙏🏻
This is perfect explanation!)
Thank u for the session professor Dave!
Thank you so much i have been doing bad in math class until I started watching your channel
You are the best professor
These videos are so helpful.
This video really helped me thank you so much
Wow sir now I understand very quickly thanks sir 😊
Does pi exist as a fraction because of the circumference and diameter of any circle.
Thank you!!! You summed it up perfectly and not in an annoying way like some.
Thank you for the chart. I think my students will understand this much better now. :)
I tried this one to my students. This also a good help.
ua-cam.com/video/lchnm_-8WgI/v-deo.html
thank you! very good explanation i’ll be subscribing
Thanks a lot, Sir
Thank you professor : )
Sir please tell that what you say after ''Support me on patreon?''
support me on patreon so i can keep making content! i need a little help with the cost.
@@ProfessorDaveExplains Sir we need more professors like you. I can give my support by watching your videos.
Thank you this is very helpful.
This is also helpful.
ua-cam.com/video/lchnm_-8WgI/v-deo.html
Thabkyou i have a test tomorrow this helps alot 👍
Educational for us👌
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!!!
Excellent!
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.
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.
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.
Who is watching this because they have an exam soon
me
Yes,please how do u know it
Me
Good explanation 5/5 💖
Thank Professor
Wait what about negative of those imaginary like negative 1 times square root of negative 1 or simply -1i is this really possible.
I wanna blow your mind
Remember when we write numbers on a line
You have to write another line with 90° and that's imaginary line
A number can be real and imaginary in the same time
Like 2+4i 1-i we write them as z
Z=a+bi
For better understanding go search complex numbers
Nice explanation
Thank you! 🖖
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??
Is 7/22 rational or irrational?
Thanks sir this help me too much
Is i ^ i rational or irrational?
This is the one i find all day mi a look for one like this thank you so much❤
Professor thank you for your hard work❤
nice one
Hello profesor dave, I didn’t know you make Math classes too. Thank you, i learned a lot from your videos
Nice explanation. Tnk u sir..
Great explanation, Prof. Dave!
Great!!!!!!!!!! 👍👍👍
Thank you so much you were really helpful 😊
When you explain what is essentially a whole lecture at school in 9 minutes
What is decimal piont 🤔
Can i get this ppt.
You are saying that 9/9 = 0.999999999... which = 1, I can't understand why that? It's just obvious that giving 9 apples to 9 number of people means everyone has one. So how decimals can exist here?
I'm waiting for the answer from you porfessor.
However, I love this channel, it really makes things as easy as 1+1. Never knew it's so easy and never felt so interested in math like now, although i was interested in it before too!
There is no number that could ever exist between 0.9 repeating and 1, therefore they are the same number.
Hey I have another explanation, if u mind checking on it
Actually
1/9 = 0. 1111...
and so if we multiply it by 9 on both sides of the equation, we have
9/9 = 0.9999...
I hope this helps u
Good work. But 9 over 9 (9/9) equals 1 not 0.9999... , right?
Those are the same thing.
Thank u smh im am struggeling with that tyyy
Sent here by my math teacher
Good job brother ☺️
As a college student, this video has been helpful to me
Thank you
Professor pardon my ignorance, but kindly explain again how can .99999........(infinity) be equal to 1 ?
there is no number between them, so they are the same number
It's a limit. The limit as sum of 9/10^k from k=1 to infinity will equal 1
Just to clarify: Do I only conjugate if there's an irrational number in the denominator?
4
Its a really great help thank you so much sir you helped me finished my module
At 6.35 min 9/9 = 0.9 bar . How??🤔🤔🤔
Dear Professor
As you said in video "pi" is irrational that is it can not be described in terms of (a/b) but why we can express "pi" as (22/7)????
that's just an approximation! you can get very very close that way, but no matter what numbers you put in that fraction, it will never be precisely pi.
Dear Sir ,
you mean to say that ''pi" is not equal to (22/7) ???? if so who make that number 3.14.... how it get discovered ? A sequence can not be made until it is not generating from a ratio.Sir that sequence 3.14.... is generated from (22/7) i.e. it is a rational because it is described in terms of a/b as you said in video.
It's a rough value made by Archimedes. There are fractions which are way closer to pi's actual value, but they are never pi. 355/113, 103993/33102, for example, are even closer
DrJimmable Thanks for clarification
Blueton Thanks for clarification
The song was amazing and explaination is more awesome.Thanks Professor
Ooh so that’s how gojo’s power works.
I wish you sir for 1 million subscribers
Sorry for bad english
Your wish came true
Thank u
Hey prof, you defined pi as the ratio of circumference of a circle to it's diameter, isn't that a ratio, so wouldn't it contradict the proven fact that pi is irrational?
Rational means that it can be expressed with the ratio of 2 integers. The ratio of a circle's circumference to its diameter is irrational as all the time, either the circle's circumference, or the diameter of the circle will be irrational, so there is no way to express pi as a ratio of 2 integers.
It is a ratio, but not a ratio of two integers, so it is not a rational number. If the radius is a rational number, then the circumference is not, and vice versa.
No, because it is not a ratio of integers, as is the definition of a rational number. You might be able to measure a circle's circumference and diameter, seemingly in integer millimeters, for instance, if you had 100 mm diameter circle, and measured its circumference as 314 mm. That may be very true, and exactly what I would get, if I were measuring the same circle. It still does not imply that it is a ratio of integers.
Your naked eye cannot tell that the 314 mm you measure is not exactly 314 mm...or that 100 mm was exactly 100 mm, which I cannot know. We cannot know for sure, because we only have a certain amount of precision we can measure.
Thanks
Isn't the square of any positive number both negative and positive ?
Nope
Minus multiplied with minus gives plus
Actually it square root of positive numbers is both positive and negative
@@abstractguy9 No, it is not. While the equation x^2 = y has two solutions, only one of those two solutions is actually called the square root.
The definition of the square root a real number y is the *non-negative* real number x such that x^2 = y, and the relation x |-> "square root of x" is a function, so the output is unique. It does not have two outputs.
thanks you are prof
finalllyyy. I finish algebra 1😊
Good video
I can’t wait to use this information in my life
Update?
@@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
@@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
@@Ilovebrushingmyhorse Sarcasm 100 lmao
So is the square root of -2 both irrational and imaginary?
Is an imaginary number.
Yes. The two classifications are not mutually exclusive. All the classifications of real numbers, also occur for imaginary numbers.
Best explanation indeed!
Best ever explanation I have ever seen Thank you so much
NICE
I’m rusty at math, but am in to philosophy, which will explain the nature of my comment.
Here’s a question I put out in a bunch of UA-cam comments regarding irrational numbers.
-------
Hopefully someone answers this question.
1: a rational number is rational because it can be represented by the ratio of any two numbers.
This is what I believe it is. If I’m wrong please let me know.
so far, this seems reasonable, as a ratio is how any two entities are related, generally speaking.
But take pi, for example.
isn’t pi the ratio of the circumference of a circle to its diameter? Those are two numbers. And I can represent pi by those two numbers. And potentially I can have an infinite set of “circumference to diameter” ratios to represent pi?
Unless what makes an irrational number irrational is that the resulting ratio is an unending and patternless decimal.
Maybe this latter half is more the definition. But the problem is why call it irrational?
Irrational suggests “no relation or connection” in that a ratio is how any two or more entities are connected or related generally speaking.
So, my first impulse was to think, ah, irrational numbers must not have ANY ratio of two
numbers that can represent them. But as with the circumference and diameter, it seems that there can be a whole bunch of pairs of two numbers that can represent Them.
Idk. What am I missing here? I may have understood this years ago, but it’s been a long time.
Because so far, it just seems to me that an irrational number is also represented by the ratio of two numbers but that it’s decimal is unending and patternless.
You're overthinking it. Rational numbers are numbers that can be expressed as ratio of two "integers" (whole numbers). Using your example, "circumference" or "diameters" are not "integers", if you get what I mean. Circumference or diameter of a circle can be measured as a whole number, but circumference or diameter of a circle are not "whole numbers".
@@FriFreeman first, I want to thank you for responding.
And hopefully we can dialogue about this just for a bit.
So, You mention integers, and whole numbers.
Well, let’s talk about that; what are the various classifications of numbers.
Off the top of my head.
1: Natural numbers: Are the counting numbers. But exclude zero, fractions/decimals and negative numbers. Probably because it’s more empirical, where zero and negative numbers are more abstract.
2: Whole numbers: This includes the natural numbers and also includes zero. but still excludes fractions, decimals and negatives
3: Integers: include what it means to be natural and whole numbers, but also includes negatives. But still excludes fractions and decimals.
4: Rational numbers: I thought included all of the above but also includes fractions and decimals.
But you say that “rational numbers can be expressed by integers”
My thought of an integer includes whole numbers and negatives, but not fractions/decimals.
But let me stop there. Do you agree with the definitions above?
@@tatsumakisempyukaku Hi, Your question intrigues me. I find it interesting that as a student of philosophy you are more intellectually positioned to satisfactorily answer your question than a mathematician. As the student of philosophy It would be natural for you to ponder 'which came first' the circle and its diameter or the values and constructs assigned to relationship between them. If the circle and its diameter were first than there was certainly nothing irrational, in the general sense of the word, that could be fairly applied to them. Rather, as you alluded to in your inquiry, that they existed in relationship would in fact argue they were indeed perfectly rational. But consider this: The irrational terminology is not grounded in relationship but in this case in mathematics. It is used to classify constructs that feature yielding values of pattern-less repeating numbers. The question of why this terminology was used, given that it presses on the general understanding of irrationality, is I believe a quite fair one. It is a question of human experience and choice. But for me, I would be more comfortable having that question answered by a student of philosophy. Anyway, thank you for your curiosity. I admire it.
Do anyone know how to plot non terminating decimals (recurring and non recurring) on a number line.
Watch at 3: 40, ua-cam.com/video/sbGjr_awePE/v-deo.html
@@MohammedAhmed-vw8hy ok it's showing what irrational numbers are. But it's still not showing how to plot them on number line or on graph.
@@KhanKhan-tp4ch There is no meaningful way to plot numbers on a real line. The idea of "plotting numbers" is more of a visual aide for learning than an actual mathematically valid concept.
I will never forget this 😉
Professor Dave explains very well 🤗
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.
👍 Now I understand
great video. I was confused weather imaginary number is irrational or not, but it was solved by you. Thank you.
where does the Chaitin's constant fit into this diagram
Uncomputable real number. Also, Chaitin's constants are not specific numbers, but a type of numbers.
Precise and well explained!! Thanks a ton!!
Your explanation is mind blowing sir
Well explained just looking for a video giving info about the numbers and found ur video very helpful..
The video image isn't good. Real numbers are special case of imaginary numbers ai+b, where a=0.
No, imaginary numbers and real numbers are both examples of complex numbers. If a = 0, it is a real number, not an imaginary number.
Lol I am in grade 12 studying applied mathematics but I came back here to understand the basis properly
Is there only one type of imaginary number, or are there significant subsets in there?
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.
I think you forgot to add about composite numbers 😅😅
Dear that way...the list goes on and on
Are all irrational numbers non-terminating and non- repeating decimals?
yes indeed
I HAVE NOW CONQUERED ALGEBRA...1. I've conquered Algebra 1. ONTO THE NEXT ADVERSARY!