inconsistent

Alternative title: How to tell your math teacher "no".

You forgot the final level "I don't know how to solve this"

“No real value” also happens to be the official mathematical classification for NFTs

Me on my math exam: The answer is left as an exercise to the reader

when your parents ask if you are lying you can just tell them "it's a complex statement"

"No solution" is used frequently in systems of equations. Two parallel lines have no intersecting points and that is the easiest form of all "no solution" problems to understand.

This guy: (flawlessly explains all the ways an equation can have no answer) My calculator: "NaN"

6:51 technically with the definition, the output is always NON NEGATIVE. An absolute value could be 0 :)

About 15 minutes in, for 2⁰, you could argue/explain the definition (not prove) that 2⁰ = 2¹⁻¹ = 2¹ * 2⁻¹ = 2 * ½ = 1.

Honestly thought this was going to be a lesson on how to stand up for yourself and reject requests you don't want to handle.

I love that doll its like Mr . Bean's doll ...

The sixth level of 'no answer' is when you are trying to answer the question "is there a sixth level of 'no answer'."

You forgot to include "No Nontrivial Solution" since every homogeneous system of equations has at least the trivial solution x=0, e.g., in a system of homogeneous linear equations

3:56 how about the most basic of situations with no solution... Solve x+1=x+2 lol

"Undefined" also works when you are using a function with an argument outside the domain of its function. Say you have "f(x) = x for x > 0". You can say that f(-4) is undefined.

In computing science, we also have "I won't tell you" (no permission or not requested), typically a NULL value.

just realised the indeterminate family is on your shirt lmao

The early explanation of complex numbers reminds me of a Top Ten list I did when I was teaching: Top Ten Lies Math Teachers Tell. It began with, "you can't substract a larger number from a smaller one," and, "you can't divide a smaller number by a larger one," and continued with things like, "You can't take the square root of a negative number." Near the top I had, "20 liters of one substance plus 10 liters of another will always yield 30 liters of the mixture," and the #1 lie was ... "You need to know this."

02:35 The case of "No real value" happens also when we calculate the roots of quadratic equation with discriminant (so-called "delta") is negative.

Not only he explains well, but you can also see how happy he is in his face alone, keep it up man. great video

A simple example for No Solution is "x + 1 = x + 2" It almost looks trivially solvable but obviously isn't, regardless of the system.

In my experience, "indeterminate" is applied whenever it refers to a test, as is "inconclusive". Limit tests, like those in the video, and some primality tests are good examples, but I most often see the term used when it has to do with convergence tests for infinite series. For example, the divergence test or nth-term test proves that an infinite series does not converge to any value if the terms in the series do not approach 0, but does not definitively prove the inverse. There are series that do not converge even though the value of their terms approach 0, so in those cases the nth-term test is indeterminate. In any case, it all just means the test cannot prove an answer and more work must be done.

I was going to close this because it was just another tab, but I loved your pacing, and stuck around. I liked your style and level of explanation. Subscribed

I think indeterminate is also used to refer to certain cases of convergence tests for integrals.

19:30 I think 0/0 is inderterminate bc per the long division you used earlier: what, when multiplied by zero is equal to zero? Basically everything. So it is not like 1/0 where we cannot supply a value, it is kinda the opposite, we have too many values.

0⁰ : Could be 1 or 0, but exponent is stronger, so the 1 wins, therefore 0⁰ = 1

I disagreed a lot with the √(x) = - 5 but then I came to understand this really well actually. When ever we want both roots, we actually mention ± which means that √x can only be other the positive or negative value. And as far as mathematics is concerned, √x is *_defined_* to give the positive value. Wow, this makes so much sense now!

Technically, the square root of a complex number is a multi-valued function; whilst the real square root of 25 is 5 by definition (as the square root of a real number, if it exists, is defined to be the positive number whose square yields that number), 25 has two complex square roots, namely 5 and -5. In fact, any nonzero complex number has two distinct square roots. Also, 1/0 can obviously be defined to be whatever you like - be it 17, -3, or even a newly invented number such as ∞. In that case, 17*0 would be 1 by definition; the trouble, however, is that this is not consistent with the algebraic structure of a field, as the distributive law would yield 1 = 17(0+0) = 17*0 + 17*0 = 1 + 1 = 2. It would also mean that the multiplicative inverse of a number is not uniquely determined and all sorts of other stuff - if you're willing to make that trade-off, though, you are free to do so, as mathematicians can literally do whatever they want. Similarly, 0^0 can be defined to be 0, 1, or whatever you want, and in fact, there are contexts where 0^0 is defined to be 1 by convention; this is often done, for instance, to avoid cumbersome situations in general formulae such as the binomial theorem. The trouble is that 0^0 cannot be defined in the real or complex numbers while remaining consistent with familiar properties of limits, such as multiplicativity. This is a crucial point; as long as you're not touching limits, you're fine doing whatever you want with 0^0. In fact, you're even fine with limits as long as you formulate all your theorems about limits while excluding all 0^0 type situations. Of course, that's a lot of work, which makes it an unusual convention. It is critical to realize that definitions can mean whatever we want them to mean; the point of definitions is to capture the essence of a certain object, aid learners in understanding a given subject, and make theorems and proofs as brief as possible. You may, for instance, define 1 to be a prime number, but if you're doing number theory afterwards, it would make your theorems longer because, as 1 has very different structural properties from what we generally consider to be the primes, you would have to keep considering it as a special case and potentially exclude it. Of course, this is all just a bunch of sounds coming out of our mouths that we decide means something, and in general, you should always be able to say "wale" instead of "prime number," "eyeshadow" instead of "cardinality," and "lightbulb" instead of "angle." All of it is arbitrary, after all. This thought is brought to its logical conclusion in predicate logic, which, simply put, is a purely syntactical type of language that starts out with only very few basic symbols. One nice way to picture translating your statement into predicate logic is that you feed it to a computer, whom you have previously given a few abbreviations (e. g. A ∧ B is a shorthand for ¬(¬A ∨¬B), A → B is a shorthand for ¬(A ∧¬B), etc., where, if you haven't seen these symbols before, ∧ means "and," ∨ means "or," ¬ means "not," and → means "implies" - normally, you'd use a different type of arrow for that last one, but the typographical limitations of my device don't allow for that), and that the computer basically "unravels" all of it by substituting in what you wrote these things should stand for and makes a rather long mess out of it. Of course, no mathematician actually thinks in those terms; however, the good thing is that all of it is unambiguous and can be deciphered and even checked based on axioms and inference rules that you are to first declare as valid or invalid.

As a software developer, I’ve had design discussions about the meaning of “null”. In a database, this is when no value is stored. You’ve supplied a useful set of mathematical meanings. Other non mathematical meanings include “not applicable”, “unknown”, “not yet determined”, “invalid”, “declined to enter”, etc. At first this seems too pedantic, but really it can make a database function better to augment a nullable field with a null reason list to express why a value is missing. Unfortunately databases are not designed to do this easily. Null tends to be the design equivalent of a blank stare.

6:25 √(x²) = |x| This would result in |x| = 25 So x = ±5 And ±5 has -5, so there you have your solution

Where would an equation like "x + 1 = x" fit?

you can prove 2^0=1 and the undefinedness of 0^0 if you define exponentiation as x^1=x; x^(n+1) = x^n * x; x^(n-1) = x^n / x; this means that 2^0= 2^1 / 2= 2 / 2 = 1, and for any x it means that x^0 = x / x, which is equal to 1 for nonzero x, but undefined for 0^0 because division by 0 is undefined edit: by this definition, inf^0 is also equivalent to inf / inf

I think 0^0 should be 1 because the exponent (0) says that you do not multiply the base (0), so you are not multiplying by 0.

I am fan of your videos... And your smile when you trying to explain something interesting.. thanks.. for sharing knowledge..

In many contexts, 0^0 is actually defined as 1 since it obeys more algebraic rules than if we were to define it as 0.

Wow this guy is still loving the comments. Salute to you 🙋‍♂️

Excited to watch this video!

Very nice explanation! As always :D

*[**14:22**]:* Misconception here. The equation b⁰ collapses to an empty product when considering repeated multiplication, which is 1 by definition. This is also applicable to 0⁰ = 1.

Thank you so much for the awesome explanation

05:40 First of all, the general reason why this equation has no solution is this: The left hand side of the equation is positive and the right hand side of the equation is negative. So easy ;)

For the no real value, I usually use a+bi

0^0 is undefined because for the x^0 rule, the logic is as follows: when multiplying powers of the same value, you add the power values together, ie. x^a * x^b = x^(a+b) Thus x^0 can be written as x^1 * x^-1. x^1 is just x and x^-1 equals 1/x. x * 1/x = x/x = 1 That means that in the term 0^0, your trying to solve 0/0, which is conflicting because x/0 is undefined, but x/x = 1

6:55 you should say non-negative for abs x ccan equal 0

Everybody gangsta until complex number can't do anything anymore

I love your videos. Both educational and charmingly presented! Btw.: When you said "hello" at the very beginning you sounded 100% German.

"it has no real value" hey look thats me

So, for computations 0^0 is undefined. For limits, 0^0 is indeterminate .

The main difference between "undefined" and "does not exist" is that anything that "does not exist" still has a definition. The lim(x→∞) sinx is defined, it's [insert definition of limit] for sinx when x approaches infinity, but when you attempt to compute it, it happens that no value can be the answer.

I would like to start this by saying that I absolutely love your channel and videos, you have inspired me to learn and enjoy math for years and so thank you! I do have a bit of confusion with the “no solution” part though, specifically the “sqrt(x) = -5” part as if you rework the equation as “sqrt(x) = i² • x” then square both sides you end up with “x = i⁴ •5²” if you take the fourth root of this you end up with the expression “quartic root(x) = Z = 0 + i•sqrt(5)” which resembles a complex number. I am absolutely no expert on this matter by any means so there is a very high probability that I made a few mistakes along the way, this may not even be a valid solution but I thought about it as soon as I saw the equation so if you could be so kind as to clarify this, it would mean the world to me as learning a new thing, especially from someone as talented and kind as yourself, is a graceful opportunity for me.

6:08 The square root operation outputs both positive and negative values. Therefore it has not one answer, but two. 5 and -5, making 25 indeed the correct answer.

"DNE" is a negation of a quantifier in logic, whereas "undefined" refers to any operation which is given an argument outside its domain. This is consistent with what he says in the video, but more general.

I study maths with arabic and french but i don t know why that man make maths easy with that innocent smile .😘

0^0 = 1 Because You can always put a "1 •" before something and it doesnt change For example with 2 1 • 2^2 = 1 • 2 • 2 =4 1 • 2^1 = 1 • 2 = 0 1 • 2^1 = 1 As you can see its always a "• 2" less So with 0 its this 1 • 0^2 = 1 • 0 • 0 = 0 1 • 0^1 = 1 • 0 = 0 1 • 0^0 = 1

Im only 1min and 8 sec in and those whitebord skills are slick

Another kind of “no solution” equation could be 2x+1=2x+5

0⁰ is 1; it's literally defined.

That's the first time I've heard long division explained like that. I always thought of it as "How many times does the outside number fit into the inside number?" So when dividing by 0 the answer would be infinity, which is not a number and so calculators could and should say NaN. The way you described division changes that. Though I'd honestly say "no solution" rather than undefined still.

Awesome video. Very helpful ❤👍

Imagine walking into this class without knowing he’s holding a mic.

for 6:00 if you let x=25i^4 (25 * 1) then sqrt(x) = 5i^2 = -5, wouldn't this count as a complex solution? I know its kind of playing a technicality but I can't find any way to contradict it

0^x is undefined for negative x (equivalent to 1/(0^-x) = 1/0). 0^0 is indeterminate, and when the exponent zero is a discrete value and not a limit, it is convenient to define all x^0 := 1, including 0^0 (this is used in expressions of polynomials as summations, for example).

5:50 sqrt(25)= + or - 5, meaning that 5 and -5 are solutions

No real value: *exists* Complex numbers: let me introduce myself

Thanks for such a great content with love from India

The first example given for "no solution" (#2) is exactly the same kind of example as for "no real solution" (#1).

When you were talking about the non existent you gave the example of an equation including -5. Now square root of 25 is 5 but it is also -5, solution of square root of 25 is +-5. So square root of 25 gives the true result -5 and an extra result +5 so solution exists

At 6:00 ish: For sqrt(x) = -5 x = 25.exp[i(2pi +k*4pi)] Would work (with k as a whole number) I think.

In algebra and combinatorics, it's generally agreed that the value of x to the zero power = 1. It is in calculus and mathematical analysis that the expression is left undefined. The polynomial "x to the zero power" is the identity element of a polynomial ring. Defining x to the zero power = 1 is necessary for many polynomial identities like the binomial theorem.

18:57 everybody happy made me lought

one of my favourite examples with the 1^infinity I.F. is when you have the limit of 1 to some function f(x), where f(x) is approaching inifinity, because to deal with it you change from lim 1^f(x) to 1^lim f(x)

1:12 If the symbol means "positive square root", then no, there is no positive square root of -9, even in complex numbers. 3i is not a positive number, as positive numbers are real numbers.

x^-a = 1/x^a by definition (for positive a, negative a, and even a=0) => x^0 = 1/x^0 => 0^0 = 1/0^0 => 0^0 is its own multiplicative inverse => 0^0 = 1, as there is no other real number that is its own multiplicative inverse.

I like how I’m the video you say that 0 to the power of zero is undefined while your shirt shows it as indeterminate

NRV was always the shorthand my teachers understood was me saying "no real value" when in school...

Can you solve this integral : Integral of t^n/(1+t)^n dt, t from 0 to infinity .

hey blackpenredpen, I'm still kinda confused, isn't 0/0 by itself indeterminate? Since if you have 0/0 = x then 0x = 0, therefore x can be any number, but if you have 1/0 just saying it is equal to a number doesnt make sense, so its undefined. Or is 0/0 only indeterminate in the context of limits?

For the indeterminate section, when you get a limit that evaluates to 0/0, you can just apply L’Hopital’s rule and get the same answer

undefined is often for function, when the input is not in the domain. define f(x) = 3x+1 if x is odd; x÷2 if x is even. we can see that the domain of f is integer. f(27)=82 f(82)=41 f(0.5) is undefined.

I would love to have the indeterminate family t-shirt jajajaja

12:07 Absolutely hilarious. Good video anyway 24:18

after 20 years thank you SO much for being the person to teach me precisely why anything divided by zero is called "undefined." it's sweet justice for my young self who kept hearing the same "well what if you had six slices of pizza and had to distribute it to zero people???" with no elaboration

This video gives me another question, how computer do limits? For example the sin doesn't have a solution and other things are just see what it seems to approximate so how computers see that?

Why sqrt(x) = -5 doesnt work ? sqrt(25) = ± 5 , not just +5

Teddy is adorable 💖

I define w as a number whose square root is -5. I define w as a number whose absolute value is -1. I define w as the limit of sin(x) as x approaches infinity. w is my new favorite number, and it's better than anyone else's favorite number.

Now thinking about it, DIV0 is a way to force a remainder, as from long division, if the last subtraction results in a non-zero positive integer, that integer becomes the remainder. Possible exploitation from NULL Remainder Numerator?

11:42 :the answer is in wheel theory YAY teddy!!!

An easy way to look at 0^0 is by just looking at the general pattern with exponents. An exponent is in the form a^b. Every time we increase b by 1, we multiply by a, and every time we decrease b by 1, we divide by a. We also say that a^1 = a. Using this, we can determine that a^0 = a/a, so 0^0 = 0/0, which is undefined. Note that for every a =/= 0, a/a = 1, which is consistent with the definition of a^0 (and arguably is where the definition comes from).

The reason for 'undefined' is because the multiplicative inverse of zero has no definition. Therefore division by zero has no answer.

When you thought it was going to be a meme video converting real life to math equations but it's actually a helpful video of math terms.

The ultimate level that finish your homework instantly: "I don't know"

√-x = i√x, where x is bigger than 0 and i is the square root of -1.

There's a bit more rigor for why dividing by 0 is undefined instead of DNE. Consider taking the limit of dividing a number by X as X approaches 0. On the positive side, you are approaching infinity. On the negative side you are approaching negative infinity. Aside from the fact of how multiplying by 0 works, the limit is contradicting, positive and negative infinity. It's because of the limit that we say undefined instead of DNE.

In those cases 1, 2, and 3, I think it makes sense to say that a solution of the equation f(x) = 0 does not exist in the real numbers or that the limit of a function as x approaches some number does not exist as a real number or that the value of a function evaluated x does not exist in the real numbers (perhaps because the function is not defined at x). For cases 4 and 5, the fact that a function is not defined at x does not mean that the function cannot be defined at x. An example is the reciprocal function a/x for some fixed real number a. There are some applications where you can define a/0 to be 0. While there may be some ambiguity in defining a/x at 0, we should not interpret "undefined" and "indeterminate" as "cannot be defined" and cannot be determined, respectively.

Can somebody tell me a function in which the limit 1^(inf) will differ from 1? Edit: (1+1/n)^n -> e

Programmers are also used to "NaN" = Not A Number.

13:24 The other distinction between DNE and undefined is that undefined values are literally that: undefined. We have not defined what x/0 is. Mathematicians haven’t settled on it. DNE is defined however, namely that it simply does not exist. Sin(x) does not approach anything and therefore we define it as DNE. We don’t say does not exist for x/0, because there is no mutual agreement on that it does not exist.

'has no real value' hit too close to home