'B is +and- 1 because if you multiply the same numbers it still equals 1 (1x1=1, -1 x -1=1)' The radical sign usually refers to the principal root.

@@thetaomegatheta ik but thanks

If you know that, then why did you claim that sqrt(1) is '+and- 1'?

'0^0 is 1 and 0 at the same time' Obviously not 'at the same time'. 'so it's undefined' When it is defined as 0 or as 1, it is obviously not undefined. '√1 is ±1 because (+1)(+1) is 1 and (-1)(-1) is 1' Usually, the radical sign will refer to the principal root.

@@thetaomegatheta does that not make sense? You have x=0^n which is x=0 and x=n^0 which is x=1, put them in a system of equations and you need to find a single value of x that is equal to 0 and 1 at the same time (there isn't one)

'You have x=0^n which is x=0 and x=n^0 which is x=1, put them in a system of equations and you need to find a single value of x that is equal to 0 and 1' There are obviously no solutions there for n > 0, and we give a special definition to 0^0. Usually it is defined as 1. It is obviously not defined in multiple non-equivalent ways at the same time.

@@thetaomegatheta 0^0 is undefined, and replying to the other thing you said earlier, if √ only had the positive value as it's result you could prove that 2+2=0 Why is 0⁰ undefined? Well find me a value that is equal to 1 and 0 at the same time and we'll settle on that being the solution

'0^0 is undefined' Unless it is given a definition in the relevant context. 'if √ only had the positive value as it's result you could prove that 2+2=0' Where's this proof of yours, then? 'Why is 0⁰ undefined?' It seems to usually be defined as 1. 'Well find me a value that is equal to 1 and 0 at the same time' Why should I? We specifically define 0^0 as 1 or we leave it undefined. We all know that it is impossible to define the functions f(x) = x^0 and g(x) = 0^x at x = 0 in a way that both of them are continuous. That doesn't prevent us from defining 0^0.

'Mathematicians accept a contradiction while it is known' This is nonsense. What 'contradictions' are you even talking about?

0^0 being 1 and undefined at the same time. But let's check with Euler's formula assuming 0 is a complex number. But zero has no real and no imaginary part. Is it a number?

'0^0 being 1 and undefined at the same time' This is nonsense. If, in a particular context, 0^0 is defined as 1, then it is not undefined. 'But zero has no real and no imaginary part' It does. Re(0) = 0 Im(0) = 0. 'Is it a number?' Yes. In particular, it is the additive identity of the field of integers.

​@@thetaomegatheta OK, it seems zero to the power of zero is undefined. But why is it undefined? If zero is an ordinary number, why can we not apply the usual arithmetic rules and calculate the result of zero to the power of zero? In other words, why is the result of this calculation undefined? And is undefined a number? I think 0 might not be a number, at least it would be a very speciall one. The brother of infinity. Is infinity a number? I would say no. There are several types of infinity, hence it is not clearly defined. To qualify as a (complex) number, in my opinion an object under test has to have a finite real part and a finite imaginary part and the absolute value has to be greater than zero. The absolute value of 0*r +0*i is still 0. Therefore, it is not a number acoording the above definition. The reason for that is, that like infinity, zero is not clearly defined. Zero has indefinitly many instances, because the direction of the zero vector is not defined. Therfore we must sadly exclude 0 from the natural numbers. In order to avoid more conflicts in the future, any number should be defined as a*x + b*y + c*z. Integers: a*x, x can be omitted to it falls back to a Complex: a*x+b*y Spheric: a*x + b*y + c*z

'OK, it seems zero to the power of zero is undefined' Unless it is defined in the relevant context. Usually it seems to be defined as 1. It's like you don't understand what the word 'undefined' means. 'But why is it undefined?' The primary motivation for leaving it undefined is that a bunch of relevant functions can't all be continuously extended to that point at the same time. 'If zero is an ordinary number, why can we not apply the usual arithmetic rules and calculate the result of zero to the power of zero?' How many times do you need to be explained that we can and do define 0^0? The motivation for leaving it undefined is that a bunch of functions can't be continuously defined at that point at the same time. 'I think 0 might not be a number' Well, you think wrong. It is the additive identity (I might have misspoken previously and called it the 'multiplicative identity' previously - will fix that if I did) of the field of integers. Integers are numbers. 0 is a number. 'The brother of infinity. Is infinity a number? I would say no. There are several types of infinity, hence it is not clearly defined' This is just nonsense. There is no element of any number field called 'infinity'. There are points that are called that or something similar in various extensions of number fields as topological spaces. When people say that there are 'different sizes of infinity', what they mean is that there is more than one infinite cardinal. None of the cardinals are called 'infinity'. All of this stuff is very clearly defined. 'To qualify as a (complex) number, in my opinion an object under test has to have a finite real part and a finite imaginary part and the absolute value has to be greater than zero' That's not what a complex number is, and you are just pulling the last requirement out of your ass. A complex number is a formal sum a+i*b, where a and b are real numbers and i is the imaginary unit with all the associated properties. 'The absolute value of 0*r +0*i is still 0' And? 'Therefore, it is not a number acoording the above definition' And nobody uses that definition, so this is ignorable. 'The reason for that is, that like infinity, zero is not clearly defined' This is nonsense. 0 is the additive identity of the field of integers. Nothing unclear about this. 'Zero has indefinitly many instances' This is just gibberish. You didn't even bother defining what an 'instance of zero' is. 'Therfore we must sadly exclude 0 from the natural numbers' 0 is often excluded from the set of natural numbers, but you provided no actual motivation or proof of the fact. That 'therefore' is misplaced. 'In order to avoid more conflicts in the future' What 'conflicts'? 'any number should be defined as a*x + b*y + c*z' This is nonsense. What are a, x, b, y, c, and z? 'Integers: a*x, x can be omitted to it falls back to a Complex: a*x+b*y Spheric: a*x + b*y + c*z' 'Spheric'? And where are quaternions, octonions, and other number algebras?

:)

Circular reasoning. You need to know this limit to find the derivative of sinx