Multiplication is iterated addition; exponentiation is iterated multiplication, tetration is iterated exponentiation, call all of those as different grades of iterated arithmetical operators.     Is there a known $n$-iterated arithmetical operator such that $\mathbb C$ is not closed under it?
120535
Considering subtraction can break out of the natural numbers and into integers, division can break out of integers and into rational numbers, and exponentiation can break out of rational numbers and into irrational and complex numbers...  Can tetration (or more specifically the inverse of tetration) 'break out' of complex numbers and into some other set? And can this be proven?  If not, are there any other hyperoperations that could do so?
104635
In a set $X$ we define a binary operation $*$ such that  $$\forall x, y \in X,\  (x*y)*y=y*(y*x)=x.$$  Is $*$ commutative and why?
24628
What is $i^i$ and $(i^i)^i$. I fond some answers  but is there a clearer explanation of this? Maybe something help me to understand.
75814
Assuming arithmetic operations on $\overline{\mathbb{C}}$ (that's the extended complex plane) are defined via arithmetic operations on the corresponding sequences, I need to give examples showing why $0\cdot \infty$, $\infty/\infty$, and $0/0$ are meaningless.  I also had to show the same thing for $\infty - \infty$, but that was easy: Consider the sequences $\{1,3,3,5,5,7,7,9,\cdots \}$ and $\{1,2,3,4,5,6,7,8,\cdots \}$, both of which belong to the class $\infty$ (i.e., the equivalence class of all sequences converging to $\infty$). Then, subtracting the second sequence from the first, term-by-term, we obtain the resulting sequence $\{0,1,0,1,0,1,\cdots\}$, which does not belong to any convergence class, since it has no limit.  However, I can't approach the three cases I'm asking on here about in the same way. I must admit, I'm a little confused as to what is being asked. For reference, it is problems 2-4 on p. 79 of Markushevich's Theory of Functions of a Complex Variable, Vol I.  If someone could furnish me with an example, at least for $0 \cdot \infty$, and maybe either $0/0$ or $\infty/\infty$, perhaps I could use that to help me figure out the other case on my own.  Thank you.
99299
If multiplication is not repeated addition, what is it? How do you define multiplication? All of us had to memorize the multiplication table in elementary school, but how did they come up with it if so many people claim that it is wrong to think of it as repeated addition? Voodoo magic?
69069
I had a difficult time in phrasing my question. But I was wondering if there is an operation that, when repeated n times, results in the addition operator. Same way as repeating addition n times results in the multiplication operator, and repeating multiplication n times results in the exponentiation operator etc.  So $n$ times addition of a number $x$ results in $x\times n$.  And $n$ times multiplication of a number $x$ results in $x^n$  Then my question is $n$ times ...what... results into the number $x+n$.  Let's call this operator: $@$.  For example, the following would then hold:  $$a\times a=a^2$$  $$a+a=a\times 2$$  $$a@a=a+2$$  My question is, does it make any sense thinking of such an operator, is there anything known about it, can it be followed through even further like $a\sim a = a@2$?
31848
I understand that the complex numbers have a geometric representation as ordered pairs of real numbers. From what I understand complex numbers are very useful for working in two dimensions mathematically. The fact that $i^2 = -1$ follows from the definition of multiplication of complex numbers:  $(a,b)(c,d) = ac-bd+(bc+ad)i$  My question is this: If one was simply working with ordered pairs of real numbers with no knowledge of $i=\sqrt{-1}$, would this definition of multiplication ever follow logically from a practically arising situation?
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For example $2+2\times2$ is $6$ not $8$.   Actually $+$ and $\times$ are binary operations on $\mathbb Z$. but here there is an triple $(2,2,2)$ which we sent to $2+2\times 2$. So we have to put and order for applying operations as binary such that $$a+b\times c \: : \: (a,b,c) \to a+(b\times c)$$ with assuming that $(\cdots)$ has priority more.  My question is why multiplication has priority more than addition? We could also define $$a+b\times c \: : \: (a,b,c) \to (a+b) \times c.$$
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Does there exist some notation that represents the iterative composition of a single-input, single-output function with itself? As in, say, $f_5(x)=f(f(f(f(f(x)))))$.  In other words, going by the above (incorrect, I'm pretty sure) notation:  $f(x)=x+1$, $f_n(m)=m+n$  I'm looking for the correct way to express the notion of "$f_n(x)$" for any $f$.
85647
I'm working in $\mathbb{Z}_n$ under multiplication  (not addition) for some time now and I'm beginning to notice something: For $n \in \mathbb{N}$ where $n$ is even and $\frac{n}{2}$ is odd, it seems to be always true that $\frac{n}{2}$ is idempotent, i.e. $(\frac{n}{2})^2 \equiv \frac{n}{2}$ mod $n$ when $n$ satisfies the aforementioned definition. So I wanna try to prove this but I don't know how to approach it. My first thought was induction since this iterates across $n \in \mathbb{N}$ satisfying my definition of $n$.  Proof Let $n \in \mathbb{N}$ such that $n$ is even and $\frac{n}{2} = 2k+1$ for some other $k \in \mathbb{N} \cup \{0\}$ (Note that $k$ could equal $0)$ Define $P(n)$ as the statement:$(\frac{n}{2})^2 \equiv \frac{n}{2}$ mod $n$ Our basis case would be $P(2): (\frac{2}{2})^2=(1)^2 \equiv 1$ mod $2 \equiv \frac{2}{2}$ mod $2$ So $P(2)$ holds as true. Assume $P(i)$ is true for some $i \in \mathbb{N}$ where $i = 2(2k+1)$. Inductively, then we have: $$P(i+1): (\frac{i+1}{2})^2 =  ?? $$  Then I get stuck here and don't know how to proceed. Obviously, I can't do much once I put $k$'s into the equation because $k$ can be any number. All we know about $k$ is that $k \leq n$ by how we defined $n$.
71867
You know, like addition is the inverse operation of subtraction, vice versa, multiplication is the inverse of division, vice versa , square is the inverse of square root, vice versa.  What's the inverse operation of exponents (exponents: 3^5)
12243
I understand that when you raise any number $x$ to a power, you multiply $x$ by itself the number of times indicated in the power. However, what happens when $i^i$ is performed? How can a number be multiplied an imaginary amount of times? Wolfram Alpha says that it is equal to $e^{{-\pi}/{2}}$, but how would you arrive at that answer? Any response will be appreciated, thanks!
55147
here says a new operation addition and new operation multiplication  x ⊕ y = x + y − 1, x ⊗ y = x + y − xy on set Z,where the operations on the right hand side are ordinary addition and multiplication of integers. Prove that the set Z equipped with these two new operations ⊕, ⊗ is a ring. Does it have a multiplicative identity?  I tried to use the addition cancellation but I do not understand why or I got the wrong understanding for this question.   I assume the there is y and y' are 2 solution for equation  x ⊕ y = x + y − 1, therefore:   x ⊕ y = x + y − 1  x ⊕ y' = x + y − 1 and I get  x ⊕ y =  x ⊕ y' after the cancellation, then I get y=y', and it means the addictive is unique. So, can I say that the set Z equipped with new addiction x ⊕ y = x + y − 1? or I just go into a wrong way for solve this problem?  ask in advance, thank you
11969
I started thinking about the number of associative (binary) operations on a set with $n$ elements today.  Looking online I found  which indicates only $113$ of the possible $19,683$ operations on a three-element set satisfy association.  So, $50\%$ of binary operations on a two-element set satisfy association, and less than $0.6\%$ of all binary operations on a three element set satisfy association.  For an $n$-element set, where $n$ denotes a natural number, is the ratio $R_{n}=A_{n}/B_{n}$, of the number of associative binary operations $A_{n}$ to the number of all binary operations $B_{n}$, in general small?  I don't mean the following questions as equivalent, but since they seem more concretely answerable in principle, does   $$ \lim_{n \to +\infty} \frac{A_{n}}{B_{n}} = 0 ? $$       Also, is $F : n \to R_{n}$ a monotonically decreasing function?  Some Background of the Question: In his Linear Algebra Problem Book 1995 on p. 6 Paul Halmos writes "The commonly accepted attitudes toward the commutative law and the associative law are different.  Many real life operations fail to commute; the mathematical community has learned to live with that fact and even to enjoy it.  Violations of the associative law, on the other hand, are usually considered by specialists only."  For all I know, Halmos might only have written that to motivate the study of Linear Algebra and doesn't quite literally mean what he appears to say.  But, if he means what he appears to say, and if $F : n \to R_{n}$ is monotonically decreasing, or $R_{n}$ is generally small, I think there's something amiss with what Halmos says, since non-associative operations seem so common that one may as well enjoy them.
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I would like to find a way to construct uncountably many non-isomorphic linear (total) orderings of natural numbers (as stated in the title).  I've already constructed two non-isomorphic total orderings. They are $$ 1 &lt; 2 &lt; 3 &lt;\ldots\\ 1 &gt; 2 &gt; 3 &gt;\ldots $$ where "$&lt;$" and "$&gt;$" are  strict total ordering relations.  Thanks in advance for your help.
12482
Most of the time I see matrix multiplication presented and defined, as a seemingly arbitrary sequence of operations. For example, the  I'm currently reading for a linear algebra course defines the product $AB$ as the $(i, j)$ entry in the $1 \times 1$ matrix that is the product of the ith row of $A$ and the jth column of $B$. Properties of matrix multiplication are subsequently proven based upon this definition. The definition is clear, but why the matrix product is useful is not clear to me as a student. A  I'm referencing defines the product $AB$ in terms of linear combinations.   The problem I have is doing matrix multiplication quickly by hand, particularly when the $A$ is $p \times 1$ and $B$ is $1 \times q$. I would like to know of how to look at or define matrix multiplication, in a manner which makes it easy (for the average student) to compute by hand, while being intuitive and consistent for use for later proofs.   wikipedia has a great .
67707
We use multiplication for repeated addition, and in turn use exponents for repeated multiplication. What topic comes after this, for repeated exponentials? Is there something my teachers are hiding from me?