Why is the expression $\underbrace{n\cdot n\cdot \ldots n}_{k \text{ times}}$ bad?

Question

For writing a (german) article about the power with natural degree I have the following question:

In school one defines the power with natural degree via

$$n^k = \underbrace{n\cdot n\cdot \ldots \cdot n}_{k \text{ times}}$$

In calculus normally recursion is used to introduce the power:

$$\begin{align} n^0 & := 1 \\ n^{k+1} & := n \cdot n^k \end{align}$$

Question: What are the disadvantages of the expression $\underbrace{n\cdot n\cdot \ldots \cdot n}_{k \text{ times}}$ and why it is not used to define the power in advanced mathematics (although it is intuitive)?

Note: I'm sorry, I had an ambiguous question before. I do not want to know, why we define the power. I want to know, why we use recursion and not the expression $\underbrace{n\cdot n\cdot \ldots \cdot n}_{k \text{ times}}$ to define the power in advanced mathematics...

My ideas so far:

• $\underbrace{\ldots}_{k \text{ times}}$ is no operator, which was introduced before.
• definitions with recursion lead naturally to a scheme, how properties of these concepts can be proved via induction

Answer 1

$n^k = \underbrace{n\cdot n \cdots n}_{k\ \textrm{times}}$ only works when $k$ is a positive integer. In order to define exponentiation more generally, we must refine the definition of exponentiation several times:

• First, we must address a suitable definition for $k = 0$ and $k \in \{-1, -2, \ldots \}$, including operations of the form $n^{k_1}n^{k_2}$ that reconcile with the "repeated multiplication" definition when $k_1, k_2$ are positive integers;
• Next, we must address a suitable definition for $k \in \{ \frac{1}{m} \mid m \in \mathbb{Z} \}$, including operations of the form $n^{k_1}n^{k_2}$ that reconcile with the similar operations defined above (and in doing so, we cover the more general case $k \in \mathbb{Q}$);
• Next, we must address a suitable definition for $k \in \mathbb{R} \backslash \mathbb{Q}$. This turns out to be more complicated than one might expect, one such approach is to use the language of Dedekind cuts.

For these reasons, the "repeated multiplication" definition fails for almost every interesting value of $k$. The goal is to find a definition that reduces to "repeated multiplication" in the special case that $k \in \mathbb{Z}^+$, but that works more generally for all real numbers when the base is positive.

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To address your edit, which is not something I've actually encountered, but I will comment nonetheless:

If you define $n^{k+1} = n \cdot n^k$, this definition is satisfactory for all values of $k$, presuming you have defined exponentiation rigorously. This differs from the repeated multiplication definition in that you may start from an arbitrary $k$. This is actually one step in the process of defining exponentiation. More generally, we wish to declare that $n^{k_1}n^{k_2} = n^{k_1+k_2}$ regardless of the classes of real number to which $k_1$ and $k_2$ belong. The "recursive" definition you presented is a special case of this.

Answer 2

Why do we write $ab$ instead of $$\underbrace{b+b+\cdots+b}_{a \textrm{ times}}~~~?$$

Answer 3

Because only multiplication of two numbers is defined at that point. Also, if it's not yet defined to be associative, that expression may be ambiguous.

Answer 4

Although this is hardly the reason that cause confusion because people have common sense, "..." is not well defined if you want to be picky.

For example $ \{1,2,3,4,...,100\} $ could mean set of natural number less or equal to 100 or the image of that set under function $f(n)=(n-1)(n-2)(n-3)(n-4)(n-100)+n$

Define things using recursive remove the "ambiguity".