Sum of Indices of Real Number/Positive Integers
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Theorem
Let $r \in \R_{> 0}$ be a positive real number.
Let $n, m \in \Z_{\ge 0}$ be positive integers.
Let $r^n$ be defined as $r$ to the power of $n$.
Then:
- $r^{n + m} = r^n \times r^m$
Proof
Proof by induction on $m$:
For all $m \in \Z_{\ge 0}$, let $\map P m$ be the proposition:
- $\forall n \in \Z_{\ge 0}: r^{n + m} = r^n \times r^m$
$\map P 0$ is true, as this just says:
- $r^{n + 0} = r^n = r^n \times 1 = r^n \times r^0$
Basis for the Induction
$\map P 1$ is true, by definition of power to an integer:
- $r^{n + 1} = r^n \times r = r^n \times r^1$
This is our basis for the induction.
Induction Hypothesis
Now we need to show that, if $\map P k$ is true, where $k \ge 1$, then it logically follows that $\map P {k + 1}$ is true.
So this is our induction hypothesis:
- $\forall n \in \Z_{\ge 0}: r^{n + k} = r^n \times r^k$
Then we need to show:
- $\forall n \in \Z_{\ge 0}: r^{n + k + 1} = r^n \times r^{k + 1}$
Induction Step
This is our induction step:
\(\ds r^n \times r^{k + 1}\) | \(=\) | \(\ds r^n \times \paren {r^k \times r}\) | Definition of Integer Power | |||||||||||
\(\ds \) | \(=\) | \(\ds \paren {r^n \times r^k} \times r\) | Real Multiplication is Associative | |||||||||||
\(\ds \) | \(=\) | \(\ds r^{n + k} \times r\) | Induction Hypothesis | |||||||||||
\(\ds \) | \(=\) | \(\ds r^{n + k + 1}\) | Definition of Integer Power |
So $\map P k \implies \map P {k + 1}$ and the result follows by the Principle of Mathematical Induction.
Therefore:
- $\forall n, m \in \Z_{\ge 0}: r^{n + m} = r^n \times r^m$
$\blacksquare$
Sources
- 1966: Richard A. Dean: Elements of Abstract Algebra ... (previous) ... (next): $\S 0.1$. Arithmetic: Example $1: \ \text{I}$