Zero is Zero Element for Natural Number Multiplication

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Theorem

Let $\N$ be the natural numbers.


Then $0$ is a zero element for multiplication:

$\forall n \in \N: 0 \times n = 0 = n \times 0$


Proof

Proof by induction.

For all $n \in \N$, let $\map P n$ be the proposition:

$0 \times n = 0 = n \times 0$


Basis for the Induction

By definition, we have:

$0 \times 0 = 0 = 0 \times 0$

Thus $\map P 0$ is seen to be true.


This is our basis for the induction.


Induction Hypothesis

Now we need to show that, if $\map P k$ is true, then it logically follows that $\map P {k + 1}$ is true.


So this is our induction hypothesis $\map P k$:

$0 \times k = 0 = k \times 0$


Then we need to show that $\map P {k + 1}$ follows directly from $\map P k$:

$0 \times \paren {k + 1} = 0 = \paren {k + 1} \times 0$


Induction Step

This is our induction step:


\(\displaystyle 0 \times \paren {k + 1}\) \(=\) \(\displaystyle \paren {0 \times k} + 0\) Definition of Natural Number Multiplication
\(\displaystyle \) \(=\) \(\displaystyle 0 + 0\) Induction Hypothesis
\(\displaystyle \) \(=\) \(\displaystyle 0\) Definition of Natural Number Addition

By definition:

$\paren {k + 1} \times 0 = 0$

So $\map P k \implies \map P {k + 1}$, and the result follows by the Principle of Mathematical Induction.


Therefore:

$\forall n \in \N: 0 \times n = 0 = n \times 0$

$\blacksquare$


Sources