0 × 0 = 0 0\times0=0

Algebra Level 3

Everybody knows that 0 × 0 = 0 0\times0=0

Is it possible to prove it?

No, it's an axiom Yes, it is possible

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2 solutions

0 = 1 × 0 = ( 0 + 1 ) × 0 = 0 × 0 + 1 × 0 = 0 × 0 + 0 = 0 × 0 \begin{aligned} 0 & =1 \times 0 \\ & = (0+1) \times 0 \\ & = 0\times 0 + 1\times0 \\ & = 0\times 0+0 \\ & = 0\times 0\end{aligned}

Note: We used that for any numbers a , b , c a, b, c , 1 × a = a ( a + b ) c = a c + b c a + 0 = a \begin{aligned} 1\times a & =a \\ (a+b)c & = ac+bc \\ a+0 & = a\end{aligned}

Lemma: a R a × 0 = 0 \forall a \in \mathbb{R}\, a \times 0 = 0

Proof:

\begin{equation*} \begin{aligned} 0a &= 0a + 0 && \quad \text{by }\textit{identity element }(+ ) \\ &= 0a + 0a + (-0a) && \quad \text{by }\textit{inverse element }(+ ) \\ &= (0 + 0)a + (-0a) && \quad \text{by }\textit{distributivity } \\ &= 0a + (-0a) && \quad \text{by }\textit{identity element }(+ ) \\ &= 0 && \quad \text{by }\textit{inverse element }(+ ) \end{aligned} \end{equation*}

Corollary: 0 × 0 = 0 0 \times 0 = 0

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