Zero Element of Multiplication on Numbers

Theorem

On all the number systems:

• natural numbers $\N$
• integers $\Z$
• rational numbers $\Q$
• real numbers $\R$
• complex numbers $\C$

the zero element of multiplication is zero ($0$).

Proof

This is demonstrated by showing that:

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

for all $n$ in all standard number systems.

$\blacksquare$

Also see

• Identity Element of Addition on Numbers
• Identity Element of Multiplication on Numbers

Sources

• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 4.3$. Units and zeros: Example $74$