# Minus One is Less than Zero

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## Theorem

- $-1 < 0$

## Proof

\(\displaystyle 0\) | \(<\) | \(\displaystyle 1\) | Real Zero is Less than Real One | ||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle -0\) | \(>\) | \(\displaystyle -1\) | Order of Real Numbers is Dual of Order of their Negatives | |||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle 0\) | \(>\) | \(\displaystyle -1\) | Negative of Real Zero equals Zero | |||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle -1\) | \(<\) | \(\displaystyle 0\) | Definition of Dual Ordering |

$\blacksquare$

## Sources

- 2000: James R. Munkres:
*Topology*(2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 4$: The Integers and the Real Numbers: Exercise $2 \ \text{(g)}$