Identity 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 identity element of multiplication is one ($1$).

Proof

This is demonstrated in these pages:

• Identity Element of Natural Number Multiplication is One
• Integer Multiplication Identity is One
• Rational Multiplication Identity is One
• Real Multiplication Identity is One
• Complex Multiplication Identity is One

$\blacksquare$

Also see

• Identity Element of Addition on Numbers

Sources

• 1964: Walter Ledermann: Introduction to the Theory of Finite Groups (5th ed.) ... (previous) ... (next): Chapter $\text {I}$: The Group Concept: $\S 1$: Introduction: $(1.3)$
• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 4.3$. Units and zeros: Example $71$
• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 4$: Neutral Elements and Inverses
• 1967: Michael Spivak: Calculus ... (previous) ... (next): Part $\text I$: Prologue: Chapter $1$: Basic Properties of Numbers: $(\text P 6)$
• 1983: François Le Lionnais and Jean Brette: Les Nombres Remarquables ... (previous) ... (next): $1$
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): identity element
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): identity element
• 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): Chapter $\S 2.4$: Composition of