Identity Element of Multiplication on Numbers
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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
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
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): identity
- 2017: Amol Sasane: A Friendly Approach to Functional Analysis ... (previous) ... (next): Chapter $\S 2.4$: Composition of