Identity Element of Addition 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 addition is zero ($0$).

Proof

This is demonstrated in these pages:

• Identity Element of Natural Number Addition is Zero
• Integer Addition Identity is Zero
• Rational Addition Identity is Zero
• Real Addition Identity is Zero
• Complex Addition Identity is Zero

$\blacksquare$

Also see

• Identity Element of Multiplication on Numbers

Sources

• 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: $(\text P 2)$
• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): zero: 1c.
• 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