Commutative Law of Addition
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Theorem
Let $\mathbb F$ be one of the standard number sets: $\N, \Z, \Q, \R$ and $\C$.
Then:
- $\forall x, y \in \mathbb F: x + y = y + x$
That is, the operation of addition on the standard number sets is commutative.
Natural Number Addition is Commutative
The operation of addition on the set of natural numbers $\N$ is commutative:
- $\forall m, n \in \N: m + n = n + m$
Integer Addition is Commutative
The operation of addition on the set of integers $\Z$ is commutative:
- $\forall x, y \in \Z: x + y = y + x$
Rational Addition is Commutative
The operation of addition on the set of rational numbers $\Q$ is commutative:
- $\forall x, y \in \Q: x + y = y + x$
Real Addition is Commutative
The operation of addition on the set of real numbers $\R$ is commutative:
- $\forall x, y \in \R: x + y = y + x$
Complex Addition is Commutative
The operation of addition on the set of complex numbers is commutative:
- $\forall z, w \in \C: z + w = w + z$
Sources
- 1951: B. Hague: An Introduction to Vector Analysis (5th ed.) ... (previous) ... (next): Chapter $\text I$: Definitions. Elements of Vector Algebra: $1$. Scalar and Vector Quantities
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 1.1$. Number Systems: $\text{I}.$
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 4.2$. Commutative and associative operations: Example $61$
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 2$: Compositions: Example $2.1$
- 1967: Michael Spivak: Calculus ... (previous) ... (next): Part $\text I$: Prologue: Chapter $1$: Basic Properties of Numbers: $(\text P 4)$