Associative Law of Addition

Theorem

Let $\mathbb F$ be one of the standard number sets: $\N, \Z, \Q, \R$ and $\C$.

Then:

$\forall x, y, z \in \mathbb F: x + \paren {y + z} = \paren {x + y} + z$

That is, the operation of addition on the standard number sets is associative.

Natural Number Addition is Associative

The operation of addition on the set of natural numbers $\N$ is associative:

$\forall x, y, z \in \N: x + \paren {y + z} = \paren {x + y} + z$

Integer Addition is Associative

The operation of addition on the set of integers $\Z$ is associative:

$\forall x, y, z \in \Z: x + \paren {y + z} = \paren {x + y} + z$

Rational Addition is Associative

The operation of addition on the set of rational numbers $\Q$ is associative:

$\forall x, y, z \in \Q: x + \paren {y + z} = \paren {x + y} + z$

Real Addition is Associative

The operation of addition on the set of real numbers $\R$ is associative:

$\forall x, y, z \in \R: x + \paren {y + z} = \paren {x + y} + z$

Complex Addition is Associative

The operation of addition on the set of complex numbers $\C$ is associative:

$\forall z_1, z_2, z_3 \in \C: z_1 + \paren {z_2 + z_3} = \paren {z_1 + z_2} + z_3$

Also see

• Associative Law of Multiplication

• Commutative Law of Addition
• Commutative Law of Multiplication

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: $\text{II}.$
• 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 4.2$. Commutative and associative operations: Example $64$
• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 2$: Compositions: $\S 2$: Example $2.1$
• 1967: Michael Spivak: Calculus ... (next): Part $\text I$: Prologue: Chapter $1$: Basic Properties of Numbers: $(\text P 1)$
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): addition
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): associative
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): addition
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): associative