Commutative Laws in Algebra

Taken and adapted from:

- Tom M. Apostol, Calculus, Volume 1 (New York: Blaisdell, 1961), pp. 13-19.
- Susanna S. Epp, Discrete mathematics: with applications, fifth edition, metric version (Australia, 2020), Appendix A.

Many fundamental operations in mathematics involve combining two elements from a set to produce another element of the same set; such rules are known as binary operations. For instance, addition and multiplication are binary operations on the real numbers, because adding or multiplying any two real numbers always yields another real number. Formally, if $ * $ is a binary operation on $S$, then $* : S \times S \to S$, meaning for any $a, b \in S$, the result $a * b$ is also in $S$. This property is called closure under the operation.

An important property that some binary operations possess is commutativity, which means that the order of the operands does not affect the result. For example, for all real numbers $a$ and $b$, both addition and multiplication satisfy $a + b = b + a$ and $ab = ba$ (see also the examples below).

Commutative Laws in Algebraic Structures

For all numbers $a$ and $b$ $(\forall \; a, b \in \mathbb{C})$,

\[a + b = b + a\] \[ab = ba.\]

The commutative laws are valid within the following sets of numbers:

Natural Numbers: $\mathbb{N},$
Integers: $\mathbb{Z}$,
Rational Numbers: $\mathbb{Q}$,
Real Numbers: $\mathbb{R}$,
Complex Numbers: $\mathbb{C}.$

Examples

Natural Numbers ($\mathbb{N}$)

Addition
$3 + 5 = 5 + 3 = 8$

Multiplication
$3 \cdot 5 = 5 \cdot 3 = 15$

Integers ($\mathbb{Z}$)

Addition
$(−3)+7=7+(−3)=4$

Multiplication
$(−2)×5=5×(−2)=−10$

Rational Numbers ($\mathbb{Q}$)

Addition
$\frac{1}{2}+\frac{3}{4} = \frac{3}{4}+\frac{1}{2} = \frac{5}{4}$

Multiplication
$\frac{2}{3} \cdot \frac{3}{5} = \frac{3}{5} \cdot \frac{2}{3} = \frac{2}{5}$

Real Numbers ($\mathbb{R}$)

Addition
$2.5 + 3.1 = 3.1 + 2.5 = 5.6$

Multiplication
$1.2 \cdot 4.5 = 4.5 \cdot 1.2 = 5.4$

Complex Numbers ($\mathbb{C}$)

Addition
$(2 + 3i) + (1 - 4i) = (1 - 4i) + (2 + 3i) = 3 - i$

Multiplication
$(2 + 3i)(1 - 4i) = (1-4i)(2 + 3i) = 14 - 5i$

Meaning of the used symbols

Symbol	Meaning
$ \forall $	for all (universal quantifier)
$ \in $	is an element of