RWM101: Foundations of Real World Math

The commutative law of addition and multiplication tells us that the order you use to add or multiply numbers does not matter. In other words, $3+2=5$, and $2+3=5$. The same is true of multiplication: $2 \times 3 = 6$, and $3 \times 2 = 6$. You obtain the same result regardless of the order you use to add or multiply numbers.

Note that the commutative property only holds for addition and multiplication. It does not work for subtraction or division. For example, $10-2=8$, but $2-10=-8$. In subtraction, the results are not the same when you change the order of the numbers. Likewise in division: $20/2=10$, but $2/20=0.10$. Again, the results are not the same when you change the order of the numbers in the division.

But familiar, old, real numbers are not the only kinds of entities we add and multiply in mathematics. For example, you will learn that matrix multiplication is not commutative when you study matrices in linear algebra and vector calculus. Similarly, we use a more-advanced, exotic kind of number called a quaternion for all sorts of tasks, such as robotics, 3-D animation, and video game design. Quaternion multiplication is noncommutative.

While quaternions and matrices are beyond the scope of this course, they provide an important lesson: rearrangement can affect useful kinds of multiplication. Sometimes order matters! The fact that real number addition and multiplication are commutative warrants an appreciative and careful understanding of the property.