Commutativity in Daily Life: A Cooking Example and Beyond

The commutative property is a fundamental concept in mathematics, stating that the order of addition or multiplication does not change the result. However, this property extends far beyond the realm of numbers into our daily lives, often in subtle and surprising ways. One great example of the commutative property in action is combining ingredients while cooking or baking. For instance, when making a smoothie, you can add fruits and liquids in any order without altering the final outcome.

The Commutative Property in Mathematics

Mathematically, the commutative property is well-defined for addition and multiplication. Let's take a closer look:

Addition

For addition, the commutative property states:

2 3 5 and 3 2 5

Multiplication

For multiplication, the commutative property comes into play:

4 times 5 20 and 5 times 4 20

In essence, whether you add bananas first or yogurt first, the final smoothie remains the same.

Non-Commutativity in Daily Life

While commutativity is a common phenomenon, many activities in our daily life do not commute. Understanding these non-commutative scenarios helps us navigate a diverse and dynamic world. Let's explore some examples:

Independent Operations

Intuitively, independent operations commute. For instance, putting on your left sock and your right shoe are independent actions and can be performed in any order without affecting the outcome. Similarly, in the context of the Rubik’s cube, rotating the left face counterclockwise and then the right face counterclockwise also commute, as they are independent actions.

Identical Operations

Interestingly, identical operations often commute. For example, taking two bites from an apple, regardless of the order, will result in two bites. In the case of the Rubik’s cube, performing two consecutive counterclockwise turns on the same face will commute with each other because they are identical operations.

Repeats and Inverses

If two operations are both repeats of the same basic action or their inverse, they also commute. As an example, turning the left face of the Rubik’s cube counterclockwise twice and turning the same face clockwise three times will commute. Similarly, "walk forward 3 steps" and "walk forward 2 steps" commute, as they are essentially actions repeated in sequence.

Geometric Commutativity

Commutativity can also be observed in geometric movements, particularly when dealing with flat surfaces. Traveling 100 miles north and then 100 miles east will result in the same final position as traveling 100 miles east and then 100 miles north, due to the approximate flatness of the Earth's surface. However, on a spherical surface, these movements would not commute, leading to different final positions.

Real-World Applications and Implications

The concept of commutativity and its counterpart, non-commutativity, has profound implications in various fields, including physics, biology, and even psychology. For instance, in quantum physics, particles do not commute, marking a significant difference from classical physics. The Nobel Prize winner Chen-Ning Yang derived the Yang-Mills equations by treating these quantities as matrices with non-commutative multiplication.

The failure of commutativity can also be seen in the diversity of life. In a commutative world, where order does not matter, the potential for diversity is limited, growing polynomially. Conversely, in a non-commutative world, such as the way our DNA is read, the order of genetic codes can lead to exponential diversity. This non-commutativity contributes to the vast diversity we see in species and even among individuals within a species.

In practical terms, non-commutative operations make our lives more diverse and interesting. Boring jobs often involve commutative operations, leading to a lack of diversity and complexity. Similarly, a completely commutative puzzle, like one restricted to only turning the left and right faces, would be trivial and lack the complexity and interest present in non-commutative puzzles.

Understanding commutativity and non-commutativity is crucial for appreciating the complexity of our world and the diverse ways in which we interact with it every day.