# Understanding Mathematical Order: A Call for Clarity

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*Dedicated to Bella L and her engaging mathematics articles*

Bella L recently highlighted a crucial aspect of teaching mathematics: the sequence of operations in expressions, often referred to as the "order of operations." In essence, multiplication and division share equal precedence, which is higher than that of addition and subtraction, both of which also have equal precedence. Moreover, parentheses and function arguments, as represented in the notation y= f(x), take precedence over all.

Alternatively, one might view it recursively: parentheses and function arguments delineate separate mathematical expressions that need resolution first. After resolving these, multiplication and division follow, and lastly, addition and subtraction. The equality of multiplication/division and addition/subtraction arises because they are inverses of each other; thus, their order of execution does not matter. However, this non-commutativity is noteworthy. If we think of subtraction as the inverse operation of addition, we realize that the expression is not symmetric; it matters whether we express it as (?a) + b or a + (?b). Similarly, with division, the inversion of the operands affects the outcome.

Bella coined the acronym PEMDAS to refer to these rules: Parentheses, Exponentiation, Multiplication, Division, Addition, and Subtraction. In my childhood, we referred to it as “BODMAS,” a term I will touch upon shortly.

As someone who has dedicated her career to mathematics and physics, I sometimes forget how perplexing these conventions can be for children. They are simply that: conventions. When a child begins to understand *why* mathematics works, conventions like these can seem arbitrary. It is vital to explain that these conventions foster a shared understanding among people.

Ultimately, there is no more *why* behind PEMDAS than the reason we add a silent -e to many English words without pronouncing it. Similarly, the final -r in German words often goes unspoken, yet it does not lead to a collapse of the language. These are conventions that can change over time. For instance, a Berlin bike rental service named "Deezer" was eventually discontinued, perhaps due to its unappealing parody of the language. Various dialects of English, outside of American and Scottish variations, also omit the final -r, resembling the German practice.

Unlike the silent letters in English or German, however, the PEMDAS convention exists to resolve genuine ambiguities in calculations.

Bella's article triggered a wave of nostalgia, prompting me to exclaim, "So, that’s what they call it now!" Fifty years ago, we chanted “BODMAS” or “BOMDAS,” remembering it as “brackets, orders, division, multiplication, addition, and subtraction.” The “O” was originally understood as “order,” a term that confused my math teacher, who mistakenly referred to it as “ordination,” a term found in archaic texts. This misnomer puzzled students, particularly when I questioned its relevance to my Anglican priest uncle.

My uncle, during a Sunday gathering, chuckled at the outdated terminology. He had encountered both "ordering" and "ordination" while studying math in 1930s England, clarifying the confusion for me. Fond memories of my uncle, his ceremonial attire, and our shared laughter come to mind when I think of operational order.

There came a point when I unconsciously adopted the convention of operation order without thought, likely around the time I began studying algebra. By then, multiplication often lost its symbol, becoming either juxtaposed or represented with a dot, while division appeared as a slash or a negative exponent. This shift likely occurred during my teenage years, approximately around 13 or 14.

At that age, I also delved into John Fraleigh's "A First Course in Abstract Algebra," where I began to perceive multiplication as the fundamental operation within groups. In contrast, children typically view multiplication as repeated addition and exponentiation as repeated multiplication, leading to a reverse learning order.

For those curious, the Knuth Up-Arrows define an infinite series of operations; a single arrow represents exponentiation, while multiple arrows imply repeated iterations of the prior operation. This hierarchy can become overwhelmingly complex, often relevant only in computer science, which was Donald Knuth’s area of expertise.

In my case, my relationship with calculators was complicated, leaning toward dislike. My sole calculator was a Polish notation device, where one inputs operands followed by the operation. This system likely ingrained the correct nesting of operations in my mind, as the absence of an equals sign requires users to navigate the order of operations thoughtfully.

This calculator, named after the Voyager space missions that inspire me, remains a cherished possession, even as I generally shy away from gadgets. I distinctly remember that I never used one in my professional career; once my undergraduate exams concluded, calculators and I parted ways.

For complex calculations, one needs a reliable setup in software like Mathematica or through custom code, allowing for thorough debugging and validation before applying real-world scenarios, especially in critical fields like aerospace.

When my son mentioned “BODMAS” two years ago, I was taken aback. “Do they at least explain what the O stands for?” I inquired.

*He genuinely believed it was called “Oxponentiation.”* His math teacher had simply stated it was an "old word for exponentiation." My son and his peers thought it was “oxponentiation”!

Perhaps that term could refer to the ordination of Anglican priests at Oxford or even the whimsical idea of dropping oxen on convoluted math teachers who complicate matters unnecessarily.

Simplicity must prevail, or we risk "Death By Ox!"

At least my daughter is taught to refer to it as AOOC, “Algebraic Operation Order Convention.” While it may not have a catchy acronym, it conveys a vital point: whether PEMDAS, BODMAS, AOOC, or any other term, these are simply linguistic conventions, not mathematics themselves.

This distinction is crucial for children to grasp, as it highlights that these conventions exist to prevent ambiguity. Despite their seemingly arbitrary nature, these agreements are essential for clear communication and understanding of mathematical concepts.