What's In A Rule: Dissecting BODMAS

oomfies solve this pic.twitter.com/0RO5zTJjKk

— #em 𖤐 (@jik99k) July 28, 2019

Ughh!! Not again! I muttered as I scroll through my Twitter feed. This post and similar virals keep popping in our Facebook and Twitter pages regularly. Every time, thousands reply. Many get it right, but most fail. I don’t curse. We do not do arithmetic in our day-to-day life, except for counting money or paying bills. Even then, we use calculators.  

By now, you must be saying- it’s elementary Watson. Don’t start a lecture on BODMAS.

No! not at all. Instead, we will explore the logic behind this rule and its origin.

BODMAS is an elementary rule that you learn in school. It is so common that it’s difficult to dig out the history of its origin or ask the logic behind this rule.

Some argue that this rule must have been formed as a ‘Convention’ to streamline arithmetic calculations, sometime in the 19^th century, when we started producing mathematics textbooks. The need for such convention became extremely crucial when we began developing computers. However, there are documented evidences of the use of primitive forms of parenthesis, even in the seventeen century. Is there any chance that this rule may have originated in ancient times and has evolved further?

Let us investigate the rule itself to get some clues.

Parenthesis, in itself, represents priority. Therefore, it is common sense to perform the operations within parenthesis first. However, to understand the rest of the rule, let us focus on addition and multiplication.

Remember, that we currently use the Indo-Arabic number system that is suggested to be invented by ancient Indian mathematicians. Suppose we want to write “Three Hundred Twenty-One.” In the Roman numeral system, this would be CCCXXI. However, in the Indo-Arabic system, we will write it as 321. How are we getting this number?

Following the rule of Indo-Arabic system: 321 = 3 Hundreds + 2 Tens + 1 Ones = 3 × 100 + 2 × 10 + 1 × 1. Though it’s elementary for us, the invention of this system of representing a number was revolutionary.

Note the sequence of operations in this example. We have multiplied first and then performed the additions. Isn’t that what BODMAS says? What if we do not follow this rule and perform the additions first? Let’s check: 3 × 100 + 2 × 10 + 1 × 1 = 3 × 102 × 11 × 1 = 3366. So, the rule of multiplication before the addition is at the heart of our numeral system. Without this rule, our number system will not work.

In a way, division and subtractions are similar to multiplication and addition, respectively. Let me elaborate: 8 ÷ 4 × 2 = 8 × (1/4) × 2 and 4 – 2 + 2 = 4 + (-2) + 2. So, the order that we have followed for multiplication and addition should also be followed for division and subtraction.

The example of 4 – 2 + 2 also explains why we must perform operations from left to right. Let’s do the operations in the opposite way, 4 – 2 + 2 = 4 – 4 = 0. But even without following the BODMAS we can say that this result is wrong as 4 – 2 + 2 = 4 + (-2) + 2 = 4. Therefore, we have to stick to the rule of doing operations from left to right.

What about the exponents then? If you pay attention, you can easily notice that exponentiated terms are actually multiple operations clubbed together, just like we do by parenthesis. For example, 2³ = (2 × 2 × 2) and $2^{\frac{3}{2}}= \sqrt{2\times 2\times 2}$. So, it is wiser to keep such terms separate from other operations and perform them beforehand. Otherwise, the calculation would get complicated.

So, in brief, the rule for multiplication and addition is at the heart of BODMAS, and as I suggested, it is a fallout of our numeral system. This numeral system originated in ancient India. The way we use symbolic algebra now was not developed at that time. But they had already started using notations for mathematical operations. Can there be any clue in the works on ancient Indian mathematicians on the order of operation? My hunch is there must be. I haven’t been able to explore much of their works. If you have any clue, do share it with me.