Why do we use BODMAS?

Question

In any of mathematical step why do we always use BODMAS rule. However if we dont use it we can prove some unequal numberss like 1=2? Specifically why we always use BODMAS rule?

Answer 1

A written language needs some rules for interpreting the signs on paper or stone: consider

that you can see on the “Boustrophedon” entry in Wikipedia.

This is written in a peculiar way: counting from the top, odd numbered lines must be read from left to right, even numbered ones from right to left.

English is read left-to-right and top-to-bottom; Arabic is right-to-left and top-to-bottom; classical Japanese top-to-bottom and right-to-left.

Each writing system has its own rules for decoding it, that is, for extracting a meaning from the signs. Such rules are conventional: there is no law of nature that imposes to read English from left to right.

Written mathematics is a language of its own; during the centuries it has developed some conventions for reducing symbol usage while not introducing ambiguities. We read $$ a+bc $$ as “the sum of $a$ with the product of $b$ times $c$” instead of “sum $a$ with $b$ and multiply the result by $c$”. There's no law of nature that requires preferring the former interpretation. When the algebraic notation was (slowly) developing, the preference went to multiplication having precedence over addition: $$ ax^2+bx+c $$ is a common expression that is interpreted as a polynomial.

Whether you call the set of conventions as BODMAS or PEMDAS or give no name (as usual in Italy), it is still a set of conventions that make an algebraic expression not ambiguous with regard to its meaning, provided everybody has learned the proper way of reading it.

Answer 2

We want to have an acceptable agreement about how to evaluate complex expressions with more than one operation on it. For instance, how would you evaluate $$5\times3+2$$ If you did the multiplication first, you'd get $15+2=17$, but if you did the addition first, you'd get $5\times5=25$. You could always use braces to indicate which operation should be given precedence, but that would get really messy when a formula had four or five or more variables in it. So we came up with BODMAS (or PEMDAS as we call it here in the States) as an agreed rule to keep the parentheses down to a minimum.

Answer 3

These rules do not involve or affect any real mathematics - they are just conventions: ways we have agreed to write things down so that we don't have to write quite so much. They are like rules for spelling and grammar - they don't affect what you are trying to say, but make it easier to communicate.

That said, it's best to avoid expressions that might be ambiguous. A careful computer programmer would use extra parentheses even though the computer language knows the conventions.

I think much too much time is spent in school teaching these rules, asking students to evaluate complicated expressions that would never occur except for testing those very conventions.

Answer 4

It's by agreement. Note that we would always have to chose in what order to carry out indicated binary operations, if we want to use our present system of notation.

Say we have the expression $2+3×4-1.$ If we didn't make such conventions, different (otherwise valid) interpretations of such would mean different things, which shouldn't be, of course. The usual way to specify this order is by using a combination of brackets. However, this quickly becomes tiresome. So we see that we can avoid the brackets if we adopt a simpler convention for interpreting certain strings of symbols (where we know where the brackets are expected to be without actually having them there).

Thus, we can give definite meaning to expressions like $2+3×4-1,$ for example, since we know that it actually means (by definition) $2+(3×4)-1,$ and not $(2+3)×4-1,$ say. If the expression was longer and more complicated, and we don't have this convention, then we'd have to use a monstrously confusing mess of nested brackets. That wastes a lot of space and time on something we can do without with a tacit agreement; so we do. And that's better, obviously.