Difference between PEMDAS and BODMAS.

Question

I didn't get the point that when the PEMDAS and the BODMAS rule are different, then how can they both yeild the same results. I have searched over google but found everywhere that they're the same. Where as I see them as according to the precedency they have in the order the put in are different

PEMDAS Parentheses > Exponents > Multiplication > Division > Addition > Subtraction

BODMAS(Also known as PEDMAS/BIDMAS) Brackets/Braces > Order > Division > Multiplication > Addition > Subtraction

The main difference I see in both if them is the change of precedency of multiplication and division. Can anyone elaborate this?

Answer 1

Division is the inverse operation of multiplication, and subtraction is the inverse of addition. Because of that, multiplication and division are actually one step done together from left to right; the same goes for addition and subtraction. Therefore, PEMDAS and BODMAS are the same thing.

To see why the difference in the order of the letters in PEMDAS and BODMAS doesn't matter, consider the following expression:

$$ a_1 / b_1 \times a_2 / b_2 \times a_3 / b_3 $$

As long as you treat the multiplication and division steps with equal precedence and read them from left to right, you will end up with the following after simplification:

$$ \frac{a_1 a_2 a_3}{b_1 b_2 b_3} $$

With that, we can acknowledge that PEMDAS and BODMAS both have the same four steps.

$$ \text{parentheses} \rightarrow \text{exponents} \rightarrow \underbrace{\begin{matrix} \text{multiplication} \\ \text{division}\end{matrix}}_{\text{from left to right}} \rightarrow \underbrace{\begin{matrix} \text{addition} \\ \text{subtraction}\end{matrix}}_{\text{from left to right}} $$

P.S. Thanks to @coffeemath and @JMoravitz for their comments improving this answer.

Answer 2

Neither of them quite accurately reflect actual usage. Here's how I'd describe the rules for grouping that they're referring to, with some of the subtle issues that are often left out:

• Parentheses/brackets group terms because that is the entire point of those symbols: to indicate groupings of terms. So this really precedes the whole question of conventions for grouping, as this is a way of explicitly indicating a grouping.
• Exponentiation binds more tightly (that is, has higher operator precedence) than addition, subtraction, multiplication, or division. So, for example, $ab^c$ means $a(b^c)$, not $(ab)^c$. Note, however, that the raised position of the exponent itself acts as a "grouping" (by position rather than with explicit parentheses), so $a^{b+c}$ means $a^{(b+c)}$, not $a^b + c$. By convention, it is also treated as right-associative, that is, $a^{b^c}$ is taken to mean $a^{(b^c)}$, not $(a^b)^c$. (This is because, for positive real numbers $a, b, c$, we have $(a^b)^c = a^{bc}$; it can be expressed without nested exponents at all.)
• Multiplication and division bind more tightly than addition and subtraction (so $a + b \cdot c$ means $a + (b \cdot c)$, not $(a + b) \cdot c$), but less tightly than exponentiation. But there are some further subtleties:
  • Fractions denoted with a long horizontal bar also group "by position" like an exponent does, so $\frac{a + b}{c}$ means $(a + b)/c$, not $a + (b/c)$, even though there are no explicitly written parentheses grouping the $a + b$ expression.
  • Multiplication "by juxtaposition", that is, indicating multiplication by putting two expressions next to each other without a multiplication symbol between them ($ab$ rather than $a \cdot b$), can create genuinely ambiguous notation: $a/bc$ could mean either $(a/b)c$ or $a/(bc)$, and therefore such expressions should be avoided if it's not obvious from context what's intended. (For example: does $1/2x$ mean $\frac{1}{2x}$ or $\frac{1}{2} x$? I'd say the only proper answer is "don't write things like $1/2x$, and if someone else has written it, ask them to clarify". The point of mathematical notation is to express things clearly, unambiguously, and concisely, not to trick people with ambiguous edge cases of notation.)
  • Division is not associative, so expressions like $a/b/c$ are also potentially confusing and should be avoided (by using fractions with a long horizontal bar or by using parentheses). However, if you do encounter an expression like that, division is treated as left-associative, so that would mean $(a/b)/c$, not $a/(b/c)$. The division sign $\div$ is much less common than $/$ outside of some countries' grade school mathematics, but the two follow the same grouping rules.
• Addition and subtraction bind less tightly than exponentiation, multiplication, and division. Also, subtraction is similar to division in that it's taken as left-associative; expressions of the form $a - b - c$ are also far more common than the analogous expressions for division, and unambiguously mean $(a - b) - c$, not $a - (b - c)$. Likewise when combined with addition: $a - b + c$ means $(a - b) + c$, not $a - (b + c)$.

Another point is that these are rules for grouping, not really for "order of operations": If you encounter an expression like $$2 + 3 + 5 \cdot 7,$$ the grouping rules tell you that this means $$(2 + 3) + (5 \cdot 7),$$ but the fact that multiplication binds more tightly than addition doesn't mean you have to evaluate multiplication first. It's perfectly legitimate, for instance, to compute $2 + 3 = 5$ and simplify the expression to $5 + (5 \cdot 7)$, then compute the multiplication, then the final addition.