# How far does BEDMAS (order of operations) go?

I was just curious to know how far does BEDMAS go? Can I always find the opposite of something and do it on both sides of an equation to remove it? I know this works for elementary math and high school math but this also work for higher level maths?

EDIT:

BEDMAS is the acronym for order of operations meanings Brackets then Exponents then Division/Multiplication then Addition/Subtraction.

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Might be good to add this to posting:BEDMAS order of operations. Each letter stands for: B - Brackets E - Exponents D - Division M - Multiplication A - Addition S - Subtraction Regards – Amzoti Apr 22 '13 at 18:06
There are other kinds of Mathematical statements like inequalities where doing something on both sides doesn't necessarily maintain the relationship. For example, compare 1 and 2 and then multiply both by -1 and note the inequality gets flipped. There are other kinds of Mathematical operators that don't necessarily use equations or inequalities and thus I'd be prepared for that. Also, not everything has an opposite as "multiplying by zero" doesn't have an opposite that can be evaluated in a meaningful way. – JB King Apr 22 '13 at 18:36
The inequality "gets flipped", but that's because multiplying by $-1$ is equivalent to adding the opposite of each side to both sides. To me, this is preferable, because it doesn't make it seem like $-1$ has some magical voodoo interaction with inequalities. – Emily Apr 22 '13 at 18:39

It looks like in your head you are mixing up the "order of operations" with the process of solving equations. These two things are completely different, but it is understandable why they might look similar to someone in a modern day primary-secondary school curriculum.

The order of operations is just a set of conventions so that we can read mathemaical expressions unabiguously. Strictly speaking, if I wrote $3+2\times 4$ without an order of operations, it could either mean $3+(2\times 4)=11$ or $(3+2)\times 4=20$. To eliminate this ambiguity, we said "OK, everybody uses the first one (multiplication before addition.)

Now, solving equations by doing "equal things to equal sides" is a process you are probably learning in algebra. The key idea of solving equations using that method is that you are applying a function to both sides of an equality. That employs the order of operations, but it certainly is a completely different process.

If a function has two inputs, and it sees that they are equal, then the two associated outputs of the function will also be equal. (That's what it means to be a function!) Lots of operations you have learned are functions. The function $f(n)=n+3$, if applied to $y-3=x$ would say "ah, these two things are equal, and so I say that $y-3+3=x+3$ as well!" After combining like terms, we have the new thing, $y=x+3$.

Similarly, subtracting a number is a funtion. Similarly multiplying by a fixed number is a function. The same can be said for division because dividing by a number is just multiplying by the inverse of that number.

The same goes for squaring both sides: $f(n)=n^2$. Because this is a function, you can apply it to both sides of $\sqrt{x}=2$, and the result will still have equality: $(\sqrt{x})^2=2^2$. After simplification, $x=4$. In a moment I will say that this one is not like the other four because undoing a power is not always a function!

You asked "can always find the opposite of something and do it on both sides of an equation to remove it?" Certainly this was true for addition, subtraction, multiplication and division. If you know about logarithms and exponentials, then you can also say it is true for them. The thing that ties all these things together is that they are pairs of inverse operations which can undo each other.

There are times, however, when the thing you want to do does not have an inverse (and so you may have to be careful when you are solving it.) I demonstrated above that it is true when raising sides to the $n$th power. Now the inverse would be taking the $n$'th root, but reversing even roots (for example square roots) is not as easy! As a simple example, consider $(-2)^2=2^2$. One would like to just knock off the two powers, but of course $-2\neq 2$.

Another example would be the sine function: $\sin(0)=\sin(\pi)$. Just knocking off sine from both sides would definitely not get you a sensical answer. Solving in situations where the operation does not have a well-defined inverse is not impossible, but I hope I've expressed that it is just not as simple as it is with the four basic arithmetic operations.

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When you have an equals sign, you assert that the thing on one side is exactly the same as the thing on the other side. Therefore, performing the same operation on both sides will always result in another equality, as long as the operations are performed according to their rules.

For more complicated operations/algebras/etc. this becomes difficult. For example, in complex analysis, you have to be careful to specify which branch of the logarithm you are taking; you cannot just say $x = y \implies \log x = \log y$ if $x, y \in \Bbb C$.

But as long as you stay within the rules of your algebra/field/whatever, order of operations always holds.

That said, there are times when the cancellation laws don't hold. A simple example is that for vectors $x,y,z \in \Bbb R^n$, $x\times y = x\times z$ does not imply that $y = z$. This is because the cross-product operation $\times$ does not form a group, and so attempting to cancel $x$ leads to something that violates the rules of the algebraic structure induced by $\times$.

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It does, of course, go without saying that some operation being valid is not the same thing as some operation being useful... – Emily Apr 22 '13 at 18:37