why should i solve expressions in parenthesis first?

Question

The title speaks for it self. I would like to know the reason why i absolutely have to solve what's in a parenthesis first. Every time i search for answers, i always get results regarding the order of operations and they all seem to skip an explanation regarding the priority of the parenthesis. They always just accept it as a first priority.

Answer 1

This is by the rules of BODMAS/PEMDAS. First letter is brackets/parentheses.

You do parentheses first because they group a set of values together.

Here's a simple example of why this holds.

Say you have three shopping bags in one hand and five in the other. Each bag has four items. How many items are there in total?

Intuitively, you would say this:

Total: eight bags. Each has four items, so total of $32$ items

and you would be correct.

Now to formulate an expression, you could do $$(3\times4)+(5\times4).$$

Without evaluating the product in each bracket, you would get $$(3+5)\times4=8\times4=32$$ as determined previously.

But that was by doing the parentheses first. If you choose to do this $$(3+5)\times4=3+5\times4=3+20=23$$ then this is clearly incorrect. This is because you have not grouped the shopping bags together.

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Another example (with units):

Trial $1$: $20$ seconds; $5$ metres per second.

Trial $2$: $10$ seconds; $5$ metres per second.

Total distance?

Correct:

$$(20\times5)+(10\times5)=(20+10)\times5=\color{red}{30\,\text{seconds}}\times\color{blue}{5\,\text{metres per second}}=150\,\text{metres}$$

Incorrect:

$$(20\times5)+(10\times5)=(20+10)\times5=\color{red}{20\,\text{seconds}}+\color{red}{10\,\text{seconds}}\times\color{blue}{5\,\text{metres per second}}$$ which gives $$20\,\text{seconds}+50\,\text{metres}$$ which clearly doesn't make sense to add.

Answer 2

That's the very purpose of parentheses. When you write $$3\ \times \ 2\ +\ 1\ $$ the usual priority rules are such that the result is $$6+1$$ If you want to indicate to the reader that here exceptionally you need the addition to be carried out first, the way to say it with symbols is to use parentheses:$$3\ \times \ ( 2\ +\ 1)\ $$ It's a convention.

Answer 3

I don't know what it means to solve for sure, but you should interpret expressions partitioned with sets of parenthesis as they were intended to be read. That is the point of the parentheses: to make it clear what exactly was meant by the expression.

Ignoring this convention is akin to deliberately misunderstanding written or spoken words. Not using parentheses appropriately is akin to not using punctuation or capitalization while writing. The result is ambiguous and hard to read.

Answer 4

In most cases it makes your life easier to evaluate what is inside parentheses first.

For example try $$ 125(35-31) + 2^4(17-15)$$ With parentheses first, we get $$ 125(4)+2^4(2)=500+32=532$$

Otherwise, $$ 125(35)-125(31) +2^4(17)-2^4(15) $$ which you need a calculator or you will be spending lots of time on it.

Answer 5

We use parentheses to distinguish between

(A) "Multiply 2 by 3 and subtract the result from 7"

and

(B) "Subtract 2 from 7 and multiply the result by 3"

If we just wrote both of these expressions as

$7-2\times3$

then there would be confusion as to which one we meant. If we just use an order of operations precedence (say multiply/divide before add/subtract) then we have no way of writing (B). So we use parentheses when we want to override the precedence of operations - so

(A) $7-2 \times 3 = 7-(2 \times 3) = 1$

and

(B) $(7-2) \times 3 = 15$

There are ways of writing arithmetical expressions that are unambiguous without needing parentheses - an example is reverse Polish notation - but you only see them used occasionally.

Answer 6

Even though order of precedence is often explained in terms of "what must be done first", that is somewhat misleading. It's not about which order you do things in when several of them are ready to do, but just a way to specify which things it is you must do.

If you want to evaluate an expression such as $$ 7^2 + 6\cdot(3+5) $$ then the misleading rule would tell you that you must start by adding $3$ to $5$. But actually there's nothing at all wrong with instead starting by squaring $7$, and then continuing to work on $6\cdot(3+5)$ afterwards -- no matter that there's a parenthesis there.

What the parenthesis tells you is something else, namely that the number you're supposed to multiply by $6$ is the result of $3+5$ rather than just the $3$.

The means that you can't start by multiplying by $6$ -- not in the sense that it is forbidden to start with the multiplication, but in the sense that it is impossible to start by multiplying, simply because the number you're going to multiply $6$ with is not there yet until you have computed $3+5$.

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Similarly, even without parenteses, $$ 7^{3-1} + 6\cdot5 $$ the glib rule that "exponentiation comes before multiplicate" doesn't mean that you can't start your work by multiplying $6$ and $5$ if that strikes your fancy. And it will be impossible to start doing the exponentiation before you do the subtraction to find out what the power is, no matter how much the glib rule claims that "exponentiation comes before subtraction".

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Where the precedence rules actually mean something is in cases like $$ 3+4\cdot 5 $$ where we need a rule to tell us that this means "add $3$ to the product of $4$ and $5$", and does not mean "multiply the sum of $3$ and $4$, by $5$".

In this simple case the precedence rule does indeed force you do do one operation before the other -- but it does so indirectly, by telling you which computation the formula means. Not by dictating how you carry it out.

Answer 7

One way of thinking about this is to recognise that expressions without parentheses are in fact shorthand which should only be used where there is no possibility of ambiguity.

In formal terms, arithmetic operations are binary, which means you need to take two terms at a time. So an expression of the form $a @ b * c$ where $@, *$ are binary operations and $a, b, c$ are numbers, has no formal meaning until we give it one.

In terms of common definitions, in cases where there is one binary operation, it is often associative, and in such cases the order of operations doesn't matter. But consider $5-4-3$ where the operation $-$ is not associative: this is ambiguous.

Where we have two operations and we are defining them formally, the relationship between the two is often expressed in terms of a distributive law, and this generally involves the meaning of terms in brackets.

So formally the use of brackets is part of the defined language of mathematics, and expressions without them are understood by convention.

Answer 8

Its because we defined that they serve that function.

The reason is because you want to make an operation to a number calculated inside the parenthasis not just some part of it.