When do you get rid of parenthesis in any equation. When do i know when parenthesis are neccessary to stay or not? [closed]

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When do you get rid of parenthesis in any equation. When do i know when parenthesis are neccessary to stay or not?

Ever since elementary school to high school, no one has stated the specific reason why and it still confused to this day.

All i know is distribution is one time parenthesis could be removed after.

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2 Answers

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Parenthesis are generally used for one of two reasons in a mathematical expression. The first reason is to indicate precedence. In this case, parenthesis are used to ensure that operations are performed in the intended order. According to the order of operations, parts of an expression inside parenthesis are to be computed before the rest of the expression.

To see this in action, take a look at the following two expressions side by side. They are identical except for the parenthesis.

• $1 + 1 \cdot 2 = 3$
• $(1 + 1)\cdot 2 = 4$

According to the order of operations the first expression evaluates to $3$, because the multiplication is done first, followed by the addition. The second expression, however, evaluates to $4$ since because of the parenthesis I am indicating that the addition should be done first, followed by the multiplication.

So in this way, parenthesis are used to specify the order in which expressions are evaluated. In the above expressions, the use of parenthesis changed the meaning of the expression. In these situations it is imperative that if you $\textit{mean}$ to use parenthesis you keep them in the expression, since the meaning changes when they are dropped. There are other situations, however, where the presence of parenthesis does not change the intended meaning. For instance:

• $1 + (2 + 3) + 1 = 7$
• $1\cdot3\cdot(2\cdot4) = 24$
• $2 \cdot 4 + (1 \cdot 3) = 11$

If the parenthesis are dropped in any of these expressions, the results do not change (try it!). In the first expression this is because addition is associative, meaning that when you are adding more than $2$ numbers together it doesn't matter which numbers you decide to add first (it doesn't matter, for instance, if I first do $2 + 3$ or if I first do $1 + 2$, etc. In the end the answer will be the same). The same goes for multiplication, which is why the second expression above is also independent of the placement of parenthesis.

In situations like this, where the parenthesis do not change the meaning of the expression, you may drop them because they are unnecessary. Note, that the above expressions are not the only situations in which this occurs. They are just examples. Spend some time writing out different expressions with and without parenthesis. See if you can get a feel for when they matter and when they do not.

So, as we have seen, the first use of parenthesis is to indicate precedence. The second is to use parenthesis as an alternative to $\cdot$ or $\times$ when indicating multiplication. For instance, we have that

$$ 2\cdot3 = 2 \times 3 = 2(3) $$

These are all equivalent ways of indicating multiplication. This becomes quite common since you get used to writing things like $2x, 5y, 4(x+2),xy$ etc, where simply writing two elements beside each other indicates multiplication. This works fine in the above expressions where at least one of the factors is a variable. But it wouldn't really make sense to do the same when both factors are numbers. For instance $23$ is clearly just read as the number twenty-three. To remedy this you may just throw in parenthesis around the $3$ to indicate multiplication.

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Parenthesis are there to ensure that a certain expression has to be calculated first regardless of order of operations. If the order of operations does not matter in a particular expression, you can omit parenthesis. For example $a \cdot (b \cdot c)=a \cdot b \cdot c$.

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