<    1      2      3    >

### Parentheses

You can use parentheses in mathematical operations. Often this is just to increase readability. Sometimes they are necessary in order to enforce an order in the calculation. You may never just calculate something in parentheses first.

The use of parentheses indicates what belongs together. Whether they are really needed is not that important. Clarity must come first.

##### Example 1

The calculation 4 × 7 = 28 can be written as

4 (7) = 28

because parentheses are an implicit multiplication.

##### Example 2

In the calculation

sin (a + b)

all is clear. If you write no parentheses, it becomes something very different, because

sin a + b = sin (a) + b

That is why you often see

sin (x)

where parentheses are used, although

sin x

is of course sufficient.

##### Example 3

In the calculation with the sine

sin (x) · a = a · sin x

all is really clear. If you use no parentheses

sin x · a = a · sin x

it is no longer clear to all what the intention is. So is

sin (x · a)

something completely different, and

sin x · (a) = (sin x) · (a) = a · sin x

is not wrong, but unnecessary tricky.

##### Example 4

The logarithm of a power is

log (an ) = n · log a

If you have no parentheses

log an = n · log a

it is not clear to all what the intention is. Very confusing is

log (a)n =  (log a)n

because the parenheses are unnecessary.

##### Example 5

In the calculation of

you must first perform the exponentiation, and only then take the square root. Thus

is completely wrong. The parentheses must be solved from inside to outside, so

##### Example 6

When calculating derivatives you can use different formats, such as

If y is a function of x we must apply the product rule on (x · y) , and the parentheses clarify this. So you'll get

##### Example 7

To write a square root you can use different formats, such as

The solid line of the root sign has the same meaning as the use of parentheses.

##### Example 8

In a power function with a negative number as base, this number must be put in paremtheses. In the calculation

(−2)4 = 16

the parentheses indicate that you work with powers of the negative number −2. In the calculation

−24 = − (+2)4 = −16

you work with powers of the positive number +2. For odd powers you get

(−2)3 = −8 = −23

##### Example 9

In the binomial formula you must calculate the square as

(a + b)2 = (a + b) (a + b) = a2 + 2ab + b2

because

##### Example 10

Sometimes parentheses cause confusion, as in the following calculation all seems clear

but the following is also explainable

If we omit the parentheses in both calculations it says

1 − 1 + 1 − 1 + 1 − 1 +  ··· = ?

and then we don't know the answer to this Grandi's series anymore.

##### History

The Italian mathematician Rafael Bombelli (1526 - 1572) introduced the round parentheses.