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### Brackets

You can use brackets 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 brackets first.

The use of brackets 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 brackets are an implicit multiplication.

##### Example 2

In the calculation

sin (a + b)

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

sin a + b = sin (a) + b

That is why you often see

sin (x)

where brackets 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 brackets

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 brackets

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 brackets 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 brackets 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 brackets 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 brackets.

##### Example 8

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

(−2)4 = 16

the brackets 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 brackets cause confusion, as in the following calculation all seems clear

(1 − 1) + (1 − 1) + (1 − 1) +  ··· = 0 but the following is also explainable

1 − (1 − 1) − (1 − 1) − (1 − 1) −  ··· = 1 If we omit the brackets in both calculations it says

1 − 1 + 1 − 1 + 1 − 1 +  ··· = ?

and then we do'nt know the answer to this Grandi's series anymore.

##### History

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