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
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 the power function is
If you have no brackets
it is not clear to all what the intention is. Very confusing is
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
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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
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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
 = y + x ddx y.gif){kind=small}
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
the brackets indicate that you work with powers of the negative number −2. In the calculation
you work with powers of the positive number +2. For odd powers you get
Example 9
In the binomial formula you must calculate the square as
because
Example 10
Sometimes brackets cause confusion, as in the following calculation all seems clear
but the following is also explainable
If we omit the brackets in both calculations it says
and then we do'nt know the answer anymore.
Additional information
The Italian mathematician Rafael Bombelli (1526 - 1572) introduced the round brackets.