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
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
If you have no parentheses
it is not clear to all what the intention is. Very confusing is
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
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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
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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
 = 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 parentheses.
Example 8
In a power function with a negative number as base, this number must be put in paremtheses. In the calculation
the parentheses 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 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
and then we don't know the answer to this Grandi's series anymore.
HistoryThe Italian mathematician Rafael Bombelli (1526 - 1572) introduced the round parentheses. |