Use of Parentheses or Brackets (The Distributive Law)
Brackets as “Packages” for Negative Numbers
Sometimes brackets are used to avoid awkwardlooking
expressions involving negative numbers. For instance, to indicate
the product of 5 and –3, we could write
(5) Ã— (3) or (5)(3)
instead of
5 Ã— –3
The brackets around the 5 in (5)(3) don’t really mean
anything, and could be dropped. By using brackets in this way, we
can also drop the ‘Ã—’ multiplication sign and just
write (5)(3) or 5(3). Dropping the ‘Ã—’ could not be
done without the use of brackets, because then the already rather
awkward expression ‘5 Ã— –3’ would become ‘5
–3,’ which most people would interpret to mean “5
subtract 3” rather than “5 times –3”.
Distributing the Minus Sign
One of the trickier situations is when a pair of brackets is
preceded by a minus sign. In such instances, regard the minus
sign as meaning “multiply by –1.” Thus, for
example,
(5 – 3 + 6) = (1) Ã— (5 – 3 + 6)
= (1) Ã— [5 + (3) + 6]
= (1) Ã— (5) + (1) Ã— (3) + (1) Ã— (6)
= 5 + 3 – 6 = 8
In effect, when we remove brackets which are preceded by a
minus sign, we need to reverse the signs of every term
that was inside the brackets.
Notice how we used two different styles of brackets in one of
the forms above so that matching pairs were easy to identify.
Nested Brackets
When the expression within a pair of brackets itself contains
brackets, we say that the brackets are nested.
In such situations, the removal of brackets must progress from
the innermost pair of brackets to the outermost pair.
For example,
5 + 3[ 4 + 6(7 – 2) ]
= 5 + 3[ 4 + 6 Ã— 5 ]
= 5 + 3[ 4 + 30 ]
= 5 + 3 Ã— 34
= 5 + 102
= 107.
In the first step, we did the (7 – 2) = 5, since this
pair of brackets was the innermost. This left just the square
brackets as the only remaining pair of brackets (hence the
innermost brackets), which we removed last.
A Caution
A little later in these notes, we will describe the general
rule for deciding which operations to do in which order when
faced with a more complication arithmetic or mathematical
expression such as in the example above. Here we just mention a
particular caution with respect to inserting brackets into an
expression.
Sometimes brackets may be inserted into an expression for
convenience:
5(6 – 2) = (5) (6) – (5) (2) = 5 Ã— 6 – 5 Ã— 2
= 30 – 10 = 20
The brackets in the first step have no effect except to act as
visual separators for the numbers.
However, inserting brackets in an invalid manner causes an
error – especially when addition or subtraction is mixed
with multiplication. For example, it is a serious error to
replace
4 + 5(6 – 2) with (4 + 5)(6 – 2).
We know that
4 + 5(6 – 2) = 4 + (5) (4) = 4 + 20 = 24.
But,
(4 + 5) (6 – 2) = (9) (4) = 36,
a quite different number. The general rule for order of
operations will state that multiplications must always be done
before additions or subtractions. When we start with
4 + 5(6 – 2)
and insert brackets to get
(4 + 5)(6 – 2),
we are putting in brackets which force the leftmost addition
to be done before the multiplication by 5. Thus, inserting
the brackets to get this last expression violates the general
rule we described that multiplications must be done before
additions.
