Download The Mathematics 11 Competency Test

The Mathematics 11
Competency Test
Use of Parentheses or Brackets
(The Distributive Law)
The Basic Idea
Pairs of parentheses or brackets (such as ( ), [ ], { }, etc) are one way that is used to group parts
of an expression together to show exactly in what order the arithmetic is to be done. Thus, for
example, when we write
2(3 + 5)
we mean “add 3 to 5 first, then multiply the result by 2.” Thus
2(3 + 5) = 2 x 8 = 16
(*)
This expression can also be interpreted to mean that the multiplication by 2 is to be done to every
term inside the brackets. Thus
2(3 + 5) = 2 x 3 + 2 x 5 = 6 + 10 = 16
(**)
which is the same final result as before. In the sequence of steps in (**), we say that we are
expanding the brackets. If we use the symbols a, b, and c to represent any three numbers, then
the overall process in (**) can be symbolized as
a(b + c) = a x b + a x c
The rule in the box above is called the distributive law for multiplication. It shows how
multiplication of the bracketed expression by ‘a’ is “distributed” to all of the terms in the brackets.
Be careful if there are minus signs inside the brackets. For example
2(3 – 5) = 2(-2) = -4
and
2(3 – 5) = 2 x 3 + 2 x (-5) = 6 + (-10) = 6 – 10 = -4
Thus, the minus sign must be retained when the brackets are removed. In general
a(b - c) = a x b - a x c
An Illustration
You can visualize the equivalence between
2(3 + 5)
and
2x3+2x5
by considering the following diagrams. We draw 16 dots arranged in two rows as follows:
David W. Sabo (2003)
Use of Parentheses
Page 1 of 4
Each row has a group of three dots and a group of 5 dots, so the number of dots in each of the
two rows is 3 + 5 = 8 dots. Now we can count the total number of dots in the diagram using the
following scheme:
= 3 + 5 = 8 dots
= 3 + 5 = 8 dots
}
total of
2 x 8 = 16
dots
That is, to get the total of 16 dots in the diagram, we count 8 dots in each of two rows, for a total
of 16 dots:
2(3 + 5) = 2 x 8 = 16.
Alternatively, we could imagine grouping the dots in a different way for counting:
2x3=
6 dots
2x5=
10 dots
16 dots in total
This last diagram counts the dots according the process
2 x 3 + 2 x 5 = 6 + 10 = 16.
By comparing the last two diagrams, you can see how the equivalence between the two
expressions
2(3 + 5)
and
2x3+2x5
comes about. In this illustration, the two expressions just amount to counting the same array of
16 dots, but tallying the dots in different orders.
David W. Sabo (2003)
Use of Parentheses
Page 2 of 4
Brackets as “Packages” for Negative Numbers
Sometimes brackets are used to avoid awkward-looking expressions involving negative numbers.
For instance, to indicate the product of 5 and –3, we could write
(5) x (-3)
or
(5)(-3)
instead of
5 x –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 ‘x’ multiplication sign and just write (5)(-3) or 5(-3).
Dropping the ‘x’ could not be done without the use of brackets, because then the already rather
awkward expression ‘5 x –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) x (5 – 3 + 6)
= (-1) x [5 + (-3) + 6]
= (-1) x (5) + (-1) x (-3) + (-1) x (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 x 5 ]
= 5 + 3[ 4 + 30 ]
= 5 + 3 x 34
David W. Sabo (2003)
Use of Parentheses
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= 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 x 6 – 5 x 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)
(4 + 5)(6 – 2).
with
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.
David W. Sabo (2003)
Use of Parentheses
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