February 11th

February 11th

Use of Parentheses or Brackets (The Distributive Law)

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) × (-3) or (5)(-3)

instead of

5 × –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 ‘×’ multiplication sign and just write (5)(-3) or 5(-3). Dropping the ‘×’ could not be done without the use of brackets, because then the already rather awkward expression ‘5 × –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) × (5 – 3 + 6)

= (-1) × [5 + (-3) + 6]

= (-1) × (5) + (-1) × (-3) + (-1) × (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 × 5 ]

= 5 + 3[ 4 + 30 ]

= 5 + 3 × 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 × 6 – 5 × 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.