Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
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= 2x5= 6 dots 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 Page 3 of 4 = 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) 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. David W. Sabo (2003) Use of Parentheses Page 4 of 4