The process of truncation involves the deletion of all the numerals to the right of a certain point in the decimal part of an expression. Some electronic calculators use truncation to fit numbers within their displays. For example, the number 3.830175692803 can be shortened as follows, depending on the number of dig- its desired in the outcome:
3.830175692803 ≈3.83017569280 ≈3.8301756928 ≈3.830175692 ≈3.83017569 ≈3.8301756 ≈3.830175 ≈3.83017 ≈3.8301 ≈3.830 ≈3.83 ≈3.8 ≈3
The wavy equality symbol (≈) means “is approximately equal to.”
ROUNDING
Rounding is the preferred method of rendering numbers in shortened form. In this process, when a given digit (call it r) is deleted at the right-hand extreme of an expression, the digit qto its left (which becomes the new rafter the old ris deleted) is not changed if 0 ≤r≤4. If 5 ≤r≤9, then q increases by 1 (round it up). Most electronic calculators use rounding. If rounding is used, the number 3.830175692803 can be shortened as follows, depending on the number of dig- its desired in the outcome:
3.830175692803
≈3.83017569280
≈3.8301756928
≈3.830175693
≈3.8301757 ≈3.830176 ≈3.83018 ≈3.8302 ≈3.830 ≈3.83 ≈3.8 ≈4
PRECEDENCE
Mathematicians agree on a certain order in which operations should be per- formed when they appear together in an expression. This prevents confusion and ambiguity. When diverse operations appear in an expression, and if you need to simplify that expression, perform the operations in the following sequence:
•
Simplify all expressions within parentheses, brackets, and braces from the inside out.•
Perform all exponential operations.•
Perform all products.•
Perform all quotients.•
For all quantities xandy, consider a difference x−yas a sum x+(−y).•
Perform all sums, proceeding from left to right.Here are two examples of the above rules of precedence. Note that the order of the numerals and operations is the same in each case, but the groupings differ.
[(2+3)(−3−1)2]2=[5×(−4)2]2 =(5×16)2 =802 =6400 [(2+3×(−3)−1)2]2=[(2+(−9)−1)2]2 =(−82)2 =642 =4096
Suppose you’re given a complicated expression and there are no parentheses, brackets, or braces in it. This is not ambiguous if the above mentioned rules are followed. Consider this example:
z= −3x3+4x2y−12xy2−5y3
If this is written with parentheses, brackets, and braces to emphasize the rules of precedence, it looks like this:
z=[−3(x3)]+{4[(x2)y]} – {12[x(y2)]} – [5(y3)]
Because we have agreed on the rules of precedence, we can do without the parentheses, brackets, and braces. Nevertheless, if there is any doubt about a crucial equation, you should use a couple of unnecessary parentheses rather than risk making a calculation error.
PROBLEM 2-7
Truncate the value of the constant pi (π), which represents the ratio of the circumference of a circle to its diameter in plane geometry, in steps from 10 digits down to six digits. Then round it off in steps from 10 digits down to six digits.
SOLUTION 2-7
First, find a reference that shows πto at least 10 digits. Most scientific calculators, including the program in the computer operating system Windows XP, have a “pi” key. This key gives the following sequence for the first 10 digits of π:
π =3.141592653
Truncating in steps down to six digits, we get this sequence of values: 3.141592653
≈3.14159265
≈3.1415926
≈3.141592
≈3.14159
Rounding in steps down to six digits produces the same end result, although a couple of the intermediate numbers are different:
3.141592653
≈3.14159265
≈3.1415927
≈3.141593
PROBLEM 2-8
What is the value of 2 +3×4 +5?
SOLUTION 2-8
First, perform the multiplication operation, obtaining the expression 2 + 12+5. Then add the numbers, obtaining the final value 19. Therefore:
2+3×4+5=19
Quick Practice
Here are some practice problems that cover the material presented in this chap- ter. Solutions follow the problems.
PROBLEMS
1. Find the arithmetic mean of 3, 4, and 20. 2. Find the geometric mean of 0, 6, and 71. 3. Find the value of 8 factorial.
4. Find the value of 6 to the 0th power (60).
5. Round off e =2.718281828459 . . . in steps down to four digits.
SOLUTIONS
1. To find the arithmetic mean, calculate as follows: (3+4+20) / 3 =27 / 3
=9
2. To find the geometric mean, calculate as follows: (0×6×71)(1/3)=0(1/3)
3. To find 8 factorial, calculate as follows:
8!=1×2 ×3 ×4 ×5×6×7×8
=40,320
4. The 0th power of any nonzero number is equal to 1. Therefore, 60=1.
5. Here is the value of eas it is repeatedly rounded off: 2.718281828459 ≈2.71828182846 ≈2.7182818285 ≈2.718281829 ≈2.71828183 ≈2.7182818 ≈2.718282 ≈2.71828 ≈2.7183 ≈2.718
Quiz
This is an “open book” quiz. You may refer to the text in this chapter. A good score is 8 correct. Answers are in the back of the book.
1. Using the product-of-sums rule, what is another expression for (x+2)(y−2)? (a) xy+2x+2y+4
(b) xy−2x+2y+4 (c) xy−2x+2y−4 (d) xy−2x−2y−4
2. Using the product-of-sums rule “in reverse,” what is another expression of the equation x2+8x+16?
(a) (x+4)(x−4) (b) (x2+4)(x2−4)
(c) (x+4)2
3. The geometric and arithmetic means of xandyare the same if and only if (a) (x +y) / 2 =(xy)1/2
(b) (x +y)2=1
(c) x2+2xy+y2=1
(d) Forget it! This can never happen. 4. The product (j+1)(j−1) is equal to
(a) 2 (b) 1 (c) −1 (d) −2
5. The product (1 +j)(1−j) is equal to (a) 2
(b) 1 (c) −1 (d) −2
6. When you want to find the arithmetic or geometric mean of two num- bers, it doesn’t matter which number is expressed “first” and which num- ber is expressed “second.” This arises from the fact that addition and multiplication are both
(a) associative. (b) commutative. (c) distributive. (d) complex.
7. Which of the following expressions is not defined for any real or com- plex value of x?
(a) (3x+3) / 2 (b) x2+10x+100
(c) x2/ (x−3)
(d) x2/ [3(x−x)]
8. In which of the following expressions must we place a constraint on the value of x, in order to make sure that the expression is defined?
(a) In the expression (3x+3) / 2. Here, the value of x must not be equal to 0.
(b) In the expression x2 + 10x +100. Here, the value of x must not be
(c) In the expression x2/(x−3). Here, the value of xmust not be equal to 3.
(d) In the expression x2/ [3(x−x)]. Here, the value of xmust be negative.
9. The expression papbcan be rewritten as
(a) p(ab)
(b) p(a+b)
(c) p(a−b)
(d) p(a/b)
10. Suppose you are confronted with the following expression that does not contain any parentheses or brackets to tell you the order in which opera- tions should be done:
26×6×52+7×8×2−3 / 4 −8 / 7
Which operation should you do first, in order to follow the rules of precedence?
(a) Multiply 26 by 6. (b) Divide 8 by 7. (c) Square 5.