Gibilisco S. Everyday Math Demystified: A Self-Teaching Guide
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SUBTRACTION
Subtraction follows the same basic rules as addition:
ð3:045 10
5
Þð6:853 10
6
Þ
¼ 304ó500 6ó853ó000
¼6ó548ó500
¼6:548500 10
6
ð3:045 10
4
Þð6:853 10
7
Þ
¼ 0:0003045 0:0000006853
¼ 0:0003038147
¼ 3:038147 10
4
ð3:045 10
5
Þð6:853 10
7
Þ
¼ 304ó500 0:0000006853
¼ 304ó499:9999993147
¼ 3:044999999993147 10
5
MULTIPLICATION
When numbers are multiplied in power-of-10 notation, the coefficients
(the numbers to the left of the multiplication symbol) are multiplied by
each other. Then the exponents are added. Finally, the product is reduced
to standard form. Here are three examples, using the same three number
pairs as before:
ð3:045 10
5
Þð6:853 10
6
Þ
¼ð3:045 6:853Þð10
5
10
6
Þ
¼ 20:867385 10
ð5þ6Þ
¼ 20:867385 10
11
¼ 2:0867385 10
12
CHAPTER 3 Extreme Numbers 61
ð3:045 10
4
Þð6:853 10
7
Þ
¼ð3:045 6:853Þð10
4
10
7
Þ
¼ 20:867385 10
½4þð7Þ
¼ 20:867385 10
11
¼ 2:0867385 10
10
ð3:045 10
5
Þð6:853 10
7
Þ
¼ð3:045 6:853Þð10
5
10
7
Þ
¼ 20:867385 10
ð57Þ
¼ 20:867385 10
2
¼ 2:0867385 10
1
¼ 0:20867385
This last number is written out in plain decimal form because the exponent is
between 2 and 2 inclusive.
DIVISION
When numbers are divided in power-of-10 notation, the coefficients are
divided by each other. Then the exponents are subtracted. Finally, the
quotient is reduced to standard form. Here’s how division works, with the
same three number pairs we’ve been using:
ð3:045 10
5
Þ=ð6:853 10
6
Þ
¼ð3:045=6:853Þð10
5
=10
6
Þ
0:444331 10
ð56Þ
¼ 0:444331 10
1
¼ 0:0444331
ð3:045 10
4
Þ=ð6:853 10
7
Þ
¼ð3:045=6:853Þð10
4
=10
7
Þ
0:444331 10
½4ð7Þ
¼ 0:444331 10
3
¼ 4:44331 10
2
¼ 444:331
PART 1 Expressing Quantities
62
ð3:045 10
5
Þ=ð6:853 10
7
Þ
¼ð3:045=6:853Þð10
5
=10
7
Þ
0:444331 10
½5ð7Þ
¼ 0:444331 10
12
¼ 4:44331 10
11
Note the squiggly equal signs () in the above equations. This symbol means
‘‘is approximately equal to.’’ The numbers here don’t divide out neatly, so the
decimal portions are approximated.
EXPONENTIATION
When a number is raised to a power in scientific notation, both the coefficient
and the power of 10 itself must be raised to that power, and the result
multiplied. Consider this example:
ð4:33 10
5
Þ
3
¼ð4:33Þ
3
ð10
5
Þ
3
¼ 81:182737 10
ð53Þ
¼ 81:182737 10
15
¼ 8:1182727 10
16
If you are a mathematical purist, you will notice gratuitous parentheses in the
second and third lines here. These are included to minimize the chance for
confusion. It is more important that the result of a calculation be correct
than that the expression be mathematically lean.
Let’s consider another example, in which the exponent is negative:
ð5:27 10
4
Þ
2
¼ð5:27Þ
2
ð10
4
Þ
2
¼ 27:7729 10
ð42Þ
¼ 27:7729 10
8
¼ 2:77729 10
7
CHAPTER 3 Extreme Numbers 63
TAKING ROOTS
To find the root of a number in power-of-10 notation, think of the root as a
fractional exponent. The square root is equivalent to the 1/2 power. The cube
root is the same thing as the 1/3 power. In general, the nth root of a number
(where n is a positive integer) is the same thing as the 1/n power. When roots
are regarded this way, it is easy to multiply things out in exactly the same way
as is done with whole-number exponents. Here is an example:
ð5:27 10
4
Þ
1=2
¼ð5:27Þ
1=2
ð10
4
Þ
1=2
2:2956 10
½4ð1=2Þ
¼ 2:2956 10
2
¼ 0:02956
Note the squiggly equal sign in the third line. The square root of 5.27 is an
irrational number, and the best we can do is to approximate its decimal
expansion because it cannot be written out in full. Also, the exponent in
the final result is within the limits for which we can write the number out
in plain decimal form, so power-of-10 notation is not needed to express the
end product here.
PROBLEM 3-3
State the rule for multiplication in scientific notation, using the variables
u and v to represent the coefficients and the variables m and n to represent
the exponents.
SOLUTION 3-3
Let u and v be real numbers greater than or equal to 1 but less than 10, and
let m and n be integers. Then the following equation holds true:
ðu 10
m
Þðv 10
n
Þ¼uv 10
ðmþnÞ
PROBLEM 3-4
State the rule for division in scientific notation, using the variables u and v to
represent the coefficients and the variables m and n to represent the exponents.
SOLUTION 3-4
Let u and v be real numbers greater than or equal to 1 but less than 10, and
let m and n be integers. Then the following equation holds true:
ðu 10
m
Þ=ðv 10
n
Þ¼u=v 10
ðmnÞ
PART 1 Expressing Quantities
64
PROBLEM 3-5
State the rule for exponentiation in scientific notation, using the variable
u to represent the coefficient, and the variables m and n to represent the
exponents.
SOLUTION 3-5
Let u be a real number greater than or equal to 1 but less than 10, and let
m and n be integers. Then the following equation holds true:
ðu 10
m
Þ
n
¼ u
n
10
ðmnÞ
Exponents m and n can be real numbers in general. They need not be
restricted to integer values. But that subject is too advanced for this chapter.
We’ll be dealing with them in Chapter 14, when we get to logarithms and
exponential functions.
Approximation and Precedence
Numbers in the real world are not always exact. This is especially true in
observational science and in engineering. Often, we must approximate.
There are two ways of doing this: truncation (simpler but less accurate)
and rounding (a little more difficult, but more accurate).
TRUNCATION
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 in steps as follows:
3:830175692803
3:83017569280
3:8301756928
3:830175692
3:83017569
3:8301756
3:830175
3:83017
3:83
3:8
3
CHAPTER 3 Extreme Numbers 65
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 q to its left (which becomes the new r
after the old r is deleted) is not changed if 0 r 4. If 5 r 9, then q is
increased by 1 (‘‘rounded up’’). Most electronic calculators use rounding
rather than truncation. If rounding is used, the number 3.830175692803
can be shortened in steps as follows:
3:830175692803
3:83017569280
3:8301756928
3:830175693
3:83017569
3:8301757
3:830176
3:83018
3:8302
3:830
3:83
3:8
4
PRECEDENCE
In scientific notation, as in regular arithmetic, operations should be per-
formed in a certain order when they appear together in an expression.
When more than one type of operation appears, and if you need to simplify
the expression, the operations should be done in the following order:
*
Simplify all expressions within parentheses, brackets, and braces from
the inside out.
*
Perform all exponential operations, proceeding from left to right.
*
Perform all products and quotients, proceeding from left to right.
*
Perform all sums and differences, proceeding from left to right.
PART 1 Expressing Quantities
66
Here are two examples of expressions simplified according to the above
rules of precedence. Note that the order of the numerals and operations is
the same in each case, but the groupings differ.
3:4 10
5
þ 5:0 10
4
¼ð3:4 10
5
Þþð5:0 10
4
Þ
¼ð340;000 þ 0:0005Þ
¼ 3:400005 10
5
3:4 ð10
5
þ 5:0Þ10
4
¼ 3:4 100;005 10;000
¼ 3;400;170;000
¼ 3:40017 10
9
Suppose you’re given a complicated expression and there are no paren-
theses, brackets, or braces in it. This is not ambiguous if the above-mentioned
rules are followed. Consider this example:
z ¼3x
3
þ 4x
2
y 12xy
2
5y
3
If this were written with parentheses, brackets, and braces to emphasize the
rules of precedence, it would look like this:
z ¼½3ðx
3
Þ þ f4½ðx
2
Þyg f12½xðy
2
Þg ½5ðy
3
Þ
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’re better off to use a couple of unnecessary paren-
theses than to make a costly mistake.
PROBLEM 3-6
What is the value of 2 þ 3 4 þ 5?
SOLUTION 3-6
First, perform the multiplication operation, obtaining the expression
2 þ 12 þ 5. Then add the numbers, obtaining the final value 19. Therefore:
2 þ 3 4 þ 5 ¼ 19
CHAPTER 3 Extreme Numbers 67
Significant Figures
The number of significant figures, also called significant digits, in a power-of-
10 number is important in real-world mathematics, because it indicates the
degree of accuracy to which we know a particular value.
MULTIPLICATION, DIVISION, AND EXPONENTIATION
When multiplication, division, or exponentiation is done using power-of-10
notation, the number of significant figures in the result cannot legitimately
be greater than the number of significant figures in the least-exact expression.
You might wonder why, in some of the above examples, we come up
with answers that have more digits than any of the numbers in the original
problem. In pure mathematics, where all numbers are assumed exact, this
is not an issue. But in real life, and especially in experimental science and
engineering, things are not so clean-cut.
Consider the two numbers x ¼ 2.453 10
4
and y ¼ 7.2 10
7
. The following
is a perfectly valid statement in pure mathematics:
xy ¼ 2: 453 10
4
7:2 10
7
¼ 2:453 7:2 10
11
¼ 17:6616 10
11
¼ 1:76616 10
12
But if x and y represent measured quantities, as in experimental physics
for example, the above statement needs qualification. We must pay close
attention to how much accuracy we claim.
HOW ACCURATE ARE WE?
When you see a product or quotient containing a bunch of numbers in
scientific notation, count the number of single digits in the coefficients of
each number. Then take the smallest number of digits. That’s the number
of significant figures you can claim in the final answer or solution.
In the above example, there are four single digits in the coefficient of x, and
two single digits in the coefficient of y . So you must round off the answer,
which appears to contain six significant figures, to two. It is important to use
PART 1 Expressing Quantities
68
rounding, and not truncation. You should conclude that:
xy ¼ 2:453 10
4
7:2 10
7
¼ 1:8 10
12
In situations of this sort, if you insist on being rigorous, you should use
squiggly equal signs throughout, because you are always dealing with
approximate values. But most folks are content to use ordinary equals
signs. It is universally understood that physical measurements are inherently
inexact, and writing squiggly lines can get tiresome.
Suppose you want to find the quotient x/y instead of the product xy?
Proceed as follows:
x=y ¼ð2:453 10
4
Þ=ð7:2 10
7
Þ
¼ð2:453=7:2Þ10
3
¼ 0:3406944444 ... 10
3
¼ 3:406944444 ... 10
4
¼ 3:4 10
4
WHAT ABOUT ZEROS?
Sometimes, when you make a calculation, you’ll get an answer that lands
on a neat, seemingly whole-number value. Consider x ¼ 1.41421 and y ¼
1.41422. Both of these have six significant figures. The product, taking signi-
ficant figures into account, is:
xy ¼ 1:41421 1:41422
¼ 2:0000040662
¼ 2:00000
This looks like it’s exactly equal to 2. In pure mathematics, 2.00000 ¼ 2.
But in practical math, those five zeros indicate how near the exact number 2
we believe the resultant to be. We know the answer is very close to a
mathematician’s idea of the number 2, but there is an uncertainty of up to
plus-or-minus 0.000005 (written 0.000005). When we claim a certain
number of significant figures, zero is as important as any of the other nine
digits in the decimal number system.
CHAPTER 3 Extreme Numbers 69
ADDITION AND SUBTRACTION
When measured quantities are added or subtracted, determining the number
of significant figures can involve subjective judgment. The best procedure is
to expand all the values out to their plain decimal form (if possible), make
the calculation as if you were a pure mathematician, and then, at the end
of the process, decide how many significant figures you can reasonably claim.
In some cases, the outcome of determining significant figures in a sum or
difference is similar to what happens with multiplication or division. Take,
for example, the sum x þ y, where x ¼ 3.778800 10
6
and y ¼ 9.22 10
7
.
This calculation proceeds as follows:
x ¼ 0:000003778800
y ¼ 0:000000922
x þ y ¼ 0:0000047008
¼ 4:7008 10
6
¼ 4:70 10
6
In other instances, one of the values in a sum or difference is insignificant
with respect to the other. Let’s say that x ¼ 3.778800 10
4
, while y ¼ 9.22
10
7
. The process of finding the sum goes like this:
x ¼ 37ó788:00
y ¼ 0:000000922
x þ y ¼ 37ó788:000000922
¼ 3:7788000000922 10
4
In this case, y is so much smaller than x that it doesn’t significantly affect the
value of the sum. Here, it is best to regard y, in relation to x or to the sum
x þ y, as the equivalent of a gnat compared with a watermelon. If a gnat
lands on a watermelon, the total weight does not appreciably change, nor
does the presence or absence of the gnat have any effect on the accuracy of
the scales. We can conclude that the ‘‘sum’’ here is the same as the larger
number. The value y is akin to a nuisance or a negligible error:
x þ y ¼ 3:778800 10
4
PROBLEM 3-7
What is the product of 1.001 10
5
and 9.9 10
6
, taking significant figures
into account?
PART 1 Expressing Quantities
70