,C h a p t e r
1
Measurement
Why This Matters: A Nurse Anesthetist’s View
Why do we care, in this day of technology, if a certified registered nurse anesthetist can do math?
Well, even today technology may fail, leaving us to use basic nursing skills of patient assessment
and math to provide the best care for our patients. If we have basic math skills, we cannot only
determine a drug dose or measure a base deficit, but we can also recognize whether the answer
we obtain is reasonable. We may recognize that the answer “just doesn’t seem right” when the
calculated dose is unexpectedly lower or higher than we anticipate. This may lead us to prevent
a potential erroneous drug dose or calculate a fluid deficit with precision. A common example
involves rechecking your calculations if you have to crack a second vial of medication to achieve
the desired dose. Most medications are packaged in common doses (i.e., spinal bupivacaine
comes as a 7.5% solution in a 2-mL vial). If your original calculation requires you to open a
second vial to draw up 22 mg (2.9 mL) of bupivacaine, it is worthwhile to double check the
dosage. If your patient is a basketball player who is 7 ft tall, it may be correct. If your patient is
5 ft, 4 in. tall, the original dosage may be catastrophic.
A REVIEW OF SOME BASIC MATHEMATICAL SKILLS
Chemistry and physics are both very logical sciences, but both depend on math to translate
concepts into application. Unfortunately, many students who struggle in these disciplines have
more trouble with the mathematics than with the scientific concepts. Therefore, let us begin
our exploration of chemistry and physics with a review of some basic math skills and concepts
that you will use throughout this course. While this chapter reviews basic math skills, it cannot
replace a basic understanding of college-level algebra.
You will need a calculator for this course. Some of you may have fancy graphing cal-
culators, but the capacity of these instruments far exceeds your mathematical needs for
1
,2 Chapter 1 Measurement
this course. Any calculator that is labeled as a “scientific calculator” will more than suffice.
You should be able to buy an adequate calculator for less than $20. When dealing with a
math problem, don’t reach for your calculator first. Whether or not the calculator gives you
the correct answer depends on your ability to give it the correct information with which to
work. Think about what operations you need to do first. See how far you can get with just
a paper and pencil. You don’t have to do arithmetic computations, such as long division,
by hand, but you should be able to perform the mathematical manipulations (e.g., solve the
equation for x) without a calculator. In fact, it is typically best to solve the equation first for
the variable of interest before plugging in numbers. As you become more adept at thinking
your way through math problems, you are more likely to get your calculator to give you the
right answer.
Order of Operations
There are four principal arithmetic operations: addition, subtraction, multiplication, and divi-
sion. When an equation involves multiplication and/or division as well as addition and/or sub-
traction, you need to do the multiplication and division operations before doing the addition
and subtraction operations. For example, what does 12 plus 3 times 10 equal?
x = 12 + 3 ⋅ 10
Since this equation mixes addition and multiplication, we need to do the multiplication first:
x = 12 + 30
x = 42
Now, you evaluate this expression:
12
x= −3⋅2+4
4
Answer: x = 1
Parentheses and division bars (fraction bars) are both symbols of enclosure. Whenever there
are symbols of enclosure, execute any arithmetic operations inside the symbol of enclosure first.
The general rule is to work from the innermost symbol of enclosure to the outermost, multi-
plying and dividing, then adding and subtracting. Consider this example, which involves both
kinds of symbols of enclosure and order of operations.
12 + 3 ⋅ (4 + 2)
x=
3.5
To solve this example, we need to first evaluate inside the parentheses, then multiply and
finally add:
12 + 3 ⋅ (4 + 2) 12 + 3 ⋅ (6) 30
x= = =
3⋅5 3⋅5 3⋅5
, A Review of Some Basic Mathematical Skills 3
Here we come to a very common error. Many students are tempted to divide 30 by 3 first,
and then multiply by 5. But the division bar is a symbol of enclosure, so multiply 3 times 5 first
and then divide 30 by 15.
30
x= =2
15
Let’s evaluate this expression:
10
+ (6 + 4) ⋅ (5 − 3)
x= 5
8
+9
4
Answer: x = 2
Negative numbers are common sources of confusion. Just remember that multiplying (or
dividing) two positive numbers or two negative numbers gives a positive number. Multiplication
or division involving a positive number and a negative number gives a negative number.
Multiplying a number by negative 1 changes the sign of that number. Likewise, a negative num-
ber can be expressed as a positive number times −1. Subtracting a negative number is the same
as adding a positive number.
Some examples are shown in the following:
4 ⋅ (−3) = −12
(−4) ⋅ (−3) = +12
−4x − (−3x) = −4x + 3x = −x
Algebra: Solving Equations for an Unknown Quantity
Addition and subtraction are inverse operators of each other, as are multiplication and division.
Usually, a problem will involve solving an equation for some variable. You want to convert the
equation into the form x = some number (or some expression). Therefore, when you want to iso-
late a variable (i.e., “solve the equation”), you need to undo any operations that are tying up the
variable by applying the inverse operation. That is, if the variable is multiplied by a number, you
need to divide both sides of the equation by that number. If a variable is divided by some number,
you need to multiply both sides of the equation by that number. This eliminates that number
from the variable by converting it into a 1. Since 1 is the “multiplicative identity element,” you
can ignore 1’s that are involved in multiplication or division. If some number is added to the
variable, you need to subtract that number from both sides of the equation. If some number is
subtracted from the variable, you need to add that number to both sides of the equation. This
eliminates that number from the variable by converting it into a zero. Since zero is the “additive
identity element,” you can ignore zeros that are involved in addition or subtraction.
The critical thing to remember, though, is that whatever you do to one side of the equation,
you have to do to the other side. Think of it as being fair. If you add a number to one side, you
1
Measurement
Why This Matters: A Nurse Anesthetist’s View
Why do we care, in this day of technology, if a certified registered nurse anesthetist can do math?
Well, even today technology may fail, leaving us to use basic nursing skills of patient assessment
and math to provide the best care for our patients. If we have basic math skills, we cannot only
determine a drug dose or measure a base deficit, but we can also recognize whether the answer
we obtain is reasonable. We may recognize that the answer “just doesn’t seem right” when the
calculated dose is unexpectedly lower or higher than we anticipate. This may lead us to prevent
a potential erroneous drug dose or calculate a fluid deficit with precision. A common example
involves rechecking your calculations if you have to crack a second vial of medication to achieve
the desired dose. Most medications are packaged in common doses (i.e., spinal bupivacaine
comes as a 7.5% solution in a 2-mL vial). If your original calculation requires you to open a
second vial to draw up 22 mg (2.9 mL) of bupivacaine, it is worthwhile to double check the
dosage. If your patient is a basketball player who is 7 ft tall, it may be correct. If your patient is
5 ft, 4 in. tall, the original dosage may be catastrophic.
A REVIEW OF SOME BASIC MATHEMATICAL SKILLS
Chemistry and physics are both very logical sciences, but both depend on math to translate
concepts into application. Unfortunately, many students who struggle in these disciplines have
more trouble with the mathematics than with the scientific concepts. Therefore, let us begin
our exploration of chemistry and physics with a review of some basic math skills and concepts
that you will use throughout this course. While this chapter reviews basic math skills, it cannot
replace a basic understanding of college-level algebra.
You will need a calculator for this course. Some of you may have fancy graphing cal-
culators, but the capacity of these instruments far exceeds your mathematical needs for
1
,2 Chapter 1 Measurement
this course. Any calculator that is labeled as a “scientific calculator” will more than suffice.
You should be able to buy an adequate calculator for less than $20. When dealing with a
math problem, don’t reach for your calculator first. Whether or not the calculator gives you
the correct answer depends on your ability to give it the correct information with which to
work. Think about what operations you need to do first. See how far you can get with just
a paper and pencil. You don’t have to do arithmetic computations, such as long division,
by hand, but you should be able to perform the mathematical manipulations (e.g., solve the
equation for x) without a calculator. In fact, it is typically best to solve the equation first for
the variable of interest before plugging in numbers. As you become more adept at thinking
your way through math problems, you are more likely to get your calculator to give you the
right answer.
Order of Operations
There are four principal arithmetic operations: addition, subtraction, multiplication, and divi-
sion. When an equation involves multiplication and/or division as well as addition and/or sub-
traction, you need to do the multiplication and division operations before doing the addition
and subtraction operations. For example, what does 12 plus 3 times 10 equal?
x = 12 + 3 ⋅ 10
Since this equation mixes addition and multiplication, we need to do the multiplication first:
x = 12 + 30
x = 42
Now, you evaluate this expression:
12
x= −3⋅2+4
4
Answer: x = 1
Parentheses and division bars (fraction bars) are both symbols of enclosure. Whenever there
are symbols of enclosure, execute any arithmetic operations inside the symbol of enclosure first.
The general rule is to work from the innermost symbol of enclosure to the outermost, multi-
plying and dividing, then adding and subtracting. Consider this example, which involves both
kinds of symbols of enclosure and order of operations.
12 + 3 ⋅ (4 + 2)
x=
3.5
To solve this example, we need to first evaluate inside the parentheses, then multiply and
finally add:
12 + 3 ⋅ (4 + 2) 12 + 3 ⋅ (6) 30
x= = =
3⋅5 3⋅5 3⋅5
, A Review of Some Basic Mathematical Skills 3
Here we come to a very common error. Many students are tempted to divide 30 by 3 first,
and then multiply by 5. But the division bar is a symbol of enclosure, so multiply 3 times 5 first
and then divide 30 by 15.
30
x= =2
15
Let’s evaluate this expression:
10
+ (6 + 4) ⋅ (5 − 3)
x= 5
8
+9
4
Answer: x = 2
Negative numbers are common sources of confusion. Just remember that multiplying (or
dividing) two positive numbers or two negative numbers gives a positive number. Multiplication
or division involving a positive number and a negative number gives a negative number.
Multiplying a number by negative 1 changes the sign of that number. Likewise, a negative num-
ber can be expressed as a positive number times −1. Subtracting a negative number is the same
as adding a positive number.
Some examples are shown in the following:
4 ⋅ (−3) = −12
(−4) ⋅ (−3) = +12
−4x − (−3x) = −4x + 3x = −x
Algebra: Solving Equations for an Unknown Quantity
Addition and subtraction are inverse operators of each other, as are multiplication and division.
Usually, a problem will involve solving an equation for some variable. You want to convert the
equation into the form x = some number (or some expression). Therefore, when you want to iso-
late a variable (i.e., “solve the equation”), you need to undo any operations that are tying up the
variable by applying the inverse operation. That is, if the variable is multiplied by a number, you
need to divide both sides of the equation by that number. If a variable is divided by some number,
you need to multiply both sides of the equation by that number. This eliminates that number
from the variable by converting it into a 1. Since 1 is the “multiplicative identity element,” you
can ignore 1’s that are involved in multiplication or division. If some number is added to the
variable, you need to subtract that number from both sides of the equation. If some number is
subtracted from the variable, you need to add that number to both sides of the equation. This
eliminates that number from the variable by converting it into a zero. Since zero is the “additive
identity element,” you can ignore zeros that are involved in addition or subtraction.
The critical thing to remember, though, is that whatever you do to one side of the equation,
you have to do to the other side. Think of it as being fair. If you add a number to one side, you
