Download Mathematical Fundamentals

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Mathematical Fundamentals
Purpose: To familiarize students with the mathematical principals and techniques to be
utilized throughout the course.
Goals: Upon successful completion of this lab assignment, students will be able to:
1. Read/write numbers in scientific notation.
2. Perform calculations using numbers expressed in scientific notation.
3. Use fundamentals of algebra to solve equations.
4. Use graphing techniques as a means of interpreting data.
Scientific Notation
An understanding of nature is built from measuring a wide variety of quantities. The
quantities involved in astronomy pose a challenge to students because the values range
from unfathomably large to infinitesimally small. For example, the distance to the star
closest to our solar system, Proxima Centauri, is nearly 25,000,000,000,000 miles away.
Such a large number can interfere with comprehension if it requires a reader to pause so
he or she can figure out whether the number is referring to 25 million, 25 billion, or 25
trillion. To make reading move more smoothly, large numbers can be rewritten in
different ways to make it easier to pick out the. One such technique is to write the
distance to Proxima Centauri as, “25 trillion miles,” thereby eliminating unnecessary
placeholders.
A mathematical approach to shortening numbers is more useful for an introductory
astronomy course. Unnecessary placeholders are eliminated mathematically by
converting the placeholders to the proper power of ten. To see how this is accomplished,
the distance to Proxima Centauri can be broken down as the multiplication of two
numbers:
25,000,000,000,000 miles = 2.5 x 10,000,000,000,000 miles
We can then make use of the fact that the number 10,000,000,000,000 (10 trillion)
can be rewritten as the number 10, raised to the 13th power (1013), and substituting it in
!
the equality
above:
25,000,000,000,000 miles = 2.5 x 1013 miles
Values expressed in scientific notation are those that are written in the form as seen
on the right hand side of the above equality. Essentially, scientific notation is the
! hand method for writing numbers that have many decimal places,
mathematical short
whether they are large or small. The main purpose for using scientific notation is to
present numbers in a more readable form. Writing numbers in scientific notation
involves separating the number into a mantissa and a power of ten. The mantissa is
found by moving the decimal point of a given number so that there will be only one non-
zero digit to the left of it, and then eliminating all unnecessary decimal place holders.
Once the mantissa is determined, it will need to be multiplied by the appropriate power of
ten. The appropriate power of ten amounts to the positive number of places the decimal
point moved to the left or the negative number of places the decimal point moved to the
right.
Example 1
Rewrite the number 32,400 in scientific notation.
Step 1: The mantissa is determined by moving the decimal point from the right of
the zero to the right of the number three. The number we get is: 3.2400.
Step 2: Eliminating unnecessary placeholders so that only three digits remain, we
end up with a mantissa of: 3.24.
Step 3: Multiply the mantissa by the appropriate power of ten by counting how
many placeholders the decimal point was moved in order to obtain the mantissa. In this
case, the power of ten is: 4.
The number 34,200 is rewritten as 3.24 x 104.
Example 2
Rewrite the number 0.0739 in scientific notation.
Step 1: We obtain the mantissa by moving the decimal point to the right of the
number seven, and then eliminating the zero. The mantissa we get is: 7.39.
Step 2: The appropriate power of ten in this case is equal to the negative number of
placeholders the decimal point was moved in order to obtain the mantissa. In this case,
the power of ten is: -2.
The number 0.00739 is rewritten as 7.39 x 10-2.
Exercise 1: Write each of the following numbers in scientific notation. When writing
each response, keep only three (3) digits in the mantissa and eliminate all other
placeholders. Be sure to following proper rounding procedures.
1. 2,370 =
4. 12,094,000,000 =
2. 0.00957 =
5. 0.000000004286 =
3. -459,090 =
6. -0.0005738 =
Calculations using numbers written in Scientific Notation
In addition to making large numbers more manageable, scientific notation is useful
in simplifying calculations, especially in cases where a scientific calculator is not
available. By making use of the distributive property of mathematics, a person can
compute a value by performing the calculation on the mantissa separately from the
powers of ten. Then the resulting mantissa and power of ten can be combined when
reporting the final answer. Below are some examples of this simplified method.
Examples
Multiplication
(1.23 x 10 2 ) " (2.46 x 10 3 ) = (1.23 x 2.46) " (10 2 x 10 3 ) = 3.03 x 10 5
Division
!
" 8.48 % " 10 9 %
(8.48 x 10 9 )
5
= $
' x $ 4 ' = 4.02 x 10
4
# 2.11 & # 10 &
(2.11 x 10 )
Raising to a Power
!
(2.92 x 10 3 ) 2 = (2.92) 2 x (10 3 ) 2 = 8.53 x 10 6
Taking a Square Root
!
1.44 x 1012 =
1.44 x 1012 = 1.20 x 10 6
Exercise 2: Perform the following calculations. Write each of your responses in scientific
notation. When writing each response, keep only three (3) digits in the mantissa and
eliminate all other placeholders. Be sure to following proper rounding procedures.
!
1. (4.32 x 10 4 ) " (1.25 x 10 3 ) =
!
!
!
2.
(3.38 x 10 7 )
=
(4.60 x 10 2 )
3.
(7.81 x 10 5 )
=
(6.20 x 10 -7 )
4. (4.35 x 1012 ) " (6.62 x 10 -5 ) =
!
5. (7.71 x 1015 ) 2 =
!
6.
!
7. (6.39 x 10 4 ) 3 =
!
8. (5.08 x 10 ) "
!
Algebra Fundamentals
2.25 x 10 8 =
4 2
1.73 x 1010
=
8.65 x 1016
A major goal of scientific research is to find how various physical properties are
related to one another. Once discovered these relationships are expressed as
mathematical formulas, which are then verified through decades, and even centuries, of
rigorous testing. Such relationships are referred to as laws of nature and we will be using
them extensively throughout the course.
The most basic use of a formula is to determine the value of one physical quantity
based on the values of others through the process of mathematical substitution. For
example, when a golf ball is hit off a tee, Newton’s 2nd law of motion (F = ma) can be
used to determine how much force (F) was applied to get the golf ball moving using.
In order to successfully use a formula to calculate a desired physical quantity, you
must follow these steps:
(1) Identify the correct formula to be used.
(2) Substitute the given values for the proper variables.
(3) Perform the calculation properly.
Exercise 3: Choose the proper formula from the list provided and use the values given in
each example to determine the value of the desired variable. Show your work by
rewriting each function with the values substituted for the appropriate variable before
recording your final result. In these examples, the variables have no physical
significance.
Formulas
f = h - g(j)
q =
r - t
u +v
" =
# - $ %µ
&
m =
n 2 + op
l2
9. Solve for q if r = 64, t = -16, u = - 11, and v = 19.
!
!
!
!
10. Solve for f if g = 2.14 x 103, h = 9.23 x 105, and j = 3.47 x 102.
11. Solve for m if l = 4, n = 5, o = 0.25, p = 44.
12. Solve for α if β = 7, µ = 32, λ = 1.5, σ = 30.
Even though scientific formulas are written in a standard form, it is possible to
determine the value of any of the physical properties within the formula by changing
around the equation. To successfully manipulate equations, basic rules of algebra must
be followed to get a single unknown in an equation by itself. This is accomplished by
applying inverse operations to both sides of an algebraic equation. Some useful inverse
operations are listed below:
Addition is the inverse operation of Subtraction (and vice versa)
x +4 = 7
x + 4 (- 4) = 7 (- 4)
x = 3
!
x - 8 = 11
x - 8 (+ 8) = 7 (+ 8)
x = 19
!
Multiplication is the inverse operation of Division (and vice versa)
x
= 14
3
x
" (3) = 14 " (3)
3
x = 42
5x = 125
5x
125
=
5
5
x = 25
Raising to a power is the inverse operation of taking a root (and vice versa)
!
!
3
x = 7
x 2 = 36
x2 =
x = 6
( x)
3
36
3
= ( 7)
3
x = 343
In many of the formulas that we will use in this course, there will be a combination
of manipulations
that will need to be applied. In these instances it will be important to
!
! to one side of an equation, the same
remember the rule that whatever operation you apply
exact operation must be applied to the other side.
x2
+ 15 = 63
3
x2
= 48
3
x 2 = 144
x = 12
Step 1: Subtract 15 from both sides.
Step 2 : Multiply both sides by 3.
Step 3 : Take the square root of both sides.
Exercise 4: Solve for y in each of
! the following examples by substituting all known
!variables and calculating a numerical
value. Show your work by writing down all
intermediate steps.
13. 19 = 3y + 5
!
14.
!
y
- 7p = m, where m = -3 and p = 2.
4
15. 3y 2 - 5x = 12, where x = 144.
!
16.
y - 5a
= c, where a = 17, b = 9, and c = -4.
b
!
17.
" #$
= 12 # % + & , where α = 4.5, δ = 7.2, θ = 6.1, and π = 3.14.
y
!
Essentials of Graphing
Determining the laws of nature first requires that experiments be conducted to test
the validity of any hypotheses that are proposed. During an experiment scientists will
look for any changes in the value of a particular quantity (called the dependent variable)
while another physical quantity (referred to as the independent variable) is adjusted.
Once the measurements are made, scientists will then look for any relationships that may
exist. The simplest method for determining any such relationships is by plotting the
collected information on a graph.
Basic graphing techniques involve being able to plot data points accurately on a
graph and then determine any mathematical relationship that may exist. Below is a
simple exercise that will illustrate how using a coordinate system can reveal information
about a physical process.
We will be using the standard Cartesian coordinate system that is taught in high
school algebra. This system has the following characteristics:
(1) A reference point called the origin (labeled as, O)
(2) One horizontal axis (labeled, x) and one vertical axis (labeled, y) that run
perpendicular to each other and cross at the origin.
(3) Coordinates that are given in the form of (x ,y) such that the ‘x’ coordinate
dictates how far along the horizontal axis a data point is from the origin, and the
‘y’ coordinate dictates how far along the vertical axis the data point is from the
origin.
18. Table 1.1 contains coordinates for six (6) random points. Plot the points onto Graph
1.1 by placing a small, but legible, point at the proper coordinates. Be sure to label
each point clearly.
Table 1.1
Graph 1.1
Point
x
y
A
B
C
D
E
F
-5
-1.5
6.5
-7.5
2
9
-4.5
3
0.5
6
8.5
-9.5
Graphing Measured Data
19. Table 1.2 lists the results of a hypothetical experiment where the distance traveled by
a car on a road trip from NYC to Washington D.C. was noted every 45 minutes (or
0.75 hours). Plot the data points from the experiment onto Graph 1.2.
Table 1.2
Time Since Beginning of Trip (t)
Distance Traveled (d)
0.75 hours
1.50 hours
2.25 hours
3.00 hours
3.75 hours
4.50 hours
40.0 miles
110 miles
140 miles
176 miles
245 miles
280 miles
20. Hopefully it is apparent that there is an orderly relationship between the distance
traveled and the amount of time traveled. The relationship gives the rate of change of
distance with time and is referred to as the speed of the car. The speed of the car can
be determined from the slope of the line that best fits the data points.
(1) Using a straightedge (like a ruler), draw a single line that best fits the data
points. This means drawing a line the best represents the relationship
between the data points, rather than connecting the data points together.
d (miles)
Graph 1.2
t (hours)
(2) Pick two arbitrary points that fall on the line you just drew and place a
small (x) on each. These points should be far apart from each other and
not include any of the data points from the table that you have already
plotted. Write the coordinates of each point below:
Arbitrary point 1: t1 = __________
d1 = __________
Arbitrary point 2: t2 = __________
d2 = __________
(3) The slope of the line will define the rate of change of distance with time.
The slope (m) of any straight line is given as the ratio of the change in
dependent variable (in this case, d) to the change in the independent
variable (in this case, t). Use the formula below to calculate the slope of
your line. Show the steps in your calculation.
m =
"d
d - d1
= 2
"t
t 2 - t1
!
m = _______________
(4) What property of the car does the slope of the line indicate? (Hint: use the
units of measurement of the slopes as a guide)