Math Help
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Scientific Notation
Many of the quantities that you will encounter in this course are astronomical, i.e., huge.
For example, the distance from the Earth to the moon is 238,857 miles; the distance from
the Sun to the Earth is 92.9 million (92,900,000) miles; and the distance light travels in one
year is 5,878,630,000,000 miles. It’s annoying and risky to deal with numbers this large. So,
we use powers of 10 to rewrite them. Note that
100 = 1
101 = 10
102 = 10 × 10 = 100
103 = 10 × 10 × 10 = 1000
104 = 10 × 10 × 10 × 10 = 10000
105 = 10 × 10 × 10 × 10 × 10 = 100000
and so on. In scientific notation, the number of zeros corresponds to the exponent of 10. If
we have 2.0 × 104 , the exponent is 4, so place 4 zeros after the 2.0, and we have 20,000. Or
you can think of the exponent as a placeholder. If we have 4.0 × 103 , just take the decimal
point and move it 3 places over, 4.^ .^ .^ ., or 4000.
For the larger values above, we have
Moon-Earth distance = 238, 857 mi ≈ 2.39 × 105 mi
Sun-Earth distance = 92, 900, 000 mi = 9.29 × 107 mi
Light-year = 5, 878, 630, 000, 000 mi ≈ 5.89 × 1012 mi.
On a scientific and graphing calculator, the EE button will enter in the equivalent of “×10n ”,
n being the exponent. For example, if you enter 4 EE 6 = , this is 4.0 × 106 and the
displayed value will be 4,000,000. You can also type 4 × 1 0 y x 6 = 4,000,000.
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The following will result in incorrect values:
4 × EE 6 = 24
4 × 1 0 EE 6 = 40,000,000.
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Laws of Algebra
For any real number a, (we use only real numbers in this course),
a×1=1×a=a
(1)
a×0=0×a=0
a
= 1.
a
(2)
For example, 2 × 1 = 1 × 2 = 2. Also,
3
4
4
(3)
= 1.
Solving Equations
Consider the equation
6x = 24.
To solve this equation, we have to find out what x is. So, we must isolate the x on one
side of the “=.” We can divide (or multiply) both sides of the equation by the same factor
without wrecking the equality. Some factors here are 6, x, and 24. If we divide both sides by
6, we have
6x
24
=
6
6
x = 4.
More examples of solving for x are shown below. Using addition and subtraction:
x + 6 = 24
x + 6 − 6 = 24 − 6
x + 0 = 18
x = 18.
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Using multiplication and division:
x
=8
x 7
(7) = 8 × 7
7
7x
= 56
7
x = 56.
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Units
Converting units is an essential skill for this course! We will convert miles to meters, meters
to kilometers, light-years to parsecs, etc. Units should be treated as quatities that obey the
laws of algebra, like x in an algebra equation. Just as 22 =1, units can cancel each other,
cm
=1, no units.
or simplify to 1. So, 22 cm
To convert units from one system to another, we’ll need a conversion factor. This factor
will come from an equality, such as 1 in = 2.54 cm. Here the conversion factor is either
1 in
2.54 cm
2.54 cm = 1 or 1 in = 1. The fact that both are equal to 1 is what allows us to change
units, since anything times 1 equals itself (refer to law 2.1)! The factor we use depends on
the conversion needed.
For example, to convert 15 cm to inches,
15 cm =
×
15 cm
15
in
2.54
= 5.9 in
=
3
1 in
cm
2.54 To convert 8 in to centimeters,
8 in = 8 in ×
2.54 cm
1 in
= 8 × 2.54 cm
= 20.32 cm
If you have any questions or would like additional help, please contact your lab instructor.
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