Changing Units (Digression) The measure
or unit of slope, speed and velocity in the last example is given
by the ratio [ km/min] or km per
minute, or km per one sixtieth of an hour. But other units
(or ratio of units) are possible. The question becomes: what units
do you like for the expression of these quantities. It is possible
to change units of time and distance.
Multiplying by the number 1 does not affect the
value of an expression, but the number 1 can be written in
many ways. Some, not all, are helpful.
These different ways can sometimes help in changing the units
used for the expression of a quantity. A few slope or velocity
based examples follow. Note that
[60min/1hr] = 1 implies the
following
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4
3 |
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km
min |
= - |
4
3 |
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km
min |
×1 |
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- |
1
2 |
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km
min |
× |
60min
hr |
= -30 |
km
hr |
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| - |
3
2 |
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km
min |
× |
60min
hr |
= -90 |
km
hr |
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In summary, multiplying by 1 = [60min/hr] does not change the speed or velocity m, but it does help
to change the units from minutes to hours. Note to do the reverse
change, multiply by
instead.
Note that care must be
taken in selecting the ratio of units whose value is 1.
A faulty choice will introduce more units instead of
permitting some to cancel.
Remark. An alternate method is to substitute
1 min = [1/60] hr into an expression. For
example
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8
3 |
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60
1 |
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km
hr |
= 160 |
km
hr |
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as before. Substitution replaces one unit by an expression
for it in terms of another. With this method, there is
little or no hazard of introducing units that don't cancel,
but the algebra or arithmetic requires a little more thought.
The choice of unit conversion method depends on where you
would like to do the work or reasoning.
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Why Slopes
and
More Math
understanding & explaining
Reason and Math
Volume 3
Printed in Canada
ISBN 0-9697564-3-7
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[ Back ] [ Home ] [ Next ]
Content Guide
Foreword
Chapter Descriptions
1. Introduction
Preview -why slopes in 1983
2. Second Preview Begins
2 Skier in Motion (V)
2 The Skier (V)
2. Position Dependent (V)
3 Slope & Extrema (V)
4 Single Factor Analysis (V)
4 Two Factor (V)
4 More Factors (V)
4 With Divisors (V)
5 Maxima & Minima Tests
6 Jumps & Discontinuities
8 Review (optional)
9 On Calculus Studies
11 Slope of Slope
13 Acceleration
14 Limits & Error Control (V)
14 Limit of a Fn.
14. Limited Error Control
14 Signif. Digits
14 Cauchy Limits
14 Sequence Limits
14 Decimal Arith.
15 What is Slope (V)
15 Slope Calculation (V)
15 Slope, a Limit
15 Tangent Lines
15 Linear Approx.,
15 Limits via Algebra (V)
15 Recap.
PS.Chain Rule for Polys
PS Chain Rule- General (V) -
PS More Chain Rule (V)
PS - Sign Analysis (V)
16 What is Velocity
17 What is Area
18 Integration
18 Area Calculation
18 Fn DefN, 6 Ways
19 Logs & Powers
19 Natural Log.
19 Exponential Fn.
20 What's Next
21 Add Vectors
22 Complex #'s
23 Complex #'s
23 Trig Identity
23 Proofs of.
24 Complex Logs etc
Units in Calculations:
7 Velocity
7 Varying Velocity Example
7. Velocity Calculation
7 Changing Units
7 Same Velocity Motions
10 Slopes without Units.
10 Units & Slopes
10 Units in Cost vs. Quantity
10 How Units Appear
10 Unit Elimination
10 Partial Elimination
10 Interest & Units
12 More on Units
Enriched material: The Appendices of Volume
3 are located in the Real
Analysis Area.
Pigeon Hole Principle
Constant Difference Thm
Continuous Functions
Rational Functions
Mean Value Theorem
One Side Range Theorem
Range On One Side
Theorem
Integration
& Lipschitz
Continuity
These appendices continue the
decimal viewpoint of limits, error
control and continuity begun
in Chapter 14. The One Sided
Range Theorem is a postscript,
not in printed version.
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