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Closeness: arithmetic and geometric progressions
1. Closeness: arithmetic and geometric progressions
Is 50 closer to 16 or 100?
What does it mean for number x to be closer to number y than number x is to number z?
2. Arithmetic mean
The
arithmetic mean of nonnegative numbers
u and
w,
u <
w, is the number
v such that the difference between
u and
v is the same as the difference between
v between and
w.
The arithmetic distance between u and v is v-u.
The arithmetic distance between v and w is w-v.
Therefore:
v - u = w - v
v - u + v = w - v + v
2 v - u = w
2 v - u + u = w + u
2 v = w + u
2 v / 2 = (w + u) / 2
v = (w + u) / 2
The arithmetic mean is the same as the mean average.
3. Geometric mean
The
geometric mean of positive numbers
u and
w,
u <
w, is the number
v such that the ratio of
v to
u is the same as the ratio of
w to
v.
The geometric distance between u and v is v/u.
The geometric distance between v and w is w/v.
Therefore:
v / u = w / v
v * v / u = v * w / v
v * v / u = w
u * v * v / u = w * u
v * v = w * u
v = SQRT(w * u)
4. Closeness
Is 50 closer to 16 or 100?
What is the arithmetic mean?
What is the geometric mean?
The arithmetic mean of 16 and 100 is (16+100)/2 = 58.
The geometric mean of 16 and 100 is SQRT(16*100) = 40.
What is the conclusion?
The conclusion is that
50 is closer to 16 arithmetically, but
50 is closer to 100 geometrically.
Thus, closeness depends on how you define and measure closeness.
Is 50 closer to 16 or 100? Explain.
5. Arithmetic series
In an
arithmetic series (or list) each term in the series is different from the preceding term arithmetically (i.e., using addition or subtraction).
Arithmetic series examples.
1, 2, 3, 4, 5, 6, ...
2, 4, 6, 8, 10, 12, ...
1, 3, 5, 7, 9, 11, ...
number of the key on a piano keyboard (MIDI (Musical Instrument Digital Interface) sequence)
Give an example of an arithmetic series.
6. Geometric series
In an
geometric series (or list) each term in the series is different from the preceding term geometrically (i.e., using multiplication or division).
1, 2, 4, 8, 16, 32, 64, 128, ...
1, 3, 9, 81, 243, ...
frequency of half tones in music (go up one octave is the tone at twice the frequency)
7. Population growth
population growth over time
8. Savings growth
savings growth with interest (7.0% is a geometric factor of 1.07)
9. National debt
national debt
social security
10. Moore's Law
power of microprocessors (Moore's Law: twice as powerful every 18 months)
11. Musical tones
frequency of half tones in music (go up one octave is the tone at twice the frequency)
Give an example of a geometric series.
12. End of page
