View
217
Download
1
Category
Preview:
Citation preview
1
Cultural ConnectionSerfs, Lords, and Popes
Student led discussion.
The European Middle Ages – 476 –1492.
2
8 – European Mathematics
The student will learn about
European mathematics from the dark ages through the renaissance.
3
§8-1 The Dark Ages
Student Discussion.
4
§8-2 Period of Transmission
Student Discussion.
5
§8-3 Fibonacci & 13th Century
Student Discussion.
6
§8-3 Fibonacci & 13th Century
More Later.
7
§8-4 Fourteenth Century
Student Discussion.
8
§8-5 Fifteenth Century
Student Discussion.
9
§8-6 Early Arithmetics
Student Discussion.
10
§8-7 Algebraic Symbolism
Student Discussion.
11
§8-7 “The Beast”
DCLXVI Roman number of the Beast.666.0000 High-precision beast.0.666 Millibeast.1010011010 Binary beast.666 + 666i Complex Beast.1-666 Area code of the Beast.00666 Zip code of the Beast1-900-666-0666 Live Beasts! One-on-one pacts!
Only $6.66 a minute!
12
§8-7 “The Beast”
$665.95 Retail price of the Beast.
Phillips 666 Gasoline of the Beast.Route 666 Way of the Beast.666k Retirement plan of the Beast.6.66% Beastly interest rate.
lbraden@sps.edu
13
§8–8 Cubic & Quartic Equations
Student Discussion.
14
§8–9 François Viète
Student Discussion.
15
§8–10 Other Mathematicians of the Sixteenth Century
Student Discussion.
16
Fibonacci 11, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 510, . . .
f 1 = f 2 = 1 and f n = f n – 1 + f n - 2
1. f 1 + f 2 + f 3 + f 4 + . . . + f n = f n + 2 - 1
2. f 12 + f 2
2 + f 32 + f 4
2 + . . . + f n2 = f n · f n + 1
3. f n2 = f n + 1 f n - 1 + ( - 1 )n – 1 for n > 1.
4. f m + n = f m - 1 · f n + f m · f n + 1
5. 5 · f n2 + 4 · ( –1)n is a perfect square.
17
Fibonacci 21, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 510, . . .
f 1 = f 2 = 1 and f n = f n – 1 + f n - 2
6. f 50 = 12,586,269,025
7. f 1+ f 3 + f 5 + . . . + f 2n - 1 = f 2n
8. f 2+ f 4 + f 6 + . . . + f 2n = f 2n + 1 - 1
9. The sum of any ten consecutive Fibonacci numbers is divisible by 11.
18
Fibonacci 31, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 510, . . .
10. f 2n2 = f 2n + 1 f 2n – 1 - 1
Area is 64 Area is 65
3
3
3
5
5
5
5
3
8
8
3
35
5 5
5
8
813
? ? ? ? ? ? ? ? ?
19
Fibonacci 41, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 510, . . .
11. ...618033989.12
15
f
flim
n
1n
n
...618033989.02
15
f
flim
1n
n
n
20
Fibonacci 51
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
. . .
11
23
5813
21
Fibonacci 61, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, . . .
f i 1 1 2 3 5 8 13 21 34 . . . .
f i2 1 1 4 9 25 64 169 441 1156 . . . .
Sum adjacent 2 5 13 34 89 233 610 1597 . . . .
f n + 1 - fn 3 8 21 55 144 377 987 . . . .
f n + 1 - fn 5 13 34 89 233 610 . . . .
f n + 1 - fn 8 21 55 144 377 . . . .
Etc.
22
Fibonacci 71, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 510, . . .
Given any four consecutive Fibonacci numbers -
n = 1 n = 2 n = 3 n =4 . . .
fn,fn+1,fn+2,fn+3 1, 1, 2, 3 1, 2, 3, 5 2, 3, 5, 8 3, 5, 8, 13 . . .
fn · fn+3 = a 3 5 16 39 . . .
2fn+1 · fn+2 = b 4 12 30 80 . . .
f 2n +3 = c 5 13 34 89 . . .
K ABC
fn · fn+1 · fn+2 · fn+3 6 30 240 1560 . . .
23
Assignment
FALL BREAK
Paper presentations from chapters 5 and 6.
Recommended