TEDGlobal 2013
Arthur Benjamin: The magic of Fibonacci numbers
Filmed:
Readability: 3.2
7,057,274 views
Math is logical, functional and just ... awesome. Mathemagician Arthur Benjamin explores hidden properties of that weird and wonderful set of numbers, the Fibonacci series. (And reminds you that mathematics can be inspiring, too!)
Arthur Benjamin - Mathemagician
Using daring displays of algorithmic trickery, lightning calculator and number wizard Arthur Benjamin mesmerizes audiences with mathematical mystery and beauty. Full bio
Using daring displays of algorithmic trickery, lightning calculator and number wizard Arthur Benjamin mesmerizes audiences with mathematical mystery and beauty. Full bio
Double-click the English transcript below to play the video.
00:12
So why do we learn mathematics?
0
613
3039
00:15
Essentially, for three reasons:
1
3652
2548
00:18
calculation,
2
6200
1628
00:19
application,
3
7828
1900
00:21
and last, and unfortunately least
4
9728
2687
00:24
in terms of the time we give it,
5
12415
2105
00:26
inspiration.
6
14520
1922
00:28
Mathematics is the science of patterns,
7
16442
2272
00:30
and we study it to learn how to think logically,
8
18714
3358
00:34
critically and creatively,
9
22072
2527
00:36
but too much of the mathematics
that we learn in school
that we learn in school
10
24599
2926
00:39
is not effectively motivated,
11
27525
2319
00:41
and when our students ask,
12
29844
1425
00:43
"Why are we learning this?"
13
31269
1675
00:44
then they often hear that they'll need it
14
32944
1961
00:46
in an upcoming math class or on a future test.
15
34905
3265
00:50
But wouldn't it be great
16
38170
1802
00:51
if every once in a while we did mathematics
17
39972
2518
00:54
simply because it was fun or beautiful
18
42490
2949
00:57
or because it excited the mind?
19
45439
2090
00:59
Now, I know many people have not
20
47529
1722
01:01
had the opportunity to see how this can happen,
21
49251
2319
01:03
so let me give you a quick example
22
51570
1829
01:05
with my favorite collection of numbers,
23
53399
2341
01:07
the Fibonacci numbers. (Applause)
24
55740
2728
01:10
Yeah! I already have Fibonacci fans here.
25
58468
2052
01:12
That's great.
26
60520
1316
01:13
Now these numbers can be appreciated
27
61836
2116
01:15
in many different ways.
28
63952
1878
01:17
From the standpoint of calculation,
29
65830
2709
01:20
they're as easy to understand
30
68539
1677
01:22
as one plus one, which is two.
31
70216
2554
01:24
Then one plus two is three,
32
72770
2003
01:26
two plus three is five, three plus five is eight,
33
74773
3014
01:29
and so on.
34
77787
1525
01:31
Indeed, the person we call Fibonacci
35
79312
2177
01:33
was actually named Leonardo of Pisa,
36
81489
3180
01:36
and these numbers appear in his book "Liber Abaci,"
37
84669
3053
01:39
which taught the Western world
38
87722
1650
01:41
the methods of arithmetic that we use today.
39
89372
2827
01:44
In terms of applications,
40
92199
1721
01:45
Fibonacci numbers appear in nature
41
93920
2183
01:48
surprisingly often.
42
96103
1857
01:49
The number of petals on a flower
43
97960
1740
01:51
is typically a Fibonacci number,
44
99700
1862
01:53
or the number of spirals on a sunflower
45
101562
2770
01:56
or a pineapple
46
104332
1411
01:57
tends to be a Fibonacci number as well.
47
105743
2394
02:00
In fact, there are many more
applications of Fibonacci numbers,
applications of Fibonacci numbers,
48
108137
3503
02:03
but what I find most inspirational about them
49
111640
2560
02:06
are the beautiful number patterns they display.
50
114200
2734
02:08
Let me show you one of my favorites.
51
116934
2194
02:11
Suppose you like to square numbers,
52
119128
2221
02:13
and frankly, who doesn't? (Laughter)
53
121349
2675
02:16
Let's look at the squares
54
124040
2240
02:18
of the first few Fibonacci numbers.
55
126280
1851
02:20
So one squared is one,
56
128131
2030
02:22
two squared is four, three squared is nine,
57
130161
2317
02:24
five squared is 25, and so on.
58
132478
3173
02:27
Now, it's no surprise
59
135651
1901
02:29
that when you add consecutive Fibonacci numbers,
60
137552
2828
02:32
you get the next Fibonacci number. Right?
61
140380
2032
02:34
That's how they're created.
62
142412
1395
02:35
But you wouldn't expect anything special
63
143807
1773
02:37
to happen when you add the squares together.
64
145580
3076
02:40
But check this out.
65
148656
1346
02:42
One plus one gives us two,
66
150002
2001
02:44
and one plus four gives us five.
67
152003
2762
02:46
And four plus nine is 13,
68
154765
2195
02:48
nine plus 25 is 34,
69
156960
3213
02:52
and yes, the pattern continues.
70
160173
2659
02:54
In fact, here's another one.
71
162832
1621
02:56
Suppose you wanted to look at
72
164453
1844
02:58
adding the squares of
the first few Fibonacci numbers.
the first few Fibonacci numbers.
73
166297
2498
03:00
Let's see what we get there.
74
168795
1608
03:02
So one plus one plus four is six.
75
170403
2139
03:04
Add nine to that, we get 15.
76
172542
3005
03:07
Add 25, we get 40.
77
175547
2213
03:09
Add 64, we get 104.
78
177760
2791
03:12
Now look at those numbers.
79
180551
1652
03:14
Those are not Fibonacci numbers,
80
182203
2384
03:16
but if you look at them closely,
81
184587
1879
03:18
you'll see the Fibonacci numbers
82
186466
1883
03:20
buried inside of them.
83
188349
2178
03:22
Do you see it? I'll show it to you.
84
190527
2070
03:24
Six is two times three, 15 is three times five,
85
192597
3733
03:28
40 is five times eight,
86
196330
2059
03:30
two, three, five, eight, who do we appreciate?
87
198389
2928
03:33
(Laughter)
88
201317
1187
03:34
Fibonacci! Of course.
89
202504
2155
03:36
Now, as much fun as it is to discover these patterns,
90
204659
3783
03:40
it's even more satisfying to understand
91
208442
2482
03:42
why they are true.
92
210924
1958
03:44
Let's look at that last equation.
93
212882
1889
03:46
Why should the squares of one, one,
two, three, five and eight
two, three, five and eight
94
214771
3868
03:50
add up to eight times 13?
95
218639
2545
03:53
I'll show you by drawing a simple picture.
96
221184
2961
03:56
We'll start with a one-by-one square
97
224145
2687
03:58
and next to that put another one-by-one square.
98
226832
4165
04:02
Together, they form a one-by-two rectangle.
99
230997
3408
04:06
Beneath that, I'll put a two-by-two square,
100
234405
2549
04:08
and next to that, a three-by-three square,
101
236954
2795
04:11
beneath that, a five-by-five square,
102
239749
2001
04:13
and then an eight-by-eight square,
103
241750
1912
04:15
creating one giant rectangle, right?
104
243662
2572
04:18
Now let me ask you a simple question:
105
246234
1916
04:20
what is the area of the rectangle?
106
248150
3656
04:23
Well, on the one hand,
107
251806
1971
04:25
it's the sum of the areas
108
253777
2530
04:28
of the squares inside it, right?
109
256307
1866
04:30
Just as we created it.
110
258173
1359
04:31
It's one squared plus one squared
111
259532
2172
04:33
plus two squared plus three squared
112
261704
2233
04:35
plus five squared plus eight squared. Right?
113
263937
2599
04:38
That's the area.
114
266536
1857
04:40
On the other hand, because it's a rectangle,
115
268393
2326
04:42
the area is equal to its height times its base,
116
270719
3648
04:46
and the height is clearly eight,
117
274367
2047
04:48
and the base is five plus eight,
118
276414
2903
04:51
which is the next Fibonacci number, 13. Right?
119
279317
3938
04:55
So the area is also eight times 13.
120
283255
3363
04:58
Since we've correctly calculated the area
121
286618
2262
05:00
two different ways,
122
288880
1687
05:02
they have to be the same number,
123
290567
2172
05:04
and that's why the squares of one,
one, two, three, five and eight
one, two, three, five and eight
124
292739
3391
05:08
add up to eight times 13.
125
296130
2291
05:10
Now, if we continue this process,
126
298421
2374
05:12
we'll generate rectangles of the form 13 by 21,
127
300795
3978
05:16
21 by 34, and so on.
128
304773
2394
05:19
Now check this out.
129
307167
1409
05:20
If you divide 13 by eight,
130
308576
2193
05:22
you get 1.625.
131
310769
2043
05:24
And if you divide the larger number
by the smaller number,
by the smaller number,
132
312812
3427
05:28
then these ratios get closer and closer
133
316239
2873
05:31
to about 1.618,
134
319112
2653
05:33
known to many people as the Golden Ratio,
135
321765
3301
05:37
a number which has fascinated mathematicians,
136
325066
2596
05:39
scientists and artists for centuries.
137
327662
3246
05:42
Now, I show all this to you because,
138
330908
2231
05:45
like so much of mathematics,
139
333139
2025
05:47
there's a beautiful side to it
140
335164
1967
05:49
that I fear does not get enough attention
141
337131
2015
05:51
in our schools.
142
339146
1567
05:52
We spend lots of time learning about calculation,
143
340713
2833
05:55
but let's not forget about application,
144
343546
2756
05:58
including, perhaps, the most
important application of all,
important application of all,
145
346302
3454
06:01
learning how to think.
146
349756
2076
06:03
If I could summarize this in one sentence,
147
351832
1957
06:05
it would be this:
148
353789
1461
06:07
Mathematics is not just solving for x,
149
355250
3360
06:10
it's also figuring out why.
150
358610
2925
06:13
Thank you very much.
151
361535
1815
06:15
(Applause)
152
363350
4407
ABOUT THE SPEAKER
Arthur Benjamin - MathemagicianUsing daring displays of algorithmic trickery, lightning calculator and number wizard Arthur Benjamin mesmerizes audiences with mathematical mystery and beauty.
Why you should listen
Arthur Benjamin makes numbers dance. In his day job, he's a professor of math at Harvey Mudd College; in his other day job, he's a "Mathemagician," taking the stage in his tuxedo to perform high-speed mental calculations, memorizations and other astounding math stunts. It's part of his drive to teach math and mental agility in interesting ways, following in the footsteps of such heroes as Martin Gardner.
Benjamin is the co-author, with Michael Shermer, of Secrets of Mental Math (which shares his secrets for rapid mental calculation), as well as the co-author of the MAA award-winning Proofs That Really Count: The Art of Combinatorial Proof. For a glimpse of his broad approach to math, see the list of research talks on his website, which seesaws between high-level math (such as his "Vandermonde's Determinant and Fibonacci SAWs," presented at MIT in 2004) and engaging math talks for the rest of us ("An Amazing Mathematical Card Trick").
Arthur Benjamin | Speaker | TED.com