TEDGlobal 2013
Arthur Benjamin: The magic of Fibonacci numbers
Arthur Benjamin: Magien i Fibonacci-tallene
Filmed:
Readability: 3.2
7,057,274 views
Matematik er logisk, funktionelt og helt igennem ... perfekt. Det matematiske legebarn, Arthur Benjamin, udforsker skjulte egenskaber, i de underlige og vidunderlige talrækker, kendt som: Fibonacci-tallene. (Og minder dig om, at, matematik også sagtens kan være inspirerende!)
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
Okay, hvorfor lærer vi matematik?
00:15
Essentially, for three reasons:
1
3652
2548
I bund og grund, af 3 årsager:
00:18
calculation,
2
6200
1628
beregning,
00:19
application,
3
7828
1900
anvendelse,
00:21
and last, and unfortunately least
4
9728
2687
og sidst, samt desværre også mindst
00:24
in terms of the time we give it,
5
12415
2105
i form af den tid vi giver den,
00:26
inspiration.
6
14520
1922
inspiration.
00:28
Mathematics is the science of patterns,
7
16442
2272
Matematik er videnskaben, der ligger bag mønstre
00:30
and we study it to learn how to think logically,
8
18714
3358
og vi studerer det, for at lære at tænke logisk,
00:34
critically and creatively,
9
22072
2527
kritisk og kreativt.
00:36
but too much of the mathematics
that we learn in school
that we learn in school
10
24599
2926
Men for meget af den matematik, vi lærer i skolen,
00:39
is not effectively motivated,
11
27525
2319
motiverer ikke effektivt nok
00:41
and when our students ask,
12
29844
1425
og når vores elever spørger;
00:43
"Why are we learning this?"
13
31269
1675
"Hvorfor bliver vi undervist i dette?",
00:44
then they often hear that they'll need it
14
32944
1961
får de tit af vide, at de skal bruge det til
00:46
in an upcoming math class or on a future test.
15
34905
3265
et kommende modul, eller en prøve ude i fremtiden.
00:50
But wouldn't it be great
16
38170
1802
Men ville det ikke være skønt,
00:51
if every once in a while we did mathematics
17
39972
2518
hvis vi til tider kastede os over matematikken,
00:54
simply because it was fun or beautiful
18
42490
2949
udelukkende fordi det var sjovt eller smukt,
00:57
or because it excited the mind?
19
45439
2090
eller fordi det stimulerede sindet?
00:59
Now, I know many people have not
20
47529
1722
Jeg ved, at mange folk ikke har haft muligheden
01:01
had the opportunity to see how this can happen,
21
49251
2319
for at se, hvordan dette kan udfolde sig -
01:03
so let me give you a quick example
22
51570
1829
så lad mig give jer et hurtigt eksempel,
01:05
with my favorite collection of numbers,
23
53399
2341
med de tal jeg holder allermest af,
01:07
the Fibonacci numbers. (Applause)
24
55740
2728
Fibonacci-tallene. (Klapsalve)
01:10
Yeah! I already have Fibonacci fans here.
25
58468
2052
Sådan! Der er Fibonacci-fans iblandt os, allerede.
01:12
That's great.
26
60520
1316
Det er skønt.
01:13
Now these numbers can be appreciated
27
61836
2116
Disse numre kan værdsættes
01:15
in many different ways.
28
63952
1878
på mange forskellige måder.
01:17
From the standpoint of calculation,
29
65830
2709
Med udgangspunkt i beregning,
01:20
they're as easy to understand
30
68539
1677
er de lige så nemme at forstå
01:22
as one plus one, which is two.
31
70216
2554
som at 1 plus 1 giver 2.
01:24
Then one plus two is three,
32
72770
2003
Efterfølgende 1 plus 2 giver 3,
01:26
two plus three is five, three plus five is eight,
33
74773
3014
2 plus 3 giver 5, 3 plus 5 giver 8
01:29
and so on.
34
77787
1525
og så videre.
01:31
Indeed, the person we call Fibonacci
35
79312
2177
Faktisk, ham vi kalder Fibonacci,
01:33
was actually named Leonardo of Pisa,
36
81489
3180
hed reelt set, Leonardo af Pisa,
01:36
and these numbers appear in his book "Liber Abaci,"
37
84669
3053
og disse tal dukkede op i hans bog; "Liber Abaci",
01:39
which taught the Western world
38
87722
1650
som lærte den vestlige verden
01:41
the methods of arithmetic that we use today.
39
89372
2827
aritmetikkens metoder - læren om tal - som vi bruger i dag.
01:44
In terms of applications,
40
92199
1721
Hvad angår anvendelsesmuligheder,
01:45
Fibonacci numbers appear in nature
41
93920
2183
ser vi Fibonacci-tal dukke op i naturen
01:48
surprisingly often.
42
96103
1857
overraskende ofte.
01:49
The number of petals on a flower
43
97960
1740
Antallet af blade på en blomst,
01:51
is typically a Fibonacci number,
44
99700
1862
er typisk et Fibonacci-tal
01:53
or the number of spirals on a sunflower
45
101562
2770
eller antallet af spiraler på en solsikke,
01:56
or a pineapple
46
104332
1411
eller en ananas
01:57
tends to be a Fibonacci number as well.
47
105743
2394
har det med også at være et Fibonacci-tal.
02:00
In fact, there are many more
applications of Fibonacci numbers,
applications of Fibonacci numbers,
48
108137
3503
Der er faktisk mange andre anvendelsesmuligheder, for Fibonacci-tal,
02:03
but what I find most inspirational about them
49
111640
2560
men det jeg finder mest inspirerende ved dem,
02:06
are the beautiful number patterns they display.
50
114200
2734
er, de smukke talmønstre, der følger med.
02:08
Let me show you one of my favorites.
51
116934
2194
Lad mig vise dig en af mine favoritter.
02:11
Suppose you like to square numbers,
52
119128
2221
Vi antager, at du nyder at kvadrere tal,
02:13
and frankly, who doesn't? (Laughter)
53
121349
2675
og ærlig talt, hvem gør ikke det? (Latter)
02:16
Let's look at the squares
54
124040
2240
Lad os kigge på kvadraterne,
02:18
of the first few Fibonacci numbers.
55
126280
1851
af de første par Fibonacci-tal.
02:20
So one squared is one,
56
128131
2030
Så, kvadratet af 1 giver 1
02:22
two squared is four, three squared is nine,
57
130161
2317
kvadratet af 2 giver 4, 3 er lig med 9
02:24
five squared is 25, and so on.
58
132478
3173
5 er lig med 25 og så videre.
02:27
Now, it's no surprise
59
135651
1901
Det er ikke nogen overraskelse,
02:29
that when you add consecutive Fibonacci numbers,
60
137552
2828
at når du ligger to på hinanden efterfølgende Fibonacci-tal sammen,
02:32
you get the next Fibonacci number. Right?
61
140380
2032
får du det næste Fibonacci-tal. Enig?
02:34
That's how they're created.
62
142412
1395
Det er grundreglen, for opbygningen.
02:35
But you wouldn't expect anything special
63
143807
1773
Men du ville ikke tro, at der ville ske noget specielt,
02:37
to happen when you add the squares together.
64
145580
3076
når du ligger kvadraterne sammen.
02:40
But check this out.
65
148656
1346
Men, kig her engang.
02:42
One plus one gives us two,
66
150002
2001
1 plus 1 giver 2
02:44
and one plus four gives us five.
67
152003
2762
og 1 plus 4 giver 5.
02:46
And four plus nine is 13,
68
154765
2195
4 plus 9 giver 13,
02:48
nine plus 25 is 34,
69
156960
3213
9 plus 25 giver 34
02:52
and yes, the pattern continues.
70
160173
2659
og ja, mønstret fortsætter.
02:54
In fact, here's another one.
71
162832
1621
Der er faktisk et mere her.
02:56
Suppose you wanted to look at
72
164453
1844
Antag at du gerne vil ligge et par,
02:58
adding the squares of
the first few Fibonacci numbers.
the first few Fibonacci numbers.
73
166297
2498
af Fibonaccis første kvadrater sammen.
03:00
Let's see what we get there.
74
168795
1608
Lad os se hvad vi ville få ud af det.
03:02
So one plus one plus four is six.
75
170403
2139
1 + 1 + 4 = 6.
03:04
Add nine to that, we get 15.
76
172542
3005
Tilføj 9 til det og vi får 15.
03:07
Add 25, we get 40.
77
175547
2213
Tilføj 25 yderligere og vi får 40.
03:09
Add 64, we get 104.
78
177760
2791
64 oveni det og vi får 104.
03:12
Now look at those numbers.
79
180551
1652
Kig engang på de tal.
03:14
Those are not Fibonacci numbers,
80
182203
2384
Det er ikke Fibonacci-tal,
03:16
but if you look at them closely,
81
184587
1879
men hvis du ser godt efter,
03:18
you'll see the Fibonacci numbers
82
186466
1883
vil du se Fibonacci-tallene,
03:20
buried inside of them.
83
188349
2178
begravet dybt i dem.
03:22
Do you see it? I'll show it to you.
84
190527
2070
Ser du dem? Lad mig vise dem for dig.
03:24
Six is two times three, 15 is three times five,
85
192597
3733
6 er 2 gange 3, 15 er 3 gange 5,
03:28
40 is five times eight,
86
196330
2059
40 er 5 gange 8,
03:30
two, three, five, eight, who do we appreciate?
87
198389
2928
1, 2, 3, 5, hvem er altid velkommen i vores hjem?
03:33
(Laughter)
88
201317
1187
(Latter)
03:34
Fibonacci! Of course.
89
202504
2155
Fibonacci! Selvfølgelig, da.
03:36
Now, as much fun as it is to discover these patterns,
90
204659
3783
Hvor sjovt det end lyder, at støde på disse mønstre,
03:40
it's even more satisfying to understand
91
208442
2482
så er det faktisk endnu mere tilfredsstillende,
03:42
why they are true.
92
210924
1958
at forstå, hvorfor de går op.
03:44
Let's look at that last equation.
93
212882
1889
Lad os kigge på den sidste ligning.
03:46
Why should the squares of one, one,
two, three, five and eight
two, three, five and eight
94
214771
3868
Hvorfor skulle kvadratet af 1, 1, 2, 3, 5 og 8, tilsammen,
03:50
add up to eight times 13?
95
218639
2545
give 8 gange 13?
03:53
I'll show you by drawing a simple picture.
96
221184
2961
Jeg vil illustrere det, med denne simple tegning.
03:56
We'll start with a one-by-one square
97
224145
2687
Vi starter med en kvadrat på 1*1.
03:58
and next to that put another one-by-one square.
98
226832
4165
Ved siden af den, også en kvadrat på 1*1.
04:02
Together, they form a one-by-two rectangle.
99
230997
3408
Sammen udgør de en 1*2 rektangel.
04:06
Beneath that, I'll put a two-by-two square,
100
234405
2549
Under den, placerer jeg en 2*2 kvadrat,
04:08
and next to that, a three-by-three square,
101
236954
2795
ved siden af den en 3*3 kvadrat,
04:11
beneath that, a five-by-five square,
102
239749
2001
under den, en 5*5 kvadrat,
04:13
and then an eight-by-eight square,
103
241750
1912
efterfulgt af en 8*8 kvadrat,
04:15
creating one giant rectangle, right?
104
243662
2572
hvor vi derved, skaber én stor rektangel, ikke?
04:18
Now let me ask you a simple question:
105
246234
1916
Lad mig nu stille dig ét simpelt spørgsmål:
04:20
what is the area of the rectangle?
106
248150
3656
Hvad er områdestørrelsen, af denne rektangel?
04:23
Well, on the one hand,
107
251806
1971
Ja, på den ene side,
04:25
it's the sum of the areas
108
253777
2530
er det summen af alle firkanterne,
04:28
of the squares inside it, right?
109
256307
1866
de kvadrater inden for området, okay?
04:30
Just as we created it.
110
258173
1359
Nøjagtig, som vi lavede den.
04:31
It's one squared plus one squared
111
259532
2172
Det er kvadratet af 1, plus kvadratet af 1,
04:33
plus two squared plus three squared
112
261704
2233
plus kvadratet af 2, plus kvadratet af 3,
04:35
plus five squared plus eight squared. Right?
113
263937
2599
plus kvadratet af 5, plus kvadratet af 8. Du er med?
04:38
That's the area.
114
266536
1857
Det er områdestørrelsen.
04:40
On the other hand, because it's a rectangle,
115
268393
2326
På den anden side, grundet den rektangulære form,
04:42
the area is equal to its height times its base,
116
270719
3648
er områdestørrelsen lig med, højden gange bunden.
04:46
and the height is clearly eight,
117
274367
2047
Højden er tydeligvis 8
04:48
and the base is five plus eight,
118
276414
2903
og bunden er lig med 5 plus 8,
04:51
which is the next Fibonacci number, 13. Right?
119
279317
3938
som er det næste Fibonacci-tal, 13. I er med?
04:55
So the area is also eight times 13.
120
283255
3363
Områdestørrelsen er altså 8 gange 13.
04:58
Since we've correctly calculated the area
121
286618
2262
Nu vi har udregnet størrelsen korrekt,
05:00
two different ways,
122
288880
1687
på 2 forskellige måder,
05:02
they have to be the same number,
123
290567
2172
må tallene være ens.
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
Det er derfor kvadratet af 1, 1, 2, 3, 5 og 8
05:08
add up to eight times 13.
125
296130
2291
giver det samme som 8 gange 13?
05:10
Now, if we continue this process,
126
298421
2374
Hvis vi fortsætter med denne metode,
05:12
we'll generate rectangles of the form 13 by 21,
127
300795
3978
vil vi danne en rektangel på 13 gange 21,
05:16
21 by 34, and so on.
128
304773
2394
herefter 21 gange 34 og så videre.
05:19
Now check this out.
129
307167
1409
Kig så her engang.
05:20
If you divide 13 by eight,
130
308576
2193
Hvis du dividerer 13 med 8,
05:22
you get 1.625.
131
310769
2043
får du 1,625.
05:24
And if you divide the larger number
by the smaller number,
by the smaller number,
132
312812
3427
Hvis du fortsat dividerer det store tal med det lille,
05:28
then these ratios get closer and closer
133
316239
2873
vil forholdet mellem disse, komme tættere og tættere
05:31
to about 1.618,
134
319112
2653
på omkring 1,618.
05:33
known to many people as the Golden Ratio,
135
321765
3301
Kendt af mange som, Det Gyldne Snit,
05:37
a number which has fascinated mathematicians,
136
325066
2596
et tal, der har fascineret matematikere,
05:39
scientists and artists for centuries.
137
327662
3246
forskere og kunstnere, gennem århundreder.
05:42
Now, I show all this to you because,
138
330908
2231
Grunden til, at jeg viser alt dette til jer,
05:45
like so much of mathematics,
139
333139
2025
som så meget af matematikken,
05:47
there's a beautiful side to it
140
335164
1967
er der en smuk side af det hele,
05:49
that I fear does not get enough attention
141
337131
2015
som jeg frygter, IKKE får nok opmærksomhed,
05:51
in our schools.
142
339146
1567
i vores skoler.
05:52
We spend lots of time learning about calculation,
143
340713
2833
Vi bruger meget tid på at lære om beregning,
05:55
but let's not forget about application,
144
343546
2756
men lad os ikke glemme anvendelsesmulighederne,
05:58
including, perhaps, the most
important application of all,
important application of all,
145
346302
3454
inklusiv den måske, vigtigste af dem alle,
06:01
learning how to think.
146
349756
2076
at lære hvordan man tænker.
06:03
If I could summarize this in one sentence,
147
351832
1957
Hvis jeg må opsummere dette i en sætning,
06:05
it would be this:
148
353789
1461
ville det være følgende:
06:07
Mathematics is not just solving for x,
149
355250
3360
Matematik handler ikke blot om, at beregne x,
06:10
it's also figuring out why.
150
358610
2925
det handler også om at finde ud af, hvorfor.
06:13
Thank you very much.
151
361535
1815
Mange tak.
06:15
(Applause)
152
363350
4407
(Klapsalver)
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