Fantastic Mathematical Beasts and Where to Find Them. Mike Naylor

Transcription

1 Fantastic Mathematical Beasts and Where to Find Them Mike Naylor

2

3 Dividing by zero

4 Ask Siri

5 1:0 =? 0:0 =??

6 1: 1 = 1 1: 0,5 = 2 1: 0,25 = 4 1: 0,1 = 10 1:0 =? 1: 0,01 = 100 1: 0,001 = :0 =? 1: 0, = : 0, = lim x 0 1 x =

7 lim x 0 1 x = 1 0 =?

8 10:2 10 candies in a bowl Split them into 2 groups How many candies in each group?

9 10:2 10 candies in a bowl Split them into 2 groups How many candies in each group? 5 in each group. 10:2 = 5 Delingsdivisjon Partition division

10 10:2 10 candies in a bowl Take out 2 candies at a time How many times can you do that until the bowl is empty?

11 10:2 10 candies in a bowl Take out 2 candies at a time How many times can you do that until the bowl is empty? 5 ganger. 10:2 = 5 Målingsdivisjon Measurement division

12 10:0 10 candies in a bowl Split them into 0 groups How many candies in each group? Delingsdivisjon Partition division

13 10:0 10 candies in a bowl Split them into 0 groups How many candies in each group?

14 Målingsdivisjon 10:0 10 candies in a bowl Take out 0 candies at a time How many times can you do that until the bowl is empty?

15 10:0 10 candies in a bowl Take out 0 candies at a time How many times can you do that until the bowl is empty?? Is the bowl empty after an infinite number of times of removing 0 candies? No. Not even close! 10:0 is not infinity. 10:0 cannot be done.

16 a = b a 2 = ab a 2 b 2 = ab b 2 (a + b)(a b) = ab b 2 (a + b)(a b) = b(a b) a + b = b b + b = b 2b = b 2 = 1

17 Dividing by zero Infinity Pi Repeating decimals Fractals Iterated function systems Complex numbers

18 Gerd Åsta Bones Mike Naylor Thomas Ramdal

19 half circle = π smaller circle C = πd = π

20 π 2

21 π 2

22 π 2

23 π 2

24 π 2

25 π 2

26 π 2

27 π 2

28 π 2

29 π 2

30 π = 2

31

32

33

34

35 Strange things happen at infinity!

36 What is the difference between 1 and 0.999? =?

37 =? An infinite number of zeros with a 1 on the end. The smallest number before 0.

38 1/3 = /3 = /3 =

39 10x = x = x = x = = 1

40 If the average of a and b is b, what can we say about a and b? mean(1, )= : 2 =

41 Repeating decimals 1/98 =

42 Repeating decimals 1/98 =

43 Repeating decimals 1/998 =

44 1/8 = 0.125??? 1/98 = /998 = /9998 =

45 1/8 = 0.125??? 1/98 = /998 = /9998 =

46 1/8 = 0.125??? 1/98 = /998 = /9998 = /7 =? 1/97 = /997 =? 1/9997 =? 1/96 = 1/95 = 1/99 =? Powers of 1?

47 /89 =

48

49 powers of

50 powers of * * * * * * * * * *10 + 1

51 powers of

52 = 1/89

53 1 Pascals trekant

54 Sierpinsky triangle Species: Triangle Genus: Sierpinski Family: Two-dimensional Order: Iterated function systems Class: Fractal Phylum: Geometry Kingdom: Math

55 steg # areal for hver totalt area Sierpinski Trekant /4 3/ /16 9/ /64 27/ /256 81/ / /1024 n

56 Øyvind

57

58

59

60 The sphinx Riddle of the Sphinx What walks on 4 legs in the morning 2 legs in the afternoon and 3 legs in the evening?

61 Geometric Riddle of the Sphinx divide the sphinx in 4 equal sphinx-shaped pieces

62 Analysis Sphinx is made of 6 triangles

63 Analysis Or 6 x 4 = 24 smaller triangles Make 4 sphinxes from 6 little triangles each

64 Repeat

65 16 sphinxes

66 64 sphinxes

67 256 sphinxes

68 1024 sphinxes

69 4096 sphinxes

70 Sphinx fractal - Mike Naylor

71 Sphinx fractal - Mike Naylor

72 Mandelbrot set Xaos

73 Speed: UI /Zooming Speed /[3] Magnification power: Calculations/Iterations/[1000+] Animate color: Filters/Palette emulator Fractal/Palette/Color cycling (or press y )

74 Real number line

75 Real number line addition = translation

76 Real number line

77 Real number line 2 2 s t r e t c h

78 Real number line 2 1/

79 Real number line 2 1/2 stretch multiplication = stretching

80 Real number line

81 Real number line 2-1 = ?

82 Real number line 2-1 = = multiplying by a negative = rotate 180

83 Real number line x 2 = 4 x =? starting with 1, what can we multiply by twice to get to 4?

84 Real number line 4 has 2 roots: 2 and We usually say the square root of 4 is 2 and not -2 so that we can define square root as a function (a function can have only one output!) But if we think of square root as reverse of multiplication of a number by the same number, we can see -2 works as a starting point as well.

85 Real number line x 2 = -1 x =? What can we multiply with twice to get to -1? (starting at 1)

86 Real number line x 2 = -1 x =? rotation, distance 1

87 Real number line x 2 = -1 x =? rotation, distance 1 done twice goes to -1

88 Real number line This number here is a root of -1 (It s not on the real number line, but ok?)

89 Real number line What is the other root of -1?

90 Real number line rotation, distance 1

91 Real number line i = i = - -1

92 Complex number plane i i

93 3i Complex number plane 2i i (1,1) 1 + i i -2i -3i

94 3i 2i Complex number plane 4 + 2i i 1 + i i (1 + i) + (3 + i) -2i -3i

95 3i 2i Complex number plane 4 + 2i i 1 + i i (1 + i) + (3 + i) -2i 4 + 2i -3i

96 3i 2i i Complex number plane angle = 45 distance = i i (1 + i) (1 + i) or (1 + i) 2-2i rotate (1+i) an additional 45-3i

97 3i 2i i Complex number plane angle = 45 distance = i i (1 + i) (1 + i) or (1 + i) 2-2i rotate (1+i) an additional 45 stretch distance 2-3i

98 (1 + i) 2 = 0 + 2i 3i 2i i Complex number plane angle = 45 distance = i (1 + i) (1 + i) or (1 + i) 2-2i -3i rotate (1+i) an additional 45 stretch distance = 2

99 (1 + i) 2 = 0 + 2i 3i 2i i angle = 45 distance = (1 + i)(1 + i) -i (1 + i) (1 + i) or (1 + i) i + i + i i + i i + (-1) 0 + 2i -2i -3i rotate (1+i) an additional 45 stretch distance = 2

100 3i 2i i i Results of multiplying -2i numbers on the unit circle are on the unit circle -3i

101 3i Cube root of 3? 2i i i -2i 3-1 = -1-3i

102 3i Cube root of 3? 2i i i -2i -3i

103 3i Cube root of 3? (1/2 + i 3/2) (1/2 + i 3/2) 2i 1/2) 2 + 2(1/2)i 3/2) + (i 3/2) 2 1/4 + i 3/2 3/4 i -1/2 + i 3/2 1 2, (-1/2 + i 3/2) (1/2 + i 3/2) -i -1/4 i 3/4 + i 3/4 + (i 3/2) /4 3/4-2i + i i

104 3i 3-1 2i i i -2i -3i

105 3i 4-1 2i i i -2i -3i

106 3i 5-1 2i i i -2i -3i

107 3i 6-1 2i i i -2i -3i

108 2 ( 1 + i ) = ( 1 + i )( 1 + i ) = 0 + 2i 0 + 2i 1 + i

109 2 ( 0 + 2i ) = 0 + 4i 2 = 4 + 0i 0 + 2i 4 + 0i

110 (-4 + 0i) 2 = i (16 + 0i) 2 = i 4 + 0i

111 2 Z orbit of (1 + i)

112 Z 2 + C orbits create the Mandelbrot set

113 Z 2 + C orbits create the Mandelbrot set

114 Choose Starting Point: C= i initial Z = i 1.0 i 0.5 i 2 Z = i 2 Z + C = i new Z = i

115 Choose Starting Point: C= i new Z = i 1.0 i 0.5 i 2 Z = i 2 Z + C = i

116 Choose Starting Point: C= i new Z = i 1.0 i 0.5 i Z 2 = i 2 Z + C = i

117 Choose Starting Point: C= i new Z = i 1.0 i 0.5 i Z 2 = i 2 Z + C = i

118 Point C = i has a stable orbit. 1.0 i Color point C black 0.5 i

119 Choose Starting Point: C= i initial Z = i 1.0 i 0.5 i 2 Z = i 2 Z + C = i

120 Choose Starting Point: C= i new Z = i 1.0 i 0.5 i 2 Z = i 2 Z + C = i

121 Choose Starting Point: C= i new Z = i 1.0 i 0.5 i 2 Z = i 2 Z + C = i

122 Choose Starting Point: C= i new Z = i 1.0 i 0.5 i 2 Z = i 2 Z + C = i

123 Choose Starting Point: C= i new Z = i 1.0 i 0.5 i 2 Z = i 2 Z + C = i

124 Point C = i has an orbit that escapes to infinity. 1.0 i 0.5 i Color point C white

125 Repeat this process for millions of points. 1.0 i The points with stable orbits (black) are the M-Set.

126

127 God jul! Takk for meg! Mike Naylor matematikkbolgen.com mike-naylor.com Facebook Matematikkbølgen