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Mathematics Behind the Internet James H. Davenport University of Bath 21 September 2009 Google a new word? Google a new word? I met this woman last night at a party and I came right home and googled her. Google


  1. “I’m feeling lucky” is often right 3��������������9����������� ����455������$����$�%$&�5���<��5 ��������������� ����������������������������������������������������������������������������������������������������������� !!"#$�������������%�&��������'()� �*&�����#���� !!+$ ������������������������� ������������ �����&����)%���%���������������%���)%���%�� ���������� ��������,������%���������������(�����������-�%�������������&����� !!.#����������������%�������(�����������-�%������� /�&������0�����1��1������2�)�3�&������������&��������������������%�4� ������ �������56&����������$ ��������� �7�������'�����%���������������&����*������ /������'�����)�%�����(��&�������/�����$ ������3&��� !!8������%�������)�&���������&��������&������������&����������������������������0������$����%�1��������������)&������&��*��������� ������������������%���&�4�����%&��������������������*�&�������������������������������&�9����������������$������������������������������������ �����������������0�&%���������������$ :������������&����*������������������������&������������;������������������������������������$����������������<�%���������������0&������ 7�����������<�%�����������&%%������-������%�=�������������������������������������������7�����������������2����������������� 57%�5 !!8#�� ����������������������������������������������������������%����������������������&%��������������%���������������&�����������'��&%�1����� 7�������5�����2�����������$���������������������� ���7����������������$�������������-����&�����������0&����������������%���-�&��$ ���%����������'�����%�����������9��:����������������������&������&����4������������$�-��������������������%�&�����&�����=*>����"+1"? )��������4�������������$�*���������%�&��������������&������������� @1 .����%�4������������������%����%�����������>���������$)%$������#$ (��3&��� !!+��������������������������&����������������������������������4�������������������$�(��3��&���5/���&���� !!8����������������-�����3������ 0�&%�����������������%�����������������%�����$�-������������������������������������������������#���������������$�7��"8�/���&���� !!8������������ ���%������������������������������&����������������%��4�������������������$�(��3&��� !!8�������������������������$A$#��������������������������4���� ��������������$ *%�����%�B���� !!+5 !!84����)��������"������&������@!!+!4�����&����*�������������@!!+85.!" @4�*����%���=���������$�(��)�������� ��� ���������������%����������@!"+@5��.! "!���������������%�������������.! !?�)�%&�����������&��������������&�����<�%��$ *%�����%�B���� !!85 !!?4����)��������"����������%�������@!!+!4�����&����*�������������@!!+85.!" @4�*����%���=���������$�(��)�������� ����������������%������������������������:�������$�)������������������������$ *%�����%�B���� !!?5 !!"!4����)��������"����������%�����CC"!"?!4��������������������%��������������%�����@!!+!4�����&����*���������� ��@!!+85.!" @4�*����%���=���������$�7��-&���������"!$".����D0 $ ���������&��������������������������%�������%�����������%���%����������$

  2. Whereas it has a lot to choose from ������������������������������������ ������������������������������������������������������ ��������� ��������������!����� ��� ������� ������� ����� ����� ��������� ����� ������ � ������������������������� F����� ��������������� � �������� ��������� ����������������� ��� ��������������� �!��"���� � �#� �� �������"�� ������ ����� �������������������� ��$ ���� ��������%� ����������������� �&����'��� ���������(���� ���� ��)���"�����������*�+��������'��,���*����������-��������.������������ '"����������*�����"����� ������� ���� ����"/0���,��0�#�)������#�������� ����� � ��������+��������1(����(����� �������������������� �2�3�4��$'"����% 5����5�������'650��������������#����� ��������+��������1(����(������ ��������������� �� ���� ��2�3�4��$'"����%��7�2888�1�� +��������)�������"��$49�8:8%��'����4��.�'!�1����,����498:8��+�������� ��� ����������������0��������0��������#�������� �(�;�6���������#�)������(�44�#�!����������������#�����24��������� ����� � +��<�������+��������)�������6������������ �������������������� �2�9� ��� 5����5�������'650��������������#����� 4=�;���4>>>� ��� �4>>>#>3#>8��+��<�������+��������)�������6������������� ��������������� � ���� ��2�9�?�$!���������%���4>>4�1���+��������)�������"��$49�8:8%� ��� ����������������0���"�����0�������)6������#�������� �(�;�6���������#�)������(�@�#�!����������������#�����@��������� ����� � ����� �&�� ��������� �6������������������������������������(���� ���� � ��� 5����5�������'650��������������#����� 22�5���4>>>� ��� �+��<�������+��������)�������6������������� ����� �&�� ��������� ��6���� ��������������������������������(���� ���� �� ���� �A�4�@�B��.������� ��� ������ ���� ����"/0���,��0+�������0����4�����#�������� �(�;&�6���������#�!��������������� ����� � 1���)���"���������)��������������"����#/�(���(��������(� ����� � ��� 5����5�������'650��������������#����� ��������������� ��6�����C)���"���*������������D��������C*�D�����������(���� ���� � ;�&� ��������� E ���� ����"/��'"����� �(�)�(��������(��1����������������� ��� ������ ���� ����"/0���,��0A�)�#����������#�������� �(�' �)�(��������(�#�!����������������#�����4���������

  3. How do we decide which pages to choose (It isn’t luck!)

  4. How do we decide which pages to choose (It isn’t luck!) The basic idea is obvious,

  5. How do we decide which pages to choose (It isn’t luck!) The basic idea is obvious, with hindsight. Choose the page with more links to it.

  6. How do we decide which pages to choose (It isn’t luck!) The basic idea is obvious, with hindsight. Choose the page with more links to it. A B ↓ ↘ ↓ C D

  7. How do we decide which pages to choose (It isn’t luck!) The basic idea is obvious, with hindsight. Choose the page with more links to it. A B ↓ ↘ ↓ C D Obviously D is more popular than C .

  8. But the Web is much more complicated!

  9. But the Web is much more complicated! A B ↓ ↘ ↓ C D ↓ ↓ E F ↓ ↓ G H

  10. But the Web is much more complicated! A B ↓ ↘ ↓ C D ↓ ↓ E F ↓ ↓ G H E and F each have only one link to them, but, since D is more popular than C , we should regard F as more popular than E (and H as more popular than G ).

  11. But the Web is much more complicated! And constantly changing.

  12. But the Web is much more complicated! And constantly changing. A B ↓ ↘ ↓ C D ↓ ↙ ↓ E F ↓ ↓ G H

  13. But the Web is much more complicated! And constantly changing. A B ↓ ↘ ↓ C D ↓ ↙ ↓ E F ↓ ↓ G H Now E is more popular than F .

  14. But the Web is much more complicated! And constantly changing. A B ↓ ↘ ↓ C D ↓ ↙ ↓ E F ↓ ↓ G H Now E is more popular than F . And G is more popular than H ,

  15. But the Web is much more complicated! And constantly changing. A B ↓ ↘ ↓ C D ↓ ↙ ↓ E F ↓ ↓ G H Now E is more popular than F . And G is more popular than H , even though nothing has changed for G itself.

  16. But the Web is much much more complicated!

  17. But the Web is much much more complicated! 1. The real Web contains (lots of) loops.

  18. But the Web is much much more complicated! 1. The real Web contains (lots of) loops. 2. The real Web is utterly massive — no-one, not even Google, really knows how big.

  19. But the Web is much much more complicated! 1. The real Web contains (lots of) loops. 2. The real Web is utterly massive — no-one, not even Google, really knows how big. 3. The real Web keeps changing.

  20. But the Web is much much more complicated! 1. The real Web contains (lots of) loops. 2. The real Web is utterly massive — no-one, not even Google, really knows how big. 3. The real Web keeps changing. 4. The real Web is commercially valuable, so there are incentives to manipulate it.

  21. The real Web contains loops

  22. The real Web contains loops Nevertheless, we could, in principle write down a set of (linear) equations for the popularity of each page,

  23. The real Web contains loops Nevertheless, we could, in principle write down a set of (linear) equations for the popularity of each page, which would depend on the popularity of the pages which linked to it,

  24. The real Web contains loops Nevertheless, we could, in principle write down a set of (linear) equations for the popularity of each page, which would depend on the popularity of the pages which linked to it, which would depend on the popularity of the pages which linked to it . . . .

  25. The real Web contains loops Nevertheless, we could, in principle write down a set of (linear) equations for the popularity of each page, which would depend on the popularity of the pages which linked to it, which would depend on the popularity of the pages which linked to it . . . . Then we could solve these equations.

  26. The real Web contains loops Nevertheless, we could, in principle write down a set of (linear) equations for the popularity of each page, which would depend on the popularity of the pages which linked to it, which would depend on the popularity of the pages which linked to it . . . . Then we could solve these equations. These equations have a name: they are the equations for the principal eigenvector of the connectivity matrix of the Web.

  27. The real Web contains loops Nevertheless, we could, in principle write down a set of (linear) equations for the popularity of each page, which would depend on the popularity of the pages which linked to it, which would depend on the popularity of the pages which linked to it . . . . Then we could solve these equations. These equations have a name: they are the equations for the principal eigenvector of the connectivity matrix of the Web. The genius of Brin and Page was to realise that these equations could be solved,

  28. The real Web contains loops Nevertheless, we could, in principle write down a set of (linear) equations for the popularity of each page, which would depend on the popularity of the pages which linked to it, which would depend on the popularity of the pages which linked to it . . . . Then we could solve these equations. These equations have a name: they are the equations for the principal eigenvector of the connectivity matrix of the Web. The genius of Brin and Page was to realise that these equations could be solved, and in a distributed and iterative manner.

  29. The real Web contains loops Nevertheless, we could, in principle write down a set of (linear) equations for the popularity of each page, which would depend on the popularity of the pages which linked to it, which would depend on the popularity of the pages which linked to it . . . . Then we could solve these equations. These equations have a name: they are the equations for the principal eigenvector of the connectivity matrix of the Web. The genius of Brin and Page was to realise that these equations could be solved, and in a distributed and iterative manner. It’s known as the “Page Rank” algorithm.

  30. The real Web contains loops Nevertheless, we could, in principle write down a set of (linear) equations for the popularity of each page, which would depend on the popularity of the pages which linked to it, which would depend on the popularity of the pages which linked to it . . . . Then we could solve these equations. These equations have a name: they are the equations for the principal eigenvector of the connectivity matrix of the Web. The genius of Brin and Page was to realise that these equations could be solved, and in a distributed and iterative manner. It’s known as the “Page Rank” algorithm. Solving these equations is what makes Google work!

  31. The real Web contains loops Nevertheless, we could, in principle write down a set of (linear) equations for the popularity of each page, which would depend on the popularity of the pages which linked to it, which would depend on the popularity of the pages which linked to it . . . . Then we could solve these equations. These equations have a name: they are the equations for the principal eigenvector of the connectivity matrix of the Web. The genius of Brin and Page was to realise that these equations could be solved, and in a distributed and iterative manner. It’s known as the “Page Rank” algorithm. Solving these equations is what makes Google work! So it’s not really “I’m feeling lucky”, it’s “I believe in eigenvectors”!

  32. Flow in the Internet Assume the routers R 1 and R 2 have total capacity 1 each. A 1 B 1 ↓ ↓ C 1 → R 1 → R 2 → C 2 ↓ ↓ A 2 B 2

  33. Flow in the Internet Assume the routers R 1 and R 2 have total capacity 1 each. A 1 B 1 ↓ ↓ C 1 → R 1 → R 2 → C 2 ↓ ↓ A 2 B 2 What is the best way of allocating bandwidth to the various flows A 1 → A 2 , B 1 → B 2 and C 1 → C 2 ?

  34. Flow in the Internet Assume the routers R 1 and R 2 have total capacity 1 each. A 1 B 1 ↓ ↓ C 1 → R 1 → R 2 → C 2 ↓ ↓ A 2 B 2 What is the best way of allocating bandwidth to the various flows A 1 → A 2 , B 1 → B 2 and C 1 → C 2 ? Of course, it all depends what you mean by “best”.

  35. Network Most Efficient

  36. Network Most Efficient A and B each get 1, and C nothing.

  37. Network Most Efficient A and B each get 1, and C nothing. A 1 B 1 ↓ 1 ↓ 1 0 0 0 C 1 R 1 R 2 C 2 − → − → − → ↓ 1 ↓ 1 A 2 B 2 Total flow 2, but C might feel aggrieved.

  38. Max–min Fairness

  39. Max–min Fairness The worst-off person gets as much as possible.

  40. Max–min Fairness The worst-off person gets as much as possible. Each flow gets 1 / 2. A 1 B 1 ↓ 1/2 ↓ 1/2 1 / 2 1 / 2 1 / 2 C 1 R 1 R 2 C 2 − → − → − → ↓ 1/2 ↓ 1/2 A 2 B 2

  41. Max–min Fairness The worst-off person gets as much as possible. Each flow gets 1 / 2. A 1 B 1 ↓ 1/2 ↓ 1/2 1 / 2 1 / 2 1 / 2 C 1 R 1 R 2 C 2 − → − → − → ↓ 1/2 ↓ 1/2 A 2 B 2 Total flow 1.5, but C is getting twice as much routing done for him as A and B are.

  42. Max–min Fairness The worst-off person gets as much as possible. Each flow gets 1 / 2. A 1 B 1 ↓ 1/2 ↓ 1/2 1 / 2 1 / 2 1 / 2 C 1 R 1 R 2 C 2 − → − → − → ↓ 1/2 ↓ 1/2 A 2 B 2 Total flow 1.5, but C is getting twice as much routing done for him as A and B are. A and B might feel aggrieved.

  43. Proportional Fairness

  44. Proportional Fairness Each flow gets the same amount of effort from the routers.

  45. Proportional Fairness Each flow gets the same amount of effort from the routers. A and B each get 2 / 3, and C gets 1 / 3. A 1 B 1 ↓ 2/3 ↓ 2/3 1 / 3 1 / 3 1 / 3 C 1 R 1 R 2 C 2 − → − → − → ↓ 2/3 ↓ 2/3 A 2 B 2

  46. Proportional Fairness Each flow gets the same amount of effort from the routers. A and B each get 2 / 3, and C gets 1 / 3. A 1 B 1 ↓ 2/3 ↓ 2/3 1 / 3 1 / 3 1 / 3 C 1 R 1 R 2 C 2 − → − → − → ↓ 2/3 ↓ 2/3 A 2 B 2 Total flow is now 5 3 ≈ 1 . 66, better than max-min, but not as good as the flow where C gets nothing.

  47. But in the real world

  48. But in the real world ▶ Routers and links have widely different capacities

  49. But in the real world ▶ Routers and links have widely different capacities ▶ The network is much more complicated, and always changing

  50. But in the real world ▶ Routers and links have widely different capacities ▶ The network is much more complicated, and always changing ▶ No-one has overall knowledge of the flows.

  51. But in the real world ▶ Routers and links have widely different capacities ▶ The network is much more complicated, and always changing ▶ No-one has overall knowledge of the flows.

  52. But in the real world ▶ Routers and links have widely different capacities ▶ The network is much more complicated, and always changing ▶ No-one has overall knowledge of the flows. Nevertheless, the purely local algorithm devised by van Jacobsen (earlier; published 1988) was shown in 1997 to converge to proportional fairness.

  53. Numbers rather than Padlocks (I) A wishes to send x to B.

  54. Numbers rather than Padlocks (I) A wishes to send x to B. A and B each think of a random number, say a and b .

  55. Numbers rather than Padlocks (I) A wishes to send x to B. A and B each think of a random number, say a and b . A’s action B’s action Message multiply x by a xa ↘ multiply message by b xba = xab ↙ divide message by a xb ↘ divide message by b

  56. Numbers rather than Padlocks (I) A wishes to send x to B. A and B each think of a random number, say a and b . A’s action B’s action Message multiply x by a xa ↘ multiply message by b xba = xab ↙ divide message by a xb ↘ divide message by b In practice, to avoid guessing, and numerical errors, x , a and b are whole numbers modulo some large prime p .

  57. Numbers rather than Padlocks (I) — snag A’s action Message B’s action multiply x by a xa ↘ multiply message by b xba = xab ↙ divide message by a xb ↘ divide message by b

  58. Numbers rather than Padlocks (I) — snag A’s action Message B’s action multiply x by a xa ↘ multiply message by b xba = xab ↙ divide message by a xb ↘ divide message by b Eavesdropper computes xa ⋅ xb xab

  59. Numbers rather than Padlocks (I) — snag A’s action Message B’s action multiply x by a xa ↘ multiply message by b xba = xab ↙ divide message by a xb ↘ divide message by b Eavesdropper computes xa ⋅ xb xab = x .

  60. Numbers rather than Padlocks (I) — snag A’s action Message B’s action multiply x by a xa ↘ multiply message by b xba = xab ↙ divide message by a xb ↘ divide message by b Eavesdropper computes xa ⋅ xb xab = x . So replacing the padlocks by numbers has given the eavesdropper the chance of doing arithmetic.

  61. Numbers rather than Padlocks (II) Let’s be more subtle.

  62. Numbers rather than Padlocks (II) Let’s be more subtle. A’s action Message B’s action raise x to power a x a ↘ raise message to power b ( x b ) a = ( x a ) b ↙ take a th root of message x b ↘ take b th root of message

  63. Numbers rather than Padlocks (II) Let’s be more subtle. A’s action Message B’s action raise x to power a x a ↘ raise message to power b ( x b ) a = ( x a ) b ↙ take a th root of message x b ↘ take b th root of message Surely this frustrates the eavesdropper?

  64. But what about logarithms? A’s action B’s action Message raise x to power a x a ↘ raise message to power b ( x b ) a = ( x a ) b ↙ take a th root of message x b ↘ take b th root of message Eavesdropper computes log( x a ) ⋅ log( x b ) log( x ab )

  65. But what about logarithms? A’s action B’s action Message raise x to power a x a ↘ raise message to power b ( x b ) a = ( x a ) b ↙ take a th root of message x b ↘ take b th root of message Eavesdropper computes log( x a ) ⋅ log( x b ) = a log( x ) ⋅ b log( x ) log( x ab ) ab log( x )

  66. But what about logarithms? A’s action B’s action Message raise x to power a x a ↘ raise message to power b ( x b ) a = ( x a ) b ↙ take a th root of message x b ↘ take b th root of message Eavesdropper computes log( x a ) ⋅ log( x b ) = a log( x ) ⋅ b log( x ) = log( x ). log( x ab ) ab log( x )

  67. But what about logarithms? A’s action B’s action Message raise x to power a x a ↘ raise message to power b ( x b ) a = ( x a ) b ↙ take a th root of message x b ↘ take b th root of message Eavesdropper computes log( x a ) ⋅ log( x b ) = a log( x ) ⋅ b log( x ) = log( x ). log( x ab ) ab log( x ) Essentially the same trick as before, but with logarithms!

  68. Do logarithms exist?

  69. Do logarithms exist? Remember that we are working modulo a large prime p .

  70. Do logarithms exist? Remember that we are working modulo a large prime p . For simplicity, I will take p = 41, since it’s small enough, and logs base 5, so that log(5) = 1.

  71. Do logarithms exist? Remember that we are working modulo a large prime p . For simplicity, I will take p = 41, since it’s small enough, and logs base 5, so that log(5) = 1. 1 2 3 4 5 6 7 8 9 10 0 1 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

  72. Do logarithms exist? Remember that we are working modulo a large prime p . For simplicity, I will take p = 41, since it’s small enough, and logs base 5, so that log(5) = 1. 1 2 3 4 5 6 7 8 9 10 0 1 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 2 31 32 33 34 35 36 37 38 39 40

  73. Do logarithms exist? Remember that we are working modulo a large prime p . For simplicity, I will take p = 41, since it’s small enough, and logs base 5, so that log(5) = 1. 1 2 3 4 5 6 7 8 9 10 0 1 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 2 31 32 33 34 35 36 37 38 39 40 So log(125) = 3, but 125 = 3 ⋅ 41 + 2

  74. Do logarithms exist? Remember that we are working modulo a large prime p . For simplicity, I will take p = 41, since it’s small enough, and logs base 5, so that log(5) = 1. 1 2 3 4 5 6 7 8 9 10 0 1 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 2 31 32 33 34 35 36 37 38 39 40 So log(125) = 3, but 125 = 3 ⋅ 41 + 2 ≡ 2 since we are working modulo 41.

  75. Do logarithms exist? Remember that we are working modulo a large prime p . For simplicity, I will take p = 41, since it’s small enough, and logs base 5, so that log(5) = 1. 1 2 3 4 5 6 7 8 9 10 0 3 1 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 2 31 32 33 34 35 36 37 38 39 40

  76. Do logarithms exist? Remember that we are working modulo a large prime p . For simplicity, I will take p = 41, since it’s small enough, and logs base 5, so that log(5) = 1. 1 2 3 4 5 6 7 8 9 10 0 3 1 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 2 31 32 33 34 35 36 37 38 39 40 And we can fill in: 10 = 2 ⋅ 5, so log(10) = 4.

  77. Do logarithms exist? Remember that we are working modulo a large prime p . For simplicity, I will take p = 41, since it’s small enough, and logs base 5, so that log(5) = 1. 1 2 3 4 5 6 7 8 9 10 0 3 1 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 2 31 32 33 34 35 36 37 38 39 40 And we can fill in: 10 = 2 ⋅ 5, so log(10) = 4. Also 4 = 2 ⋅ 2 so log(4) = 3 + 3 = 6.

  78. Do logarithms exist? Remember that we are working modulo a large prime p . For simplicity, I will take p = 41, since it’s small enough, and logs base 5, so that log(5) = 1. 1 2 3 4 5 6 7 8 9 10 0 3 6 1 9 4 11 12 13 14 15 16 17 18 19 20 12 7 21 22 23 24 25 26 27 28 29 30 2 31 32 33 34 35 36 37 38 39 40 15

  79. Do logarithms exist? Remember that we are working modulo a large prime p . For simplicity, I will take p = 41, since it’s small enough, and logs base 5, so that log(5) = 1. 1 2 3 4 5 6 7 8 9 10 0 3 6 1 9 4 11 12 13 14 15 16 17 18 19 20 12 7 21 22 23 24 25 26 27 28 29 30 2 31 32 33 34 35 36 37 38 39 40 15 40 = 2 ⋅ 20, so log(40) = log(2) + log(20) = 3 + 7 = 10.

  80. Do logarithms exist? Remember that we are working modulo a large prime p . For simplicity, I will take p = 41, since it’s small enough, and logs base 5, so that log(5) = 1. 1 2 3 4 5 6 7 8 9 10 0 3 6 1 9 4 11 12 13 14 15 16 17 18 19 20 12 7 21 22 23 24 25 26 27 28 29 30 2 31 32 33 34 35 36 37 38 39 40 15 10

  81. Do logarithms exist? Remember that we are working modulo a large prime p . For simplicity, I will take p = 41, since it’s small enough, and logs base 5, so that log(5) = 1. 1 2 3 4 5 6 7 8 9 10 0 3 6 1 9 4 11 12 13 14 15 16 17 18 19 20 12 7 21 22 23 24 25 26 27 28 29 30 2 31 32 33 34 35 36 37 38 39 40 15 10 80 = 2 ⋅ 40, so log(80) = 13, but 80 ≡ 39, and so on

  82. Do logarithms exist? Remember that we are working modulo a large prime p . For simplicity, I will take p = 41, since it’s small enough, and logs base 5, so that log(5) = 1. 1 2 3 4 5 6 7 8 9 10 0 3 6 1 9 4 11 12 13 14 15 16 17 18 19 20 12 7 21 22 23 24 25 26 27 28 29 30 2 31 32 33 34 35 36 37 38 39 40 15 19 16 13 10

  83. Do logarithms exist? Remember that we are working modulo a large prime p . For simplicity, I will take p = 41, since it’s small enough, and logs base 5, so that log(5) = 1. 1 2 3 4 5 6 7 8 9 10 0 3 6 1 9 4 11 12 13 14 15 16 17 18 19 20 12 7 21 22 23 24 25 26 27 28 29 30 2 31 32 33 34 35 36 37 38 39 40 15 19 16 13 10 But 2 ⋅ 33 = 66 ≡ 25, so we deduce that log 25 ought to be 22.

  84. Logs aren’t as simple as we thought!

  85. Logs aren’t as simple as we thought! If we continue this process, we find that we have logarithms of only half the numbers, but each one has two values, e.g. 25 seems to be 2 and 22.

  86. Logs aren’t as simple as we thought! If we continue this process, we find that we have logarithms of only half the numbers, but each one has two values, e.g. 25 seems to be 2 and 22. A fatal snag?

  87. Logs aren’t as simple as we thought! If we continue this process, we find that we have logarithms of only half the numbers, but each one has two values, e.g. 25 seems to be 2 and 22. A fatal snag? Not really. ▶ There’s a workround, which is messy, but not really difficult.

  88. Logs aren’t as simple as we thought! If we continue this process, we find that we have logarithms of only half the numbers, but each one has two values, e.g. 25 seems to be 2 and 22. A fatal snag? Not really. ▶ There’s a workround, which is messy, but not really difficult. ▶ If we’d chosen a different base, say 7, then we would have logarithms of every non-zero number.

  89. Logs aren’t as simple as we thought! If we continue this process, we find that we have logarithms of only half the numbers, but each one has two values, e.g. 25 seems to be 2 and 22. A fatal snag? Not really. ▶ There’s a workround, which is messy, but not really difficult. ▶ If we’d chosen a different base, say 7, then we would have logarithms of every non-zero number.

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