1. Clara got back her last math exam today. She is disappointed because she failed. She thought she had done well and that’s what she told her parents, but now she has to find a way to tell them totally the opposite. As she walks out the school door, she bumps into Socrates who was going in.
2. Hi Clara!
3. Hi Socrates…
4. What’s wrong? You look disappointed.
5. I just got back my math exam and I failed it. I thought I had done so well, I don’t know what happened, I knew my course by heart…
6. Don’t worry about that, it’s just one grade; you have plenty of time to catch up.
7. But I always have bad grades in math! Even when I study hard, I’m really dumb I think.
8. Look, I have to give a paper to the principal of the school, when I come back, I’ll show you that you can discover a principal that the Greeks discovered more than two thousands years ago, what do you say?
9. Ok, I’ll wait for you, but don’t be too long.
10. I’ll be back in less than two minutes.
11. Socrates comes back and sits next to Clara, with a pen and a paper in hand.
12. I am going to ask you to solve a problem that Plato talks about in one of his dialogues. The problem, simply put, is the following: how do you double the area of a square?
13. But I won’t be able to do it. I can’t even solve problems that my teacher gives me, how will I ever solve a problem more than 2000 years old?!
14. Don’t worry. You’ll solve this one.
15. Ha-ha! Ok, but I’m telling you, don’t be surprised if I can’t.
16. We’ll try and if you really can’t at least we’ll be sure that you were right. For now, all we can do is try. So there we go: if I give you this square for example (fig.1), how will find the square whose area is double the area of the original square?
17.
18. I am not sure to understand what you’re asking me, Socrates…
19. Ok. Let’s say the area of the square I have just given you is 1. How would you build the square whose area is 2, starting with the first square?
20. I can’t picture what you’re saying, Socrates. I told you before, I’m dumb!
21. No problem, I’ll state it again in a new way. So I am going to draw a second square, identical with the first one. Now, we have two identical squares whose area is 1 (fig.2). Do you agree with me so far, Clara?
22.
23. Eh…yes, it’s true, they are identical. They have the same lengths.
24. Good. Now, what is their respective area?
25. If the first one has an area of 1, then the second one will also have an area of 1.
26. Exactly, because they are identical, then their area is identical too. Now, if I add their area together, how much will I get?
27. Eh…I would say 2. Is that right?
28. Yes, how did you do it?
29. Ha-ha! I just added 1+1 and I got 2! Are you playing with me Socrates? This is not a two-thousand-year-old problem!!
30. It is, trust me. You have to be patient though; you will see how this unfolds in the end. Ok, now I take two squares of area 1, I add them together and I get an area of two. So how much is the second area greater than the first one?
31. It’s twice as big! Because we went from an area of 1 to an area of 2, so I doubled the area!
32. Exactly. Now I am going to ask you a very simple question Clara. What is the geometric shape of the first figure (fig.1)?
33. Are you kidding me again, Socrates? It’s a square!! What kind of question is that?
34. Ha-ha! Thank God one of us is patient! All right, let’s continue. So we started with a square of area 1, then we added an identical square to it and we arrived at a total area of 2. Now, what geometric shape do we get when we add the two squares together?
35. Eh…ok, this sounds like a harder question. Let’s think. If I take the first square and I stick it to the second square, like this (fig.3), then I get a… What’s the name again? I know it, wait a second… Yes, it’s a rectangle!! Am I right, Socrates?
36.
37. Excellent, Clara. So if I take two identical squares and I put them together, I get a rectangle. Now, what is the area of that rectangle?
38. Well, it’s going to be the area of the first square plus the area of the second square. So it’s 1+1=2. The area of the rectangle is 2.
39. Great. We are about to take a crucial turn in the resolution of this problem. You have been very subtle and acute in your investigation so far, Clara. Now is not the time to give up. You are on the verge of a breakthrough and it’s important that you keep your acuteness as high as possible. And don’t worry; I’ll be there to help you if you need any help.
40. Ok. I’m a little tired, but I can continue. Go ahead Socrates.
41. So the last question is the final one, and if you are able to answer it, you would have solved a problem that originated in Ancient Greece, and you will be a part of all the people, throughout history who have made the same discovery. So this is the final question: is there another way to add the two squares together, so that, in the end, you get a square instead of a rectangle? In other words, what would you need to do to the two squares that you have in order to transform them in a larger square?
42. If I understand you well, Socrates, you are asking me to take the two squares that I have and add them in a certain so as to make a new square (fig.4) instead of a rectangle like we had before (fig.3)?
43.
44. That is exactly what I am asking you Clara.
45. All right. If I take the first square and stick it on the left or right, or on top or underneath the second square, I always end up with a rectangle. The only thing that changes is its orientation (fig.5).
46.
47. It’s true…
48. Now, if I stick their vertex together instead of their sides, I get another shape and it’s still not a square (fig.6)! Well, I don’t see how you can do it.
49.
50. So what you are saying is that no matter how you stick them together side by side or vertex to vertex, you will never end up with a square, is that right?
51. That’s it.
52. I agree with you, we will never get a square in that way, even if we stick them a little bit tilted (fig.7). But, think about this question: is it the only way to add two things together?
53.
54. I don’t understand what you are saying.
55. I will give you an example. What do you when you want to put two slices of ham in your sandwich, but both slices are too large for your bread?
56. Ha-ha! I don’t see the link between the squares and the ham, but I will answer that question just to please you Socrates. In order to put the two big slices in the sandwich, I will fold them.
57. Yes, or else?
58. I can cut them too.
59. Exactly. Now, let’s go back to the squares. What else can you do to add them together?
60. By cutting them?
61. Why not?
62. But, you never said I could cut the squares!!!
63. It’s true. But did I ever say that you couldn’t cut them?
64. No, but how am I supposed to know that I can do it?
65. When I posed the problem, the only thing I said was: how do you double the area of a square?
66. But…
67. So if I don’t give you restrictions, you have to assume that you can do anything you want. The limit of what you can do in that sense is your imagination. You’re not at school anymore, remember!!
68. Ok, ok, next time I’ll know.
69. So what can you do if you know that you can cut them?
70. Well, I will take the first square, cut it in half vertically (fig.8) and stick both parts on the sides of the second square (fig.9). There you go.
72. Ok. Good idea. Now what shape is that? Is it a square?
73. Yes! Oh, no! It’s not a square; there is a part missing (fig.10). Damn it! It’s not a big deal, I’ll just cut a horizontal stripe from the part on top and a vertical stripe from the part on the right and cut them again so that I can fill the hole (fig.11)!
74.
75. Do you end up with a square?
76. No! Not again! I still have a hole, and there is a band that is coming out of the shape! Well, I can make smaller bands again then!
77. How much smaller?
78. I don’t know exactly, but I can keep on trying until it fits.
79. I might be very long and tedious. And remember, we want exactly a square, we don’t want any band coming out, everything has to be perfect.
80. Well, to know exactly I think it is impossible, maybe if I take a ruler, I might be able to do it perfectly.
81. You don’t need a ruler to do it.
82. But with the stripes like that, I can’t do it. I need a ruler or a calculator or something, otherwise I can’t know exactly.
83. Forget the ruler and calculator, all you need for this problem is your brain. Obviously if we are unable to solve the problem where we are right now, we made a mistake in our path earlier. Let’s go back to the time when we decided to cut the squares. Do you remember how you suggested to cut them?
84. Yes, I said that I should cut the first square in half vertically and…
85. Yes! Exactly. Now, all we did from here didn’t seem to work out. Maybe we need to change the way we cut it. How about instead of vertically, you cut it horizontally?
86. Come on Socrates! We’re going to have the same problem, the shape is going to be identical, instead of being vertical, it’s going to be horizontal, so we’ll run into the same problem.
87. Ok, vertically doesn’t work. Horizontally doesn’t work. What are we left with that we didn’t try?
88. Yes! I know, let’s cut it in diagonal!
89. Ok, go ahead.
90. I take the first square, I cut it in half diagonally (fig.12). Then I take the two parts and I stick them on two sides of the second square (fig.13). But it doesn’t work! It’s not a square! And no matter on what side I stick the two parts, I never end up with a square, it just gives me another strange shape.
91.
92. Think about your sandwich, Clara. You only cut one square out of the two.
93. I should cut the second square too?
94. Why not?
95. Ok, let’s do it (fig.14). It’s becoming exciting Socrates! It makes me think of when I used to play with Lego or when I do some puzzle. I have four identical shapes and I have to assemble them to form a new square, let’s see…
96.
97. Your turn:
Can you do it? You can take a piece of paper, draw two identical squares and cut them. Then cut them both diagonally, and try to build the greater square.
Additional paths of investigation: what is the length of the sides of your new square (its area is 2)? What is the difference between this number and the natural numbers (1, 2, 3, 34, 746, etc.) and the rational numbers (0.5, 13/2, 1/3, etc.)?
Would you have been able to measure that length with your ruler, like Clara suggested?
In answering these questions, you have just discovered the nature of irrational numbers.
Also, you have entered into the nature of space and quantity: the square root of 2 doesn’t exist as an independent straight line, but only as the result of a unitary quantity doubly extended.
La Monade 