TOP > Report > Creative Café Vol 7 Grasp it with an approximate number 3/3

Creative Café Vol 7 Grasp it with an approximate number 3/3pict


T:Sorry, that's really basic! But how did the very first person to work out 'Pi' actually do so?
Murakami:that was by comparing the circumference and the diameter.
T:so they measured the lengths
Murakami:yes. But if you say measurement, you introduce an even more fundamental principle-what actually is length? With a straight line, we can understand the concept of length. Therefore, by segmenting the circle into similar shapes of a pentagon, hexagon, decagon and so on we can measure the lengths, and as the number of sides increases we compare the ultimate limit with the diameter. This is formally called the Greek letter 'pi'. However we can't measure the length of distorted objects. Therefore if we think about Pi, it wasn't decided as 3.14 from the beginning; when people first wanted to calculate the ratio between the circumference and the diameter, they first carried out the calculations on the similar polygon shapes. Drawing a polygon to fit inside the circle, and drawing one to fit from the outside, clearly the one inside is shorter, and the one outside longer. The first idea to measure Pi was thus to gradually make this difference smaller and smaller.
U:in Japan, when I was in primary school, we just learned that Pi was approximately 3.14, this wasn't just a first value. Abroad, I believe Pi is taught accurately at Middle School.
Murakami:in Japan, what is taught in Primary fourth and fifth year in the initial textbooks is, let's try and measure the circumference of a drawn circle, and through the actual measurement, the teacher can say "hey look the circumference is 3.14 times the diameter isn't it". Thus you're not actually taught that Pi is 3.14. It appears there is a misunderstanding but that's not really the case. Provisionally, you are comparing the circumference and diameter of the circle, so the version of Pi which emerges is just that comparative number. In this case if we talk about Pi, we learn that it can be approximately three, or approximately 3.14. Or around that figure. But primary school teachers include many who dislike arithmetic, and see it as troublesome, so teaching in many cases becomes " remember 3.14" or " if you multiply the diameter by 3.14 you will get the circumference of the circle". It is not actually such a bad thing.
W:as with this example, subtracting and adding according to rules, in fact not just addition and subtraction but all four arithmetic operations carried out according to rules, provides a pattern where relationships are expected, like Pi to be expressed as a fraction (1/number).
Murakami:that is broadly true. Perhaps, you might say you were not taught, or that that wasn't arithmetic and so on.
W:from the old times?
Murakami:yes from long ago. In the era where there were no calculators, whatever we can say comparatively, what remains are accurate calculations in many tens of columns. As has been said before, the length of a polygon can be calculated by the four arithmetic operations.

Mathematics that I can appreciate

b: just a little earlier, you said that fractions were beautiful but that decimals were not, and sometimes gave you a bad feeling. I believe there are some who will share that feeling, but others who don't, also some exactly like you.
Murakami:Yes, yes.
b:as a teacher, how many people do think share the view you expressed before that mathematics can be thought of as beautiful?
Murakami:that's a good question. I think the majority don't think about it at all. Therefore to explain what is the square root of 2 would take considerable time. I always thought this, but it would be really good if mathematics could be appreciated in the same way as art. The square root of two is part of that I think, but it is really different from a picture. Therefore, for root 2, we have to express as a fraction, and can compare that with expressing it as a decimal, how many would share the feeling which was prettier? Perhaps no one. Basically this is not just Japan, but the trend in the whole World is, I believe, for arithmetic and science etc to gradually becoming avoided. Whatever one says, because it is difficult, and because it's troublesome. This may well be the root of "just memorise it".
To really understand or make someone understand is really troublesome, so some may prefer just to make them memorise. However with just a little more patience, if only we could help people appreciate and return to seeing the beauty of numbers, the whole mood would change. Therefore the previous example of how can we express the square root of minus one; clearly there must fundamentally be an answer, but in my primary school days, as I've said before, some strange people expressed their enjoyment in a circulating 'Oh' when expressing fractions as decimals. Why? Because from these people's point of view they thought 'Oh'! Therefore if they can return to that simple time when we thought it was interesting or beautiful, we could perhaps change. I think I've perhaps slipped from the point of the question but if I tried to answer the question in this vein, it is not such a hopeless place to start. I think we mathematicians must try and increase our efforts in this direction.

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