By David Flannery
The sq. root of two is an interesting quantity – if rather less recognized than such mathematical stars as pi, the quantity e, the golden ratio, or the sq. root of –1. (Each of those has been commemorated by means of at the least one contemporary book.) right here, in an imaginary discussion among instructor and scholar, readers will examine why v2 is a crucial quantity in its personal correct, and the way, in complicated out its targeted traits, mathematicians received insights into the illusive nature of irrational numbers. utilizing not more than simple highschool algebra and geometry, David Flannery manages to express not only why v2 is interesting and important, yet how the complete firm of mathematical pondering will be performed out in a discussion that's resourceful, fascinating, and interesting. unique and informative, The sq. Root of two is a unique advent to the excitement and playful fantastic thing about mathematical pondering.
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Extra resources for The Square Root of 2
That’s right. You also said earlier that rectangles of these dimensions are the only ones having this property. I did. Only those rectangles, the length of whose long side to the length of the short side are in the ratio 2 : 1, have this property. I’d be interested to see why. It will require some algebra. I thought it might! Let us begin with a labeled diagram of a rectangle: Here l stands for the length of the longer side and b stands for the length of the shorter side. I suppose one side has to be longer than the other for there to be any hope of success.
So we are witnessing a connection between a term in this sequence and the paper industry? Yes. You are saying that these dimensions are purposely chosen so that the ratio of the longer side of an A4 sheet to its shorter side is very nearly in the ratio 2 : 1. As I said, ideally, they should be in this exact ratio but, as we now know, this cannot be achieved with each side being an exact natural number of millimeters in length. No doubt there are good reasons for insisting on trying to achieve this ratio.
The calculation 2 Ê 140 ˆ = 19600 Ë 99 ¯ 9801 19602 - 2 =! 9801 2 =29801 2 of 2. shows that the square of 140 is within 9801 99 Well done! I see you haven’t forgotten the idea of that trick we used so well earlier. It took me a while to recall that I should write 19600 = 19602 - 2 so that I could divide 9801 into 19602 to get 2 exactly. You’ll be quite the old hand in no time! You’ll give me a big head. Anyway, we can say that the ratio of the long side of an A3 page to its shorter side is also approximately in the ratio 2 : 1.