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Walmart Services. Get to Know Us. Customer Service. Throughout the book, they play around with different equations and uncommon series related to the square root of two. They also not only tackle a few proofs of the number's irrationality, but discuss the nature of proof itself. The only real world examples they show are the diagonal of a square and as usual this leads into some discussion of the ancient Greeks and the discovery of irrational numbers in general , and how the square root of 2 relates to the business standard A4 paper and why it was created a great example that I wasn't aware of.
Again, this wasn't what I was expecting at all. But I'll give it 4 stars because it is a nice introduction to mathematical thinking, and how mathematicians go about examining and solving problems, and the reasoning behind the approaches. December 4, - Published on Amazon. This is an excellent book for anyone interested in mathematical reasoning using the value of square root of 2 as a test case.
The format is akin to a tutor-student Socratic? Anyone with basic math skills can grasp it's contents. The text does require concentration to get the most out of it. There are quite a few editing errors in this book which is a shame because it does take away from it's quality and disrupts one's train of thought. These errors were quite annoying after a while. Some of the errors are howlers like the one on pi.
square root of 2
All in all a good read. November 17, - Published on Amazon. This book takes the form of a conversation between a wise and patient teacher and a curious former mathematics student. It begins by motivating the need for a number that is the square root of two and quickly works its way to the embarrassing at least to the ancients fact that one cannot find a rational number that is exactly the square root of two.
There are many adventures connected with a sequence of rational numbers that approaches, but never quite reaches, the square root of two.
square root of 2
The student is making many of the discoveries himself rather than being given the answers by the teacher. I would jump at any further mathematics books by this author. September 24, - Published on Amazon. In school we learned that to divide by a fraction, you turn it upside down and multiply.
Yes, and what do you get in this case? Thinking time again. Is it saying that there is some fraction that when squared gives 2?
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Remember that, for us, the numerator and the denominator of a fraction are whole numbers. We never said that l and b had to be whole numbers. Which we know is impossible. Could they both be fractions? No, because when you divide one fraction by another. And a fraction squared can never give 2. So what is the equation telling us? Our mission was to discover a rectangle with the property that when it is folded along its longer side to give two smaller rectangles, the ratio of the longer to the shorter side of each of these smaller rectangles would be exactly the same as the ratio of the longer side of the original rectangle to its shorter side.
Am I right? And we succeeded in this. What have we discovered? That such rectangles exist and that the ratio of the length of the longer side of such a rectangle to the length of its shorter side must be exactly 2 : 1. It is only the ratio that matters and not the actual dimensions. But What About the Tiling Problem? Me, prove it?
A good plan. So are you about to start laying tiles? Mentally, but before I do, I had better decide on the dimensions of an individual tile. Here s stands for one length and every length at the same time. Is this the right approach? One for all. Very sophisticated. Let me suppose that I lay exactly m tiles along the long side of length 2, and exactly n tiles along the short side of length 1.
I see. Seems logical to me since m tiles, each of length s, measure exactly ms units, with n such tiles measuring ns units.
Now let me think. What did we do previously at this stage? Ah, yes. I would like to expand a little in a visual manner on the fact that twice the square of a natural number is never the square of another natural number. Twice the square of n is 2n2, and it can never be equal to another square such as m2.
Where m and n are positive integers. Because if there were an instance of this happening, then 2 would be rational. If you were a drill sergeant in command of a perfect square number of soldiers, then you could parade your squadron in a square formation. Okay, a bit of exercise would be welcome.
In perfect formation. What do you make of that? The perfect square numbers, again, are: 1,4,9,16,.
Yes, the squares of the natural numbers. But when you double this number of soldiers, I get cranky because I can no longer parade them in a square formation. Because the proof of the irrationality of 2 shows that twice a square is never a square.
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So, we might say, that here we have another consequence of the irrationality of 2? Yes, or perhaps a consequence of the proof of the irrationality of 2. I notice that the number 50, which is the size of my original enlarged squadron, is nearly a perfect square. So maybe I can cheer myself up a little bit when you come around doubling up the number of soldiers I must parade. With this particular setup for the 50 soldiers, I suppose we could say that you achieve the next best thing to the unattainable ideal of a perfect square formation. But what if, instead of having an extra soldier, you are short a soldier?
And lead from the front. So, for the sake of impartiality, we could say that one soldier above or below a perfect square number of soldiers is the next best thing to the ideal because they can be paraded in a nearperfect square formation, to coin a phrase.
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Not a bad description. Those perfect square numbers that are either one more or one less than twice another perfect square number? The parading of a doubled squadron that is one soldier in excess or short of a perfect square in a near-perfect square formation is just a way of visualizing the numerical relationship.
You are absolutely correct. Each is the other in a different guise. This means that I already know quite a few squadrons that can parade in the shape of a square and that, if doubled, can march in a near-perfect square formation.
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