above equation is the clue required to solve the mystery. Let's start with a simple guess for 3, say 2.
Burkard Polster teaches mathematics at Monash and is the university's resident mathemagician, mathem.
, to the extent it can. The simple and widely applicable procedure that we have outlined is known as the method of bisection. It is just one method to approach square roots through fractions, and not the best. The Babylonians employed a Womens Adidas Gazelle Og Trainers Sale
. He was right: once we leave the cosy company of fractions, the world of numbers becomes very, very strange. Let's step back and consider a much simpler example. If we ask about 4, (almost) all secondary students will correctly answer that it is 2. If we then go on to ask why, a depressingly large number will be stumped. However, most will answer correctly that it's because 22 = 4. That easy example makes clear the one fundamental fact about 3: it is the (positive) number that solves the equationThis equation still doesn't tell us what 3 is, but at least it tells us what 3 does. The equation provides a classic example of an inverse problem, where the process and the output are known, but the input X is a mystery to be solved. In our case the process is squaring and the output is 3. Writing 3 doesn't solve the mystery; it is just introducing a new name, an alternative to X, for our mystery input. (The same mystery is hidden in the equilateral triangle above. )The Adidas Gazelle Dark Red
much better technique, a thousand years before Pythagoras came along. However, the mathematical past is a foreign country: they do things differently there. Do we really expect today's students to go through all this? Yes, at least on occasion. Approximating by fractions is not just important for understanding square roots. It is the key to understanding logarithms and trigonometry and the whole lot. Adidas Gazelle Khaki Womens
That's obviously too big, since 22 = 4, so we'll have to go smaller. A simple way to proceed is to make a second guess that we know is too small, say 1, and then average our two guesses. That gives a new guess of 3/2. Then, 3/2 x 3/2 = 9/4, which is less than 3, indicating that 3/2 must be too small. So, we average 3/2 and 2, to give a new guess of 7/4. We can continue: squaring 7/4 Adidas Gazelle Og Leather confirms that it is too big, and so our next guess is the average of 3/2 and 7/4, namely 13/8. And on we go: 27/16, 55/32, 111/64, 221/128. Of course this can go on forever, and in some sense it has to. 3 can only be properly understood in terms of an infinity of fractions that almost but don't quite solve the defining equation. It's not very satisfying but it's the best we can do. Such approximating fractions are also the key to explaining 1. 732.
It is at the heart of any proper contemplation of our real number system. We cannot expect today's students to spend five minutes calculating fractions every time they need a square root, but we must acknowledge that something has been lost. There was a reason for the use of such methods and a meaning attached to those methods. The reason may no longer exist but the necessity of meaning remains.
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