Square Roots of Numbers between 0 and 1Date: 07/08/99 at 11:00:32 From: eric s adair Subject: Square Roots of Numbers between 0 and 1 Why do square roots always come closer to the number one? If I take the square root of 5 and keep taking the square root of the square root of the square root etc., it eventually equals one. Take the square root of .5 and keep taking the square root of the square root etc., it also eventually equals one. One number gets smaller until it equals one and one number gets bigger until it equals one. Ouch. Date: 07/08/99 at 13:02:47 From: Doctor Rick Subject: Re: Square Roots of Numbers between 0 and 1 Hi, Eric. I don't personally find this painful, but it is interesting. I will show you a graphical way to think about what you are doing. This graph shows the functions y = x (the straight line) and y = sqrt(x) (the curve). y / | / | / | / | / | / | / | / ............... * | /..: * : | /.: * : | * : | *.../ : : | * : / : : | *...../ : : | * : / : : | * : / : : | * / : : | * / : : : | * / : : : | * / : : : |* / : : : | / : : : +-----------------------+---------------------------------- x 0.5 1 2 Your operation of taking the square root of a number repeatedly is depicted graphically by the dotted lines. Start at 2 and take its square root: the height of the dotted line at x = 2 is y = sqrt(2). Set x to this value, that is, x = y. This corresponds to moving horizontally until you hit the line y = x. Then take the square root of x again: move vertically until you hit the curve, y = sqrt(x). Repeat this operation and you can see that the horizontal and vertical lines bounce around between the straight line and the curve, getting closer and closer to 1. If you start at x = 0.5, the same thing happens, but this time you move right towards 1. You can see that the reason for this convergence is that 1. y = sqrt(x) is above y = x for x between 0 and 1, and it is below y = x for x greater than 1. 2. The slope of y = sqrt(x) is positive, that is, sqrt(x) increases as x increases. You always converge on 1 because that is where the lines y = x and y = sqrt(x) cross. Are you surprised that the square root of a number less than 1 is greater than the number? The graph shows that it is. You can also see it this way: If you multiply a number by a number less than 1, the result is smaller than the original number. For instance, 8 * 0.5 = 4, which is less than 8. If you square a number less than 1, you are multiplying it by a number less than 1 (namely, itself), so the result is less than the original number. For instance, 0.5 * 0.5 = 0.25, which is less than 0.5. The square root of a number is the number whose square is the first number. If the number is less than 1, then the original number (which is the square root squared) must be less than the square root. Thus, the square root is greater than the original number. For instance, the square root of 0.25 (which is 0.5) is greater than 0.25. Does either of these approaches to the question shed more light on it for you? Maybe they raise more questions - if so, feel free to ask them! - Doctor Rick, The Math Forum http://mathforum.org/dr.math/ |
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