Decimal Square Roots
Date: 10/19/98 at 19:44:43 From: Ron Flitcraft Subject: Decimal answers in square roots Dr. Math, Recently my Algebra 2 teacher stated in front of the class that decimal points cannot be the answers to square roots. He said that any decimal point that is the answer for a square root will be an approximation. I doubted it and put a few numbers into my calculator. When my friend and I found a couple of numbers that didn't fill the number of the calculator space completely, our Algebra teacher admitted he was wrong. A couple of days later he put those same numbers in and multiplied them by themselves. They came close to but were not the exact number - omething like 46.999999999. I have devoted myself to finding a square root that comes out to a decimal point for an answer and checks when it is squared. Is there such a number? I have tried for quite some time and have not found one yet. If there isn't, why not?
Date: 10/19/98 at 20:01:25 From: Doctor Sam Subject: Re: Decimal answers in square roots Ron, A very interesting question, but I'm not exactly sure what you mean. I am guessing that by "a decimal point" you mean a terminating decimal like .6 rather than a decimal with infinitely many digits. (Do you see my confusion? All decimals have "decimal points" but some decimals have a fixed number of digits to the right of the decimal point while others do not.) If you mean "a fixed number of decimal digits" then there are many. Here is an easy way to create such a number: pick a decimal and square it. So for example, (1.5)^2 = 2.25 and the square root of 2.25 is exactly 1.5 But maybe you mean that you want to take the square root of a whole number, not a number with decimal digits to the right of its decimal point. If that is what you mean, then your teacher is correct: the only whole numbers that have a perfect and exact square root are numbers that are perfect squares, like 4, 9, 16, 25, 49. Any whole number that is not a perfect square will have infinitely many decimal digits in its square root. Your calculator can only display 9 or 10 digits and so rounds the answer. This gives a very good approximation of the square root, but it is only an approximation. For example, my calculator gives the square root of 2 as 1.414213562. Now if I enter (square root of 2)^2 the calculator gives 2. But if I take the displayed answer and square it I get: (1.414213562)^2 = 1.99999999999 Whole numbers either have whole number square roots or their square roots are irrational. An irrational number has infinitely many digits in its decimal expansion, so no matter how accurate your calculator or computer is it will never be able to compute all those digits and will give you only an approximation. I hope that helps. - Doctor Sam, The Math Forum http://mathforum.org/dr.math/
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