Exact Answers for Square Roots
Date: 05/11/2004 at 14:47:04 From: Scott Subject: Exact answers for square roots Hi Dr. Math - My teacher says that one of the problems with using calculators when doing square roots is that you don't get exact answers. I don't understand why. Aren't calculators accurate? - Scott
Date: 05/11/2004 at 16:09:27 From: Doctor Annie Subject: Re: Exact answers for square roots Hi Scott - Thanks for writing to Dr. Math. That's an interesting question. Yes, calculators are very accurate, but most calculators give answers in decimal form, and there are certain situations where the decimal form of an answer is not the most exact form. Square roots of non-perfect squares is a classic example. Keep in mind that the square root of a number is the number you square to make the original number. In other words, the square root of 25 is 5 because 5^2 = 25. Let's think about the square root of 10 now. It should be a little more than 3 since 3^2 is 9. If I ask my calculator what the square root of 10 is, it returns 3.1622776. Now suppose I round that to 3.16 and try squaring it. Will I get 10? (3.16)^2 = 9.9856 So 3.16 isn't the exact square root of 10, is it? What if I round my calculator's answer to 4 places instead and try 3.1623? (3.1623)^2 = 10.00014129 So that's not exact, either. How about rounding to 6 places? (3.162278)^2 = 10.00000215 Notice that the answers are getting closer to 10. But no matter how many decimal places I calculate for the square root of 10, I'll never actually get 10 when I square it. Square roots of non-perfect squares are called irrational numbers - they are decimals that continue infinitely and never go into a repeating pattern. You can get a sense of that by looking at the square root of 10 to 15 decimal places: 3.162277660168379.... If I had a calculator that could square those 15 decimal places and display the whole answer, it would still not be 10. Now, because the calculator can display only so many numbers, it eventually has to round things off to make them fit the display. So if you take the square root of 10 on your calculator and get a decimal form like the ones above and then square it, the calculator will say it's 10. But it's only getting that answer by rounding off. On the other hand, what happens if we square the square root of 10 in radical form? We know that we can multiply radicals: _ _ ___ \/A * \/B = \/A*B So squaring the square root of 10, we'd get: __ __ __ _____ ___ (\/10)^2 = \/10 * \/10 = \/10*10 = \/100 = 10 We get exactly 10! In other words, leaving the square root of 10 in radical form gives an exact answer, a number that when squared makes 10. Going to any form of decimal answer is not exact - it's an approximation, even if it's a very good approximation. One final thought - besides being more exact, working with numbers in radical form is much tidier and easier than getting involved with all those messy rounded decimal approximations. Calculators are wonderful devices, but many students rely on them way too much, choosing to do all their calculations with them, when often a non-decimal form of the number is much cleaner and certainly more accurate. Does that help? Please write back if you have further questions on this. - Doctor Annie, The Math Forum http://mathforum.org/dr.math/
Search the Dr. Math Library:
Ask Dr. MathTM
© 1994- The Math Forum at NCTM. All rights reserved.