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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:


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 

Associated Topics:
High School Calculators, Computers
High School Square & Cube Roots
Middle School Square Roots

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