Square Roots and Calculator Rounding

Date: 01/12/99 at 13:07:30
From: The dancing bears classroom
Subject: Square root of 43

Our teacher asked us to find the square root of 43. We came up with 
three answers: 6.557438524, 6.5574386, and 6.5574385. We each tried 
squaring them on our calculators, and some people's calculators said 
that these were 43 when squared and some said these weren't 43 when 
squared. Can you explain why the calculators showed different answers? 
Are all of them right or are they wrong? If so which ones are wrong? 
What is the real answer?

Thank you, 
The Dancing Bears Sixth Graders

Date: 01/12/99 at 13:14:27
From: Doctor Rob
Subject: Re: Square root of 43

Thanks for writing to Ask Dr. Math!

The answer is approximately 6.5574385243020006523441... How you round
this off to eight or ten decimal significant figures is the issue, 
isn't it?

The square root of 43 is not a terminating decimal, nor does it
eventually repeat (like 2/11 = 0.18181818...). The calculator only
returns an approximate answer, rounded or truncated to as many figures
as the display will show. When the approximation is squared, you 
couldn't possibly get 43, and when the square is rounded off to fit in
the display, you might get 43 or not.

None of the answers is exactly right, and none could possibly be right,
because a calculator can't give you infinitely many decimal places. On 
the other hand, if you have only eight or ten places in the display, 
all the calculators did a pretty good job of approximating to that 
degree of accuracy.

- Doctor Rob, The Math Forum
  http://mathforum.org/dr.math/