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/