Square Root of 3

Date: 5/31/96 at 2:49:5
From: Marie Langley
Subject: Square root of 3

My granddaughter wants to know, what is the square root of 3?  

Thanks, Marie Langley

Date: 6/2/96 at 17:46:25
From: Doctor Brian
Subject: Re: Square root of 3

Hello-

In a rare situation, three of our math doctors have addressed your 
question!  Since the three responses have three very different 
flavors, you're going to get all of them.  Enjoy!

The square root of 3 is not a "nice" square root such as saying that 
the square root of 81 is 9 or the square root of 4 is 2. In fact, it 
can't even be expressed as a fraction, and if we do it as a decimal, 
then the decimal "goes on forever" because no finite decimal 
gives us the precise answer.  The number 1.732051 is very close, but 
when squared, you'll get a number just *slightly* larger than 3.
  
Numbers like this, where there isn't a fraction to describe them, and 
where the decimals don't end or even repeat, are called IRRATIONAL.  
Usually you have two options - write 'the square root of 3' as an 
answer, or round off the decimal to 1.73 or 1.732051 or as many places 
as you wish.

-Doctor Brian,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 6/2/96 at 17:47:51
From: Doctor Tom
Subject: Re: Square root of 3

The square root of three is a number that cannot be written down
exactly as a decimal.  Just as if you try to write 1/3 as a decimal, 
you can say it's approximately (but not exactly) .3333, or a closer 
approximation is .33333333, or better still, 
.33333333333333333333333333333333333333333333333333333.  But none
is EXACTLY 1/3.  You can get as close as you want, however.

So, the square root of 3 is about 1.732, or, more accurately, it is 
about 1.73205080756887729353, or, with more decimal places, you can 
get more accurate approximations.

The square root of three, unlike 1/3, never repeats its decimal 
representation.

If you would like to calculate it yourself, here's an easy way:

Make a guess (which we'll call "g"), and then calculate the average of 
g and 3/g.  Make this your first guess, and repeat the process as many 
times as you want to get an answer as accurate as you want.

It works even with a bad first guess.  Let's see what happens if my 
first guess is 1:

second guess = 1/2(1 + 3/1) = 2

third guess = 1/2(2 + 3/2) = 7/4 = 1.75

fourth guess = 1/2(1.75 + 3/1.75) = 1.732142857142857142857...

fifth guess = 1/2(1.732... + 3/1.732...) = 1.7320508100147275405,

which is already accurate to almost 8 decimal places.  The next guess 
will be accurate to almost 16 decimal places, and so on.

The reason this works is that if your guess is a little too low,
then three divided by your guess must be a little too high, and
their average will be closer to the real answer.  And vice-versa.

Of course, you have to understand some higher mathematics to see why 
the method is so ridiculously good - that 5 simple calculations, 
starting from a bad guess, give almost 8 decimal places of accuracy.

-Doctor Tom,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 6/2/96 at 17:49:40
From: Doctor Ceeks
Subject: Re: Square root of 3

I just wanted to add something regarding the square root of 3. The 
(positive) square root of 3 is:

  The unique number which multiplies itself into three...

  spans the distance between opposite corners of a cubic foot...

  compares the longer leg to the shorter leg of a 30-60-90 triangle,

  and is the area of an equilateral triangle with side 2.

-Doctor Ceeks,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/