It is natural for students to ask how to find the square root of a number, much as they would want to calculate, say, the quotient of two numbers through long division.
We endeavor to work out the square root of six with simple arithmetic.
Find sqrt 6
sqrt 6 = sqrt (6 x 4/4)
= sqrt ([6/4] x 4)
= sqrt (6/4) x sqrt (4)
= 2 sqrt (6/4)
= 2 sqrt 1.5
We seek a number whose square is reasonably close to, but does not exceed, 1.5
1.2 x 1.2 = 1.44, so
2 sqrt 1.5 = 2 sqrt (1.5 x 1.44/1.44)
= 2 sqrt (1.5/1.44 x 1.44)
= 2 sqrt 1.5/1.44 x sqrt 1.44
= 2 x 1.2 x sqrt 1.5/1.44
Carry out the division:
1.04166666666.... = 144 |150.000000000000
It is immediately evident that a number whose square will be reasonably close to 1.041666...., but does not exceed it, is 1.02. That's because the square of 1 plus a small number is approximately 1 plus twice the small number. Conversely, the square root of one plus a small number is 1 plus half the small number. This is a rule of thumb, but the analysis could be made more rigorously.
Thus, 1.02 x 1.02 = 1.0404.
2 x 1.2 x sqrt 1.5/1.44
= 2 x 1.2 x sqrt 1.0416666....
= 2 x 1.2 x 1.02 x sqrt 1.04166666.../1.0404
= 2 x 1.2 x 1.02 x sqrt 1.001217..etc
By our rule of thumb, we can immediately say that 1.0006 is a number whose square will be close to, but will not exceed, 1.001217..etc
Thus, we have:
2 x 1.2 x 1.02 x 1.0006 sqrt 1.001217..etc/1.00120036
Let's see how close the numbers outside the radical sign approach sqrt 6:
2 x 1.2 x 1.02 x 1.0006 = 2.4494688.
Remembering that the square of our result should be less than 6 by a small amount, we carry out the multiplication, which yields:
By continuing our calculations, we could have found a number that differs even less from the square root we seek.
This method, which I developed about a year ago, is accessible to those with rudimentary mathematical skills, including long division, multiplication of fractions, the concept of square root, and the rules governing the square roots of products.
The method itself is transparent in terms of the way it works, dependent as it is on the basic rules of mathematics, so it should not be confusing to students. I call this method "Beating the square root out of the radical sign with the number one."
This method is certainly not in wide use, and one would be hard pressed, as I have been, to find it in old text books or anywhere on the internet. I have heard claims that this method is the same as Newton's method, or the Babylonian method, etc. etc., but no clear demonstration of such claims. This method is based on ratios, whereas other methods generally rely on differences. The same approach may be used to also find cube roots and fourth roots, etc.