Amaze all you friends by extracting square roots of irregular numbers in your head BEFORE they've punched the numbers into a calculator. This apparently amazing trick, and believe me whem people see it done they cannot believe their ears, is based upon a simple piece of geometry and an elementary bit of algebra. Consider the square shown above:
For the square, Area =XxX which is known: this is the number we wish to extract the square root of.
Also, if the width of the strip at the end is denoted by D:
1) AxA +2(AxD) =AxA +2AD (very nearly, the error being the second order shaded corner DxD)
2) Also, X=A+D
Rewriting 1) for D:
3) D=(XxX-AxA)/2A
And combining 2) and 3):
4) X=A + ((XxX)-(AxA))/2A
In equation 4) XxX is known (it is the given number), and A is estimated: the better the estimate, the more accurate the answer obtained (which is usually better than 1%). For numbers less than ten, multiply by a hundred first and divide the answer by ten: for numbers greater than a thousand, divide by a 10En first & multiply the answer by 10E(n/2).
To put the process in words - and it is very simple to execute as a piece of mental arthmetic once you have absorbed the form by practicing on a few examples - the approximate square root of a number is the estimated square root plus (the number minus the square of the estimate) divided by twice the estimate.
Example: what is the square root of 67?
guess (A) less than answer = 8
AxA = 64
X=8 + (67-64)/16
Answer 8-3/16 (8.1875)
Calculator answer: 8.1853 (error is 0.027%)
Example: what is the square root of 67?
guess (A) more than answer = 9
AxA = 81
X=9 + (67-81)/16
Answer 8-2/16 (8.125)
Calculator answer: 8.1853 (error is 0.737%)
Example: what is the square root of 150?
guess (A) less than answer = 12
AxA=144
X=144 + (150-144)/24
Answer 12-6/24 (12.25)
Calculator answer: 12.2474 (error is 0.0208%)
Tip: if you are going for speed, always give the answer as a fraction (eight and three-sixteenths, eight and a quarter, twelve and a quarter in the above examples) since it will often take you far longer to calculate the decimal equivalent than the square root!
