To estimate the

square root of a number using only simple

arithmetic, the first-order

Taylor series of the square root

function provides a convenient method. As any

calculus student knows, the first-order Taylor expansion around

*x*^{2} is given by

sqrt(*x*^{2} + *a*) ~ *x* + *a* / 2*x*

In practice, this can provide a good, quick

estimate of a number's square root, which we illustrate by calculating the square root of 40. Obviously the nearest integer square is 36, so in the equation above we can put

*x* = 6 and

*a* = 40-36 =4. This gives sqrt(40) as approximately 6+4/12 = 6.3333..., whereas the true value is around 6.324 (note that this method always yields a slight over-estimate, since we're approximating a concave-down curve by its

tangent at

*x*^{2}).

Of course, if the number in question is slightly below the nearest integer square, we simply use a negative value of *a* in the formula above; for example, to find the square root of 250 we note that 16^{2} is 256, so a good approximation is 16 - 6/32 = 15.8125, as compared to the true value of 15.81138...