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Methinks the good doctors have been resting on their carriage return a bit too much.

I would also like to note that calculators generally use series approximations to calculate each significant figure for complex calculations. So for example, when you type in sin(2.345) the calculator does the Taylor series for sine which is sin(2.345) = 2.345 - (2.345)^3/3! - (2.345)^5/5! - (2.345)^7/7! - ...As you can see you could continue this with odd integers and the longer the series the more accurate the answer becomes with more significant figures.

Yup.Actually a lot of calculators are computers, they were one of the very early users of microprocessors. The very simplest ones used just hard-wired electronics and not computers, but were often incapable of doing more complicated operations like sin and cos. Once you need to do calculations like that, then using a microprocessor to do it in software is much more desirable.

(Am I the only one here who remembers Olivetta mechanical calculators and Friden electronic calculators, both of which were plugged into the wall?)

The definition of an algorithm is one which always halts after a finite number of steps.Taylor series and Newton's method rarely converge to the "correct" answer, and then stop (except for e^0=1, or if you guess the exact answer to start Newton's method). With series which converge quickly, you can stop when each successive term adds a number smaller than the smallest digit displayed on the calculator.More troublesome are those series which converge very slowly - series which generate pi are notorious for this; even after a lot of calculations, the answer is very inaccurate. With each calculation, rounding errors increase.For series like this, it is common to rearrange the equation so it converges with very few calculations, but perhaps over a reduced range of values.Suitable mathematical relations for the sine function (angles in radians):sin(x)=sin(x+2*pi)cos(x)=sin(x+pi/2)Reducing the value of x closer to zero with methods like these means that you need fewer calculations to obtain the same accuracy.