1. Calculators don't have brains. It can't outsmart humans in theory. However, how does calculators find the solution for something like sin(21.3)? Also, how can it find the square root of a number?
Does it have a chart with all the possible values? Or does it use some advanced equation?

I heard some people say that it uses Taylor series, but what exactly is that? Some explanations and theories would be helpful.

2.

3. Try Wiki for the Taylor series, but more generally, a calculator is a little computer. It stores the numbers in its memory and applies algorithms to those numbers as the user pushes buttons. I don't know which algorithms different calculators use, but the square root is probably done using something like Newton's Method. Trig functions probably use the Taylor series. Actually, this question would probably be better answered in the Computer Science section, since it's really more about the algorithms rather than the math.

4. I'm mostly in agreement with Magimaster; for certain trigonometric functions however, the number of terms in the series is finite, and therefore we use a Taylor polynomial (although this is for special angles, such as the quadrantal). Square roots are likely taken through applying the quadratic formula to an equation of the form , where c is the number that we'd like to take the square root of.

5. You can't use the quadratic equation to find the square root, since the quadratic equation uses square roots. I think the usual think is to apply Newton's method (or some other root finding method) to that polynomial instead.

6. Good point. Although you should recall that Newton's method doesn't work every time, and therefore I'm skeptical that this technique is applied.

7. For the square root, it's easy enough to pick a starting point that will always give an accurate answer (something like taking the upper half of the digits). Still, they might not use Newton's method, but I'm confident they use some kind of root finder.

8. One would expect that quadratic, cubic, quintic, and higher-order roots would all be derived in the same fashion (this limits the algorithmic processes used). It's rather obvious that a root-finder is used to derive roots, but I simply find it unlikely that Newton's method is used. I guess we can agree to some extent, though.

9. Most calculators don't have a 5th root button though. For higher roots, I'd bet they use the exponential and logarithm functions and the identity . Actually, some calculators might even use that for the square root. I haven't checked which would be faster.

10. Magimaster,
Most calculator's I've seen have a "^" function in which they can raise a number to a power; raising a number to the power of one-fifth is equivalent to finding its fifth root (as you make later reference to in your post).

11. Yeah, but they don't have a 5th root button. Most have a separate square root button. Some have a separate cubic root button. For those buttons, I doubt they use the exponential to solve it since special purpose algorithms could be made to work faster.

12. If the basic equation looks like this: any number(X) powered to any number(Y)
If Y is an integer then you will have: the square, cube and so on... of X
If Y is in form 1/n (where n is: 1,2,3,4....), you will have: the square root, cube root,..nth root of X.

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