Find a quadratic approximation to :

Find algebraic approximants of any order:

There is no simple quadratic approximation to N:

At degree 4, there is a simple answer, which turns out to be equal to :

Machine precision is not enough to recover the Root object equal to :

30 digits of precision suffice to recover the exact value here:

When given exact input, RootApproximant uses machine-precision approximations:

RootApproximant works with complex numbers:

Find successive algebraic approximations to :

A complicated radical expression:

Use FindRoot to find the real root near 2.5:

Use RootApproximant to find an algebraic number close to the root:

Check whether the result is a root of the original expression:

RootApproximant gives a Root object close to a given algebraic number:

The Root object found may not be exactly equal to the input:

Use RootReduce to find exact Root object representations of algebraic numbers:

Specifying a linear polynomial effectively finds a rational approximation to x:

Rationalize also gives a rational approximation, but it need not be the same:

Interestingly, the approximations can be found among continued fraction convergents:

Use LatticeReduce to recognize linear combinations of more general functions:

The final relationship :

Recognizing an algebraic number may require using higher precision:

The result is not equal to a:

Providing the correct degree improves the chances of recognizing an algebraic number:

A penalty may be used to lower the degree; here it does not help in recognizing the number:

Using a higher-precision approximation allows the algebraic number to be recognized: