Alternately, computer algebra systems can operate on irrational quantities exactly by manipulating symbolic formulas for them (such as \sqrt{2}, \arctan 5, or \int_0^1 x^x \,dx) rather than their rational or decimal approximation. But exact and symbolic arithmetic also have limitations: for instance, they are computationally more expensive; it is not in general possible to determine whether two symbolic expressions are equal (the constant problem); and arithmetic operations can cause exponential explosion in the size of representation of a single number (for instance, squaring a rational number roughly doubles the number of digits in its numerator and denominator, and squaring a polynomial roughly doubles its number of terms), overwhelming finite computer storage.
