Computer algebra systems like FriCAS implement a very large number of mathematical algorithms. By that we mean:

mathematical

adj.

1. Of or relating to mathematics.
2. Meaning

   a Precise; exact. b Absolute; certain.

Ref: http://www.answers.com/mathematical&r=67

algorithm

n.

A step-by-step problem-solving procedure, especially an established, recursive computational procedure for solving a problem in a finite number of steps.

Ref: http://www.answers.com/algorithms&r=67

Links to other pages on related subjects:

• FriCAS Algebra
• Interval Arithmetic
• Manipulating Expressions
• Real Numbers
• Simplifying Expressions
• RischImplementationStatus
• Group Algorithms

Core algorithms in FriCAS deal with commutative algebra:

• polynomial arithmetic
• polynomial GCD and factorization
• Groebner bases and triangular system

Support for calculus uses notion of kernel: kernels represent algebraic and transcendental functions. In arithmetic transcendental kernels behave like variables, but for example have interesting derivatives. Algebraic kernels in arithmetic are simplified using defining relations, for example `\sqrt(2)' squared simplifies to 2. Algebraic simplification in charactristic 0 implies specific derivative, so there is no need to separately give derivative.