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Expressions and Symbols
Expressions basically consist of symbols, numbers and functions combined using the standard mathematical operations of addition, subtraction, multiplication, division, and raising to powers.
Symbols consist of a group of up to 15 consecutive letters, digits, and underscore characters, provided that the first character is not a digit. Algebra UltimaCalc distinguishes between upper and lower case. All pre-defined functions, and almost all pre-defined symbols, are in lower case.
Example:
ab + Ab + aB + AB is an expression containing four distinct symbols. The automatic simplifier tries
to put symbols into alphabetical order, with capital letters before lower case letters, so it changes the order of the terms to AB + Ab + aB + ab
The multiplication operator, *, should never be omitted. Mathematical texts usually omit this, leaving the operation to be implied by the context. This approach usually works well enough when all symbols are of only a single letter, but it can lead to confusion and error in some cases. Algebra UltimaCalc therefore complains if an operator is missing.
The exponentiation operator may be entered as either ^ or as **, whichever is more convenient. When the exponent is a fraction, take care to enclose it inside parentheses.
Example:
2^3 + 3**2 simplifies to 17
27^2/3 is simplified to 243 whereas 27^(2/3) is 9, though it does
not get simplified.
-3**4 is simplified to -81 whereas (-3)**4 gives 81
2^3 - 3**4 is simplified to -73 whereas 2^3 + (-3)**4 gives 89
5^3^2 means the same as 5^(3^2) which is 1953125,
whereas (5^3)^2 gives 15625
Special Symbols
Some symbols are given special treatment:
pi represents the ratio of the perimeter of a circle to its diameter. It is not permitted to change the value of pi.
last is a symbol whose value is always that of the last expression returned, or Undefined if there is none.
Undefined is a symbol which represents an invalid expression,
for example the result of 1/0.
Infinity and Minus Infinity only appear occasionally. If they appear
inside an expression, rather than as its result, they will be converted to Undefined. For
example, the result of finding the degree of the zero
polynomial, degree(0, x) is Minus Infinity, by definition, but
calculating 1 + degree(0, x) will give the result Undefined.
Assignment
The assignment operator := (a colon followed by an equals sign) allows you to
specify symbol values and define functions.
Expression Types
UltimaCalc Algebra has some functions designed specifically for use with certain types of expressions.
A polynomial is a sum of products of powers of a symbol with coefficients. There may be only a single symbol, with the coefficients being all numeric, or the coefficients may be expressions containing other symbols as well as numbers.
A rational expression is a ratio of two polynomials. To simplify rational expressions, use ratsimp
A trigonometric expression is an expression that consists largely of trigonometric or hyperbolic functions.
An exponential expression is an expression that consists largely of exp functions.
Click on the relevant links above for more information.