| ||||||||||||||||||
Options
To select the options available for UltimaCalc's Algebra module, click on the Options button on the Algebra window. The following window will open.
0 ^ 0
There is some disagreement about whether or not the
expression 0^0 should be considered to have a value. If you go to
UltimaCalc's main window and calculate x^x for various values of
x all greater than zero, you will find that the value of x^x
approaches 1 as x approaches 0. However,
when x is negative, the value of x^x is defined only
when x is an integer.
This option allows you to tell UltimaCalc Algebra whether it should consider
0^0 to be undefined, or take it to be 1.
Max digits shown
There is no fixed upper size for integers. You can calculate the factorial of 1000 and view the 2568 digit result to the very last digit. Usually, however, you will not be too interested in the exact values of very large numbers. This option allows you to select the point at which Algebra UltimaCalc decides to only show an approximate value. For example, the factorial of 1000 can be displayed as "about 4.023872600770e2567".
Evaluation depth
You can assign expressions to symbols, so that when an expression is simplified (evaluated), any symbol in it which has been assigned a values will be replaced by the simplified (evaluated) version of that value, which might well contain symbols which have assigned values. The 'evaluation depth' option allows you to specify how deep to go in this repeated evaluation. The 'None' check box allows you to turn off this recursive evaluation of symbols withut having to change the maximum depth to 0. Note that the maximum depth value is limited to 20. For more information, see eval.
Square Root of -1
This option allows you to choose whether or not Algebra should handle complex numbers. Mathematicians
choose to use i to represent the object whose square is -1. In electronics, the
letter i represents current, so the letter j is used instead.
If you prefer to not use complex numbers at all, select the 'Do not use'
option. The letters i and j will then represent ordinary symbols.
Max run time
Some calculations can take quite some time to perform. This option allows you to specify the maximum time to allow. If the 'Use' check box is not ticked, there will be no time limit enforced.
Example:
To evaluate the 10th order Taylor series expansion of atan(x)
around the value x=1, you could use the expression
taylor(atan(1+x), x, 0, x, 10)
This will calculate the series almost immediately. However, if you want more
terms, and increase the number 10 to 15, you may notice a small delay.
Increasing the number further, to 16, 17, 18, 19, 20, you will find that the
running time seems to increase exponentially. Eventually, if the run
time exceeds the set limit, calculation will stop. The result to date will
be returned, and a message will warn that the result may have been
truncated. (There is in fact a better approach to finding this series -
see taylor.)
Enclose function operands inside square brackets
A complicated expression can look like a mess of brackets (parentheses). Ticking this check box will cause the operand(s) of functions to be shown inside square brackets. This helps with readability. When read in, square brackets will be treated as parentheses when they enclose a list of expressions immediately following a symbol name. This allows you to copy text from the output text box and paste it into the input text box.
Use decimals not fractions
Algebra UltimaCalc normally uses numbers in the form of integers and fractions. This allows calculations to be exact. However, there are times when decimal numbers are preferred over complicated fractions. Ticking this box makes Algebra UltimaCalc use decimals instead of fractions.
Snags with decimals
Some functions may not be able to handle decimals. An example is factors which factorises single-variable polynomials with numeric
coefficients. To handle decimals, they would have to be converted into fractions first. Another snag with using decimals is that the automatic simplifier will refrain from carrying out certain simplifications. For example, sin(1/2 * pi) is not simplified to 1 when decimals are used.
Font...
This button opens the standard font requestor, to allow you to choose your preferred font for the two text boxes in the Algebra window.