0.4 Using FriCAS as a Symbolic Calculator

In the previous section all the examples involved numbers and simple functions. Also none of the expressions entered were assigned to anything. In this section we will move on to simple algebra (i.e. expressions involving symbols and other features available on more sophisticated calculators).

0.4.1 Expressions Involving Symbols

Expressions involving symbols are entered just as they are written down, for example:

xSquared := x^2

\[x^2\]

_{Type: Polynomial Integer}

where the assignment operator := represents immediate assignment. Later it will be seen that this form of assignment is not always desirable and the use of the delayed assignment operator == will be introduced. The type of the result is Polynomial Integer which is used to represent polynomials with integer coefficients. Some other examples along similar lines are:

xDummy := 3.21*x^2

\[3.21 x^2\]

_{Type: Polynomial Float}

xDummy := x^2.5

\[{{x} ^ {2}} \ {\sqrt {x}}\]

_{Type: Expression Float}

xDummy := x^3.3

\[{{x} ^ {3}} \ {{{\root {{10}} \of {x}}} ^ {3}}\]

_{Type: Expression Float}

xyDummy := x^2 - y^2

\[x^2-y^2\]

_{Type: Polynomial Integer}

Given that we can define expressions involving symbols, how do we actually compute the result when the symbols are assigned values? The answer is to use the eval function which takes an expression as its first argument followed by a list of assignments. For example, to evaluate the expressions XDummy and {xyDummy} resulting from their respective assignments above we type:

eval(xDummy,x=3)

\[37.540507598529552193\]

_{Type: Expression Float}

eval(xyDummy, [x=3, y=2.1])

\[4.59\]

_{Type: Polynomial Float}

0.4.2 Complex Numbers

For many scientific calculations real numbers aren’t sufficient and support for complex numbers is also required. Complex numbers are handled in an intuitive manner and FriCAS, which uses the %i macro to represent the square root of -1. Thus expressions involving complex numbers are entered just like other expressions.

(2/3 + %i)^3

\[-4627+13\,i\]

_{Type: Complex Fraction Integer}

The real and imaginary parts of a complex number can be extracted using the real and imag functions and the complex conjugate of a number can be obtained using conjugate:

real(3 + 2*%i)

\[3\]

_{Type: PositiveInteger}

imag(3+ 2*%i)

\[2\]

_{Type: PositiveInteger}

conjugate(3 + 2*%i)

\[3-2\,i\]

_{Type: Complex Integer}

The function factor can also be applied to complex numbers but the results aren’t quite so obvious as for factoring integer:

144 + 24*%i

\[144+24\,i\]

_{Type: Complex Integer}

0.4.3 Number Representations

By default all numerical results are displayed in decimal with real numbers shown to 20 significant figures. If the integer part of a number is longer than 20 digits then nothing after the decimal point is shown and the integer part is given in full. To alter the number of digits shown the function digits can be called. The result returned by this function is the previous setting. For example, to find the value of π to 40 digits we type:

digits(40)

\[20\]

_{Type: PositiveInteger}

%pi::Float

\[3.141592653589793238462643383279502884197\]

_{Type: Float}

As can be seen in the example above, there is a gap after every ten digits. This can be changed using the outputSpacing function where the argument is the number of digits to be displayed before a space is inserted. If no spaces are desired then use the value 0. Two other functions controlling the appearance of real numbers are outputFloating and outputFixed. The former causes FriCAS to display floating-point values in exponent notation and the latter causes it to use fixed-point notation. For example:

outputFloating(); %

\[0.3141592653589793238462643383279502884197E1\]

_{Type: Float}

outputFloating(3); 0.00345

\[0.345E-2\]

_{Type: Float}

outputFixed(); %

\[0.00345\]

_{Type: Float}

outputFixed(3); %

\[0.003\]

_{Type: Float}

outputGeneral(); %

\[0.00345\]

_{Type: Float}

Note that the semicolon ; in the examples above allows several expressions to be entered on one line. The result of the last expression is displayed. remember also that the percent symbol % is used to represent the result of a previous calculation.

To display rational numbers in a base other than 10 the function radix is used. The first argument of this function is the expression to be displayed and the second is the base to be used.

radix(10^10,32)

\[9A0NP00\]

_{Type: RadixExpansion 32}

radix(3/21,5)

\[0.\overline{032412}\]

_{Type: RadixExpansion 5}

Rational numbers can be represented as a repeated decimal expansion using the decimal function or as a continued fraction using continuedFraction. Any attempt to call these functions with irrational values will fail.

decimal(22/7)

\[3.{\overline {142857}}\]

_{Type: DecimalExpansion}

continuedFraction(6543/210)

\[31 + \frac{1}{+ \frac{1}{+ \frac{1}{+ \frac{1}{3}}}}\]

_{Type: ContinuedFraction Integer}

Finally, partial fractions in compact and expanded form are available via the functions partialFraction and padicFraction respectively. The former takes two arguments, the first being the numerator of the fraction and the second being the denominator. The latter function takes a fraction and expands it further while the function compactFraction does the reverse:

partialFraction(234,40)

\[6 -{3 \over {{2} ^ {2}}}+{3 \over 5}\]

_{Type: PartialFraction Integer}

padicFraction(%)

\[6 -{1 \over 2} -{1 \over {{2} ^ {2}}}+{3 \over 5}\]

_{Type: PartialFraction Integer}

compactFraction(%)

\[6 -{3 \over {{2} ^ {2}}}+{3 \over 5}\]

_{Type: PartialFraction Integer}

padicFraction(234/40)

\[{117} \over {20}\]

_{Type: PartialFraction Fraction Integer}

To extract parts of a partial fraction the function nthFractionalTerm is available and returns a partial fraction of one term. To decompose this further the numerator can be obtained using firstNumer and the denominator with firstDenom. The whole part of a partial fraction can be retrieved using wholePart and the number of fractional parts can be found using the function numberOf FractionalTerms:

t := partialFraction(234,40)

\[6 -{3 \over {{2} ^ {2}}}+{3 \over 5}\]

_{Type: PartialFraction Integer}

wholePart(t)

\[6\]

_{Type: PositiveInteger}

numberOfFractionalTerms(t)

\[2\]

_{Type: PositiveInteger}

p := nthFractionalTerm(t,1)

\[-322\]

_{Type: PartialFraction Integer}

firstNumer(p)

\[-3\]

_{Type: Integer}

firstDenom(p)

\[22\]

_{Type: Factored Integer}

0.4.4 Modular Arithmetic

By using the type constructor PrimeField it is possible to do arithmetic modulo some prime number. For example, arithmetic module 7 can be performed as follows:

x : PrimeField 7 := 5

\[5\]

_{Type: PrimeField 7}

x^5 + 6

\[2\]

_{Type: PrimeField 7}

1/x

\[3\]

_{Type: PrimeField 7}

The first example should be read as:

Let x be of type PrimeField(7) and assign to it the value 5

Note that it is only possible to invert non-zero values if the arithmetic is performed modulo a prime number. Thus arithmetic modulo a non-prime integer is possible but the reciprocal operation is undefined and will generate an error. Attempting to use the PrimeField type constructor with a non-prime argument will generate an error. An example of non-prime modulo arithmetic is:

y : IntegerMod 8 := 11

\[3\]

_{Type: IntegerMod 8}

y*4 + 27

\[7\]

_{Type: IntegerMod 8}

Note that polynomials can be constructed in a similar way:

(3*a^4 + 27*a - 36)::Polynomial PrimeField 7

\[{3 \ {{a} ^ {4}}}+{6 \ a}+6\]

_{Type: Polynomial PrimeField 7}