MathAction changed pages

[Nature of expressions]

Tue, 21 Jul 2026 04:45:50 +0000

++added: Important property of Expression(Integer) is that FriCAS does not enforce having differential field. FriCAS allows dependent root, which as shown above lead to zero divisors. FriCAS allows both 'exp(x)^2' and 'exp(2*x)' in a single expression and by default treats them as different quantities. Note that using differential calculus one can prove that 'exp(x)^2' and 'exp(2*x)' are equal. This means that when both are used in a single expression then laws of differential field my by violated. FriCAS has 'abs' and 'diracDelta' and few other expressions of similar nature. In arithmetic such expression are treated as transcendental quatities. This is sound in sense that succesfull computation can not produce false equality. But many such expression does not correspond to things in analitic world. FriCAS does not have complete knowledge about such expressions, so in some cases can not discover when in analytic world given expression is 0. Worse, such unrecognized 0 can appear in the denominator. If some later transformation manages to simplify the denominator, it may lead to division by 0. Computations which depend on having differential field usually call 'normalize' or similar function. This removes various redundancies which for transcendental elementary functions means that computations with transformed expression will work within a differential field. In principle algebraic quantities (like square roots) and 'abs' can be treated by considering possible cases, but currently in few simple cases FriCAS takes single choice and other cases are not handled at all.

[Division by zero during evaluation]

Tue, 21 Jul 2026 04:16:30 +0000

??changed: -expression domain is doing this auitomatically: expression domain is doing this automatically: ??changed: -if final result is sensible (in partucular does not lead to division by if final result is sensible (in particular does not lead to division by

[AxiomSources]

Wed, 01 Jul 2026 21:03:43 +0000

??changed: -For up-to-date information about Axiom see http://axiom-developer.org Axiom is no longer actively developed. Last version is available as a Debian package. Copy of old Axiom website is preserved at https://www.nongnu.org/axiom

[FriCAS Library Examples]

Wed, 01 Jul 2026 20:37:14 +0000

??changed: - ExampleTrigonometricConstants ExampleTrigonometricConstants

[RecurrenceRelationOperator]

Wed, 01 Jul 2026 20:13:07 +0000

++added: RecurrenceOperator is included in FriCAS, content below is kept for historical reasons.

[OtherComputerAlgebraSystems]

Wed, 01 Jul 2026 19:44:08 +0000

++added: Sympy -- See: https://sympy.org Due to size and complexity access to Sage (see: https://sagemath.org) is not feasible.

[BugTracker]

Mon, 22 Jun 2026 23:01:49 +0000

??changed: -FriCAS bugs should be reported in FriCAS bug tracker at SourceForge: -http://sourceforge.net/p/fricas/bugs/ FriCAS bugs should be reported in FriCAS bug tracker at Github: https://github.com/fricas/fricas/issues/

[FriCASIntegration] another silly algebraic case, totally different form from WA

Thu, 04 Jun 2026 18:27:18 +0000

??changed: -Subject: another silly algebraic case, totally different form from WA Subject: another silly algebraic case, totally different form from WA (it's not obvious that (t-x)(t+x)=1)

[#218 New version Reduce 3.8 includes MathML]

Sun, 31 May 2026 22:43:59 +0000

++added: From test1 Sun May 31 22:43:59 +0000 2026 From: test1 Date: Sun, 31 May 2026 22:43:59 +0000 Subject: Message-ID: <20260531224359+0000@wiki.fricas.org> Category: Reduce => MathAction

[ExampleTrigonometricConstants]

Mon, 02 Mar 2026 18:17:05 +0000

??changed: - Example of computations with trigonometric constants Consider the problem: Let $t := (5/9)\pi$. Simplify $tan(t) + 4sin(t)$. We can solve it by noting that involved numbers can be expressed in terms of roots of $1$, Namely, first we express trigonometric functions in terms of complex exponentials. Since $(5/9)\pi$ is a rational multiple of $\pi$ we get root of $1$, in this case root of degree $18$. But we also need $i$, which leads to root of degree $36$. Corresponding field is extention by cyclotomic polynomial, so we build it: \begin{axiom} cyclotomic(36)$CyclotomicUtilities )set output algebra on )set output tex off cF := SAE(FRAC(INT), SUP(FRAC(INT)), %) )set output tex on )set output algebra off \end{axiom} Now, we can express all ingredients in terms of generator: \begin{axiom} et := (generator()$cF)^10 ct := (et + 1/et)/2 st := (et - 1/et)/(2*(generator()$cF)^9) \end{axiom} and our expression is: \begin{axiom} ex := (4*ct + 1)*st/ct \end{axiom} Now we can find minimal polynomial and find its roots: \begin{axiom} minimalPolynomial(%) \end{axiom} and we see that the result is $\sqrt(3)$. We could also observe that the expression above is a member of smaller cyclotomic field and from that infer expression $2\cos(\pi/6)$. Consder now expression $$ i(5\sin((5\pi)/7)-5\sin((4\pi)/7)+5\sin((3\pi)/7) -5\sin((2\pi)/7)) -5\cos((5\pi)/7)+5\cos((4\pi)/7) $$ $$ -5\cos((3\pi)/7)+5\cos((2\pi)/7) -10\cos(\pi/7)+6. $$ Now we need root of $1$ of degree $28$, so we build the field: \begin{axiom} cyclotomic(28)$CyclotomicUtilities )set output algebra on )set output tex off cF2 := SAE(FRAC(INT), SUP(FRAC(INT)), %) )set output tex on )set output algebra off \end{axiom} Root of degree $4$ that is generator to power $7$ gives us imaginary unit $i$, Suare of generator gives us primitive root of degree $7$, that is $\exp(\pi/7)$: \begin{axiom} im2 := generator()$cF2^7 e1 := generator()$cF2^2 \end{axiom} We express trigonometric functions in terms of this \begin{axiom} c1 := (e1 + 1/e1)/2 e2 := e1^2 c2 := (e2 + 1/e2)/2 s2 := (e2 - 1/e2)/(2*im2) e3 := e1^3 c3 := (e3 + 1/e3)/2 s3 := (e3 - 1/e3)/(2*im2) e4 := e1^4 c4 := (e4 + 1/e4)/2 s4 := (e4 - 1/e4)/(2*im2) e5 := e1^5 c5 := (e5 + 1/e5)/2 s5 := (e5 - 1/e5)/(2*im2) \end{axiom} Using this we can compute the expression: \begin{axiom} im2*(5*s5 - 5*s4 + 5*s3 - 5*s2) - 5*c5 + 5*c4 - 5*c3 + 5*c2 - 10*c1 + 6 \end{axiom}