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11.3.2 Algebraic Simplifications

This section describes all simplifications that are performed by the algebraic simplification mode, which is the default simplification mode. If you have switched to a different simplification mode, you can switch back with the m A command. Even in other simplification modes, the a s command will use these algebraic simplifications to simplify the formula.

There is a variable, AlgSimpRules, in which you can put rewrites to be applied. Its use is analogous to EvalRules, but without the special restrictions. Basically, the simplifier does ‘a r AlgSimpRules’ with an infinite repeat count on the whole expression being simplified, then it traverses the expression applying the built-in rules described below. If the result is different from the original expression, the process repeats with the basic simplifications (including EvalRules), then AlgSimpRules, then the built-in simplifications, and so on.

Sums are simplified in two ways. Constant terms are commuted to the end of the sum, so that ‘a + 2 + b’ changes to ‘a + b + 2’. The only exception is that a constant will not be commuted away from the first position of a difference, i.e., ‘2 - x’ is not commuted to ‘-x + 2’.

Also, terms of sums are combined by the distributive law, as in ‘x + y + 2 x’ to ‘y + 3 x’. This always occurs for adjacent terms, but Calc's algebraic simplifications compare all pairs of terms including non-adjacent ones.

Products are sorted into a canonical order using the commutative law. For example, ‘b c a’ is commuted to ‘a b c’. This allows easier comparison of products; for example, the basic simplifications will not change ‘x y + y x’ to ‘2 x y’, but the algebraic simplifications; it first rewrites the sum to ‘x y + x y’ which can then be recognized as a sum of identical terms.

The canonical ordering used to sort terms of products has the property that real-valued numbers, interval forms and infinities come first, and are sorted into increasing order. The V S command uses the same ordering when sorting a vector.

Sorting of terms of products is inhibited when Matrix mode is turned on; in this case, Calc will never exchange the order of two terms unless it knows at least one of the terms is a scalar.

Products of powers are distributed by comparing all pairs of terms, using the same method that the default simplifications use for adjacent terms of products.

Even though sums are not sorted, the commutative law is still taken into account when terms of a product are being compared. Thus ‘(x + y) (y + x)’ will be simplified to ‘(x + y)^2’. A subtle point is that ‘(x - y) (y - x)’ will not be simplified to ‘-(x - y)^2’; Calc does not notice that one term can be written as a constant times the other, even if that constant is -1.

A fraction times any expression, ‘(a:b) x’, is changed to a quotient involving integers: ‘a x / b’. This is not done for floating-point numbers like ‘0.5’, however. This is one reason why you may find it convenient to turn Fraction mode on while doing algebra; see Fraction Mode.

Quotients are simplified by comparing all terms in the numerator with all terms in the denominator for possible cancellation using the distributive law. For example, ‘a x^2 b / c x^3 d’ will cancel ‘x^2’ from the top and bottom to get ‘a b / c x d’. (The terms in the denominator will then be rearranged to ‘c d x’ as described above.) If there is any common integer or fractional factor in the numerator and denominator, it is canceled out; for example, ‘(4 x + 6) / 8 x’ simplifies to ‘(2 x + 3) / 4 x’.

Non-constant common factors are not found even by algebraic simplifications. To cancel the factor ‘a’ in ‘(a x + a) / a^2’ you could first use j M on the product ‘a x’ to Merge the numerator to ‘a (1+x)’, which can then be simplified successfully.

Integer powers of the variable i are simplified according to the identity ‘i^2 = -1’. If you store a new value other than the complex number ‘(0,1)’ in i, this simplification will no longer occur. This is not done by the basic simplifications; in case someone (unwisely) wants to use the name i for a variable unrelated to complex numbers, they can use basic simplification mode.

Square roots of integer or rational arguments are simplified in several ways. (Note that these will be left unevaluated only in Symbolic mode.) First, square integer or rational factors are pulled out so that ‘sqrt(8)’ is rewritten as 2 sqrt(2)’. Conceptually speaking this implies factoring the argument into primes and moving pairs of primes out of the square root, but for reasons of efficiency Calc only looks for primes up to 29.

Square roots in the denominator of a quotient are moved to the numerator: ‘1 / sqrt(3)’ changes to ‘sqrt(3) / 3’. The same effect occurs for the square root of a fraction: ‘sqrt(2:3)’ changes to ‘sqrt(6) / 3’.

The % (modulo) operator is simplified in several ways when the modulus ‘M’ is a positive real number. First, if the argument is of the form ‘x + n’ for some real number ‘n’, then ‘n’ is itself reduced modulo ‘M’. For example, ‘(x - 23) % 10’ is simplified to ‘(x + 7) % 10’.

If the argument is multiplied by a constant, and this constant has a common integer divisor with the modulus, then this factor is canceled out. For example, ‘12 x % 15’ is changed to ‘3 (4 x % 5)’ by factoring out 3. Also, ‘(12 x + 1) % 15’ is changed to ‘3 ((4 x + 1:3) % 5)’. While these forms may not seem “simpler,” they allow Calc to discover useful information about modulo forms in the presence of declarations.

If the modulus is 1, then Calc can use int declarations to evaluate the expression. For example, the idiom ‘x % 2’ is often used to check whether a number is odd or even. As described above, ‘2 n % 2 and ‘(2 n + 1) % 2’ are simplified to ‘2 (n % 1)’ and ‘2 ((n + 1:2) % 1)’, respectively; Calc can simplify these to 0 and 1 (respectively) if n has been declared to be an integer.

Trigonometric functions are simplified in several ways. Whenever a products of two trigonometric functions can be replaced by a single function, the replacement is made; for example, ‘tan(x) cos(x)’ is simplified to ‘sin(x)’. Reciprocals of trigonometric functions are replaced by their reciprocal function; for example, ‘1/sec(x)’ is simplified to ‘cos(x)’. The corresponding simplifications for the hyperbolic functions are also handled.

Trigonometric functions of their inverse functions are simplified. The expression ‘sin(arcsin(x))’ is simplified to ‘x’, and similarly for cos and tan. Trigonometric functions of inverses of different trigonometric functions can also be simplified, as in ‘sin(arccos(x))’ to ‘sqrt(1 - x^2)’.

If the argument to sin is negative-looking, it is simplified to ‘-sin(x)’, and similarly for cos and tan. Finally, certain special values of the argument are recognized; see Trigonometric and Hyperbolic Functions.

Hyperbolic functions of their inverses and of negative-looking arguments are also handled, as are exponentials of inverse hyperbolic functions.

No simplifications for inverse trigonometric and hyperbolic functions are known, except for negative arguments of arcsin, arctan, arcsinh, and arctanh. Note that ‘arcsin(sin(x))’ can not safely change to ‘x’, since this only correct within an integer multiple of 2 pi’ radians or 360 degrees. However, ‘arcsinh(sinh(x))’ is simplified to ‘x’ if ‘x’ is known to be real.

Several simplifications that apply to logarithms and exponentials are that ‘exp(ln(x))’, e^ln(x)’, and 10^log10(x)’ all reduce to ‘x’. Also, ‘ln(exp(x))’, etc., can reduce to ‘x’ if ‘x’ is provably real. The form ‘exp(x)^y’ is simplified to ‘exp(x y)’. If ‘x’ is a suitable multiple of pi i’ (as described above for the trigonometric functions), then ‘exp(x)’ or ‘e^x’ will be expanded. Finally, ‘ln(x)’ is simplified to a form involving pi and i where ‘x’ is provably negative, positive imaginary, or negative imaginary.

The error functions erf and erfc are simplified when their arguments are negative-looking or are calls to the conj function.

Equations and inequalities are simplified by canceling factors of products, quotients, or sums on both sides. Inequalities change sign if a negative multiplicative factor is canceled. Non-constant multiplicative factors as in ‘a b = a c’ are canceled from equations only if they are provably nonzero (generally because they were declared so; see Declarations). Factors are canceled from inequalities only if they are nonzero and their sign is known.

Simplification also replaces an equation or inequality with 1 or 0 (“true” or “false”) if it can through the use of declarations. If ‘x’ is declared to be an integer greater than 5, then ‘x < 3’, ‘x = 3’, and ‘x = 7.5’ are all simplified to 0, but ‘x > 3’ is simplified to 1. By a similar analysis, ‘abs(x) >= 0’ is simplified to 1, as is ‘x^2 >= 0’ if ‘x’ is known to be real.