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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
‘

Square roots in the denominator of a quotient are moved to the
numerator: ‘`1 / sqrt(3)`’ changes to ‘

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 ‘

Trigonometric functions of their inverse functions are
simplified. The expression ‘` sin(arcsin(x))`’ is
simplified to ‘

`cos`

and `tan`

.
Trigonometric functions of inverses of different trigonometric
functions can also be simplified, as in ‘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

Several simplifications that apply to logarithms and exponentials
are that ‘` exp(ln(x))`’,
‘

`pi`

and
`i`

where ‘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.