Question

I am going throgh algorithms. It is mentioned that one of the application of algorithms is symbolic mathematics. And I found following defintion from dictionary as below.

The use of computers to manipulate mathematical equations and
expressions in symbolic form, as opposed to manipulating the numerical
quantities represented by those symbols. Such a system might be used
for symbolic integration or differentiation, substitution of one
expression into another, simplification of an expression, change of
subject etc. One of the best known symbolic mathematics software
packages is Mathematica.

My question what does statement “equations and expressions in symbolic form, as opposed to manipulating the numerical quantities represented by those symbols.” mean?

Thanks!

Answer 1

My question what does statement “equations and expressions in symbolic
form, as opposed to manipulating the numerical quantities represented
by those symbols.” mean?

By the second, something like this is meant:

>>> x = 2.3
>>> y = 9.8
>>> z = x+2*y
>>> z
21.900000000000002
>>> type(z)
<type 'float'>

where you’re treating x, y, and z as names for numbers. You’re using the computer as an old-fashioned calculator, where it only does arithmetic. z = x+2*y performs the arithmetic operations described on the right and associates the resulting number to z.

By the “symbolic form”, something more like this is meant:

sage: x, y, z = var("x y z")
sage: z == x+2*y
z == x + 2*y
sage: eq = z == x+2*y
sage: eq
z == x + 2*y
sage: type(z)
<type 'sage.symbolic.expression.Expression'>
sage: parent(eq)
Symbolic Ring
sage: eq.solve(y)
[y == -1/2*x + 1/2*z]

where x,y, and z can be expressions, or variables in some structure, rather than merely names for specific numbers, and higher-level operations can be performed.