# Values for Symbols

When

*Mathematica* transforms an expression such as

into

, it is treating the variable

in a purely symbolic or formal fashion. In such cases,

is a symbol which can stand for any expression.

Often, however, you need to replace a symbol like

with a definite "value". Sometimes this value will be a number; often it will be another expression.

To take an expression such as

and replace the symbol

that appears in it with a definite value, you can create a

*Mathematica* transformation rule, and then apply this rule to the expression. To replace

with the value

, you would create the transformation rule

. You must type

as a pair of characters, with no space in between. You can think of

as being a rule in which "

goes to

".

To apply a transformation rule to a particular

*Mathematica* expression, you type

. The "replacement operator"

is typed as a pair of characters, with no space in between.

This uses the transformation rule

in the expression

.

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You can replace

with any expression. Here every occurrence of

is replaced by

.

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Here is a transformation rule.

*Mathematica* treats it like any other symbolic expression.

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This applies the transformation rule on the previous line to the expression

.

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expr/.x->value | replace x by value in the expression expr |

expr/.{x->xval,y->yval} | perform several replacements |

Replacing symbols by values in expressions.

You can apply rules together by putting the rules in a list.

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The replacement operator

allows you to apply transformation rules to a particular expression. Sometimes, however, you will want to define transformation rules that should

*always* be applied. For example, you might want to replace

with

whenever

occurs.

As discussed in

"Defining Variables", you can do this by

*assigning* the value

to

using

. Once you have made the assignment

,

will always be replaced by

, whenever it appears.

This assigns the value

to

.

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Now

will automatically be replaced by

wherever it appears.

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This assigns the expression

to be the value of

.

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Now

is replaced by

.

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You can define the value of a symbol to be any expression, not just a number. You should realize that once you have given such a definition, the definition will continue to be used whenever the symbol appears, until you explicitly change or remove the definition. For most people, forgetting to remove values you have assigned to symbols is the single most common source of mistakes in using

*Mathematica*.

x=value | define a value for x which will always be used |

x=. | remove any value defined for x |

Assigning values to symbols.

The symbol

still has the value you assigned to it.

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This removes the value you assigned to

.

Now

has no value defined, so it can be used as a purely symbolic variable.

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A symbol such as

can serve many different purposes in

*Mathematica*, and in fact, much of the flexibility of

*Mathematica* comes from being able to mix these purposes at will. However, you need to keep some of the different uses of

straight in order to avoid making mistakes. The most important distinction is between the use of

as a name for another expression, and as a symbolic variable that stands only for itself.

Traditional programming languages that do not support symbolic computation allow variables to be used only as names for objects, typically numbers, that have been assigned as values for them. In

*Mathematica*, however,

can also be treated as a purely formal variable, to which various transformation rules can be applied. Of course, if you explicitly give a definition, such as

, then

will always be replaced by

, and can no longer serve as a formal variable.

You should understand that explicit definitions such as

have a global effect. On the other hand, a replacement such as

affects only the specific expression

expr. It is usually much easier to keep things straight if you avoid using explicit definitions except when absolutely necessary.

You can always mix replacements with assignments. With assignments, you can give names to expressions in which you want to do replacements, or to rules that you want to use to do the replacements.

This assigns a value to the symbol

.

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This finds the value of

, and then replaces

by

in it.

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This finds the value of

for a different value of

.

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This finds the value of

when

is replaced by

Pi, and then evaluates the result numerically.

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