# Making Definitions for Indexed Objects

In many kinds of calculations, you need to set up "arrays" that contain sequences of expressions, each specified by a certain index. One way to implement arrays in *Mathematica* is by using lists. You can define a list, say , then access its elements using , or modify them using . This approach has a drawback, however, in that it requires you to fill in all the elements when you first create the list.

Often, it is more convenient to set up arrays in which you can fill in only those elements that you need at a particular time. You can do this by making definitions for expressions such as .

This defines a value for

.

Out[1]= | |

This defines a value for

.

Out[2]= | |

This shows all the values you have defined for expressions associated with

so far.

You can define a value for

, even though you have not yet given values to

and

.

Out[4]= | |

This generates a list of the values of the

a[i].

Out[5]= | |

You can think of the expression a[i] as being like an "indexed" or "subscripted" variable.

a[i]=value | add or overwrite a value |

a[i] | access a value |

a[i]=. | remove a value |

?a | show all defined values |

Clear[a] | clear all defined values |

Table[a[i],{i,1,n}] or Array[a,n] | convert to an explicit List |

Manipulating indexed variables.

When you have an expression of the form , there is no requirement that the "index" i be a number. In fact, *Mathematica* allows the index to be any expression whatsoever. By using indices that are symbols, you can for example build up simple databases in *Mathematica*.

This defines the "object"

with "index"

to have value

.

Out[6]= | |

This adds another result to the

"database".

Out[7]= | |

Here are the entries in the

database so far.

You can use these definitions wherever you want. You have not yet assigned a value for

.

Out[9]= | |