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TI-83 Basic 68k TI-Basic TI-Nspire Basic
TI-83

TI-83 Basic is the most commonly used, because the TI-83/84 series has been heavily marketed by Texas Instruments to high school students needing a graphing calculator for math and science classes.

At the same time, it is the least powerful language, lacking many of the complex commands and programming capabilities.

TI-89

68k TI-Basic is much more powerful than TI-83 Basic, with support for calculus, indirection, local variables and functions, advanced picture manipulation, and several other features that make it a very rich language.

However, it isn't used nearly as much as TI-83 Basic, so it doesn't have as big of a community developed around it.

TI-Nspire

TI-Nspire Basic is quite similar to 68k TI-Basic, but not as programmer friendly: it has poor I/O support, rigid constraints on program execution, and documents are used instead of programs.

In addition, it is still relatively unknown to the TI community, so there isn't much documentation available yet.

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