The more I read about the TI-57, the more I wanted to fix mine.  Of course, these are easily found on a-large-well-known-online-auction-site for modest prices - typically $40 or less. Like most hobby projects, my goals were entirely devoid of practicality.  Vive les fous!

The heart of the TI-57 is a calculator-on-a-chip executed in 4-phase PMOS. If you've not yet gone down this rabbithole, I strongly recommend a read through Ken Shirriff's excellent coverage of this subject.  So, this means multiple supply rails and oddball voltages (at least, oddball in this day and age) are required. The IC designers at TI built the substrate bias voltage generator into the TMC1500 family, vastly simplifying things by allowing operation from a single -9V rail.  Of course, we can call -9V "ground" and that makes the TI-57 ground "+9V". It's why when you power one of these calculators from a 9V battery, you need to connect the battery's POSITIVE terminal to the BLACK wire, and NEAGATIVE terminal to the RED wire. Um...thanks, TI.

Much of how the TMC1500 works can be learned from studying the incredibly detailed and completely helpful US Patents that TI applied for prior to the production of the TI-57.  Jeff Parsons has collected the most important ones here, and it is well worth studying them - if not only as standout examples of well-written patents!  Because my goal was to replace the TMC1500 IC with a microcontroller and some discrete components, I had some questions about the "rest" of the calculator - mostly the 12-digit LED display. Digging around online in 1970's eta TI optoelectronic databooks was marginally helpful, but nothing beats an actual measurement of the LED in action to gather some critical items:

• LED segment ON time
• LED segment forward current 
• LED segment voltage drop

The TI-57 has a lot in common with a cheaper model, the TI-55. It's the same TMC1500 family process, with a different keyboard layout and ROM content.  These are even cheaper to obtain, so I picked a couple up for experimental purposes.