How do すなわち vs 要するに vs つまり differ? In which contexts are they interchangeable? Please provide some example sentences to illustrate.
すなわち and つまり both mean "that is to say", "i.e." or "which means", but すなわち sounds more bookish, technical or formal. すなわち is preferred in serious essays, academic articles, politicians' speeches, legal documents, etc. つまり is usually preferred in everyday business conversations, oral presentations, etc. They are used to describe the same thing in two different ways to make your explanation more understandable. The explanation following すなわち/つまり can be shorter or longer.
要するに is a way to say "to sum it up", "to make a long story short", "in simpler terms", etc. This is mainly used when the existing conversation has become too complicated or difficult to follow, or when you want to just say the conclusion part.
In this example, it's possible to use all of すなわち, つまり and 要するに:
Pi is an irrational number, i.e., a number that cannot be expressed in fractional form.
すなわち sounds most natural here since this sounds like an academic sentence. つまり is also natural, and tends to be preferred in speech. 要するに gives the impression that the listener cannot or does not need to understand the term "irrational number", which can often sound condescending.
In the following example, you can only use 要するに.
It's a long story if I go into details, but in short, I failed the exam.