Septenary

Septenary (also known as septimal or base 7) is a positional number system structured around the number 7 and its powers.

Usage
Septenary sees little use today, it is by far the least used non-transdecimal base. It can serve as an example, along with heptadecimal, of why prime bases are bad.

A human world built on sevens would stifle our ability to work with our cherished pairs, significantly hinder our arithmetic abilities with integer multiples of five, and have no effect on our decimal relationship with triples. We'd be more reliant on "omega" relationships and "digit sum" divisibility tests, which we'd use to divide 2, 3, and 6. The "alpha" relationships, with their "alternating sum" checks, would be far more useful than in decimal, as they would be able to check for up to 3 powers of two.

The biggest, and probably only, use of septenary in our world today is in timekeeping. A week has seven days. Septenary is the fourth mixed base starting from the least significant digit in chronary.

Irrational numbers
Here are the most commonly used irrational numbers expressed in septenary: