What's Special About This Fraction?

• −1 is its own reciprocal
• −¹⁄₂ is the real part of ω, a complex cubic root of 1
• 0 is the only rational number with no reciprocal
• ^137174210⁄_1111111111 = 0.12345678901234567890123456789012345678901234567890... (see Sloane's A010879)
• ¹⁄₃ has the same representation in binary as in negabinary: 0.010101010101010101010101...
• ¹⁄₂ is the real part of all nontrivial zeroes of the zeta function (assuming the Riemann hypothesis)
• ⁶⁹⁸⁵⁴⁄₇₀₁₂₃ is the greatest fraction not greater than 1 such that in base 10 each digit from 0 to 9 occurs precisely once in either the numerator or the denominator (but not both)
• 1 is its own reciprocal
• ¹⁹⁄₇ is a rational approximation to e
• ⁸⁷⁄₃₂ is a rational approximation to e
• ³³³⁄₁₀₆ is a rational approximation of π
• ³⁵⁵⁄₁₁₃ is a rational approximation of π
• ²²⁄₇ is a rational approximation of π