Two Proofs of a0 = 1


  1. A quotient of two equal exponential variables an equals one because all factors cancel. A quotient of all variables an and am, where m < n will have all factors cancel leaving the exponent n-m, if an is the numerator and am is the denominator. Therefore if m = n all factors will cancel leaving an exponential with n-n = 0 factors or a0. But as previously stated a quotient of two equal exponentials equals 1. Therefore a0 = 1. Q. E. D.


  1. Exponentials of the same base are unique in that an has one and only one value for each value of the exponent. Therefore a0 has one and only one value. Assume that a0 does not equal 1. Then there will be some real number y = ax that will have two exponential representations in the same base. But this contradicts the uniqueness of exponential numbers in the same base. So a0 must be equal to 1.