The infinite root of unity is unity. The infinite root of the number 2 which can be inductively derived by repeated application of any handheld electronic calculator is also unity. In fact, as the number becomes bigger and bigger the rooting process becomes slower and slower to reach the infinite root of the number which by induction is also unity. Therefore, by these processes of inductive reasoning to arrive at a certain mathematical truth, the final conclusion is that the infinite root of infinity is unity. Obviously, the rooting process is truly convergent for any given positive real number approaching infinity and the limit of convergence is always positive unity.

Surprisingly, the infinite root of negative unity can either be positive and negative unity, or pure imaginary. In fact, as the number becomes more and more negative the rooting process can only alternately gives between real negative numbers, complex numbers, or imaginary numbers depending whether the root is odd or even. Odd roots give real negative numbers while even roots give either complex numbers or pure imaginary numbers. However, for all complex roots as the rooting process approaches infinity the real part approaches positive unity while the imaginary part approaches zero giving again the inductive assertion that the infinite root of negative infinity is also positive unity iff the rooting process is even infinity while the infinite root of negative infinity is negative unity iff the rooting process is odd infinity. These assertions depend on the operational definition of odd and even infinity.