The mathematical concept of strongly real functions of positive and negative types is introduced to network theory for the first time. In this letter we show that the driving point reactance/susceptance of a pure Foster network, made up of only ideal positive inductance and capacitance element, is a strongly real function of real frequency of positive type. As a corollary, for a pure non-Foster network made up of only ideal negative inductance and capacitance elements, we show that the driving point reactance/susceptance is a strongly real function of real frequency of negative type. It is shown that a condition for a purely reactive passive network to exhibit a positive or negative reactance/susceptance-frequency gradient is that the driving point immittance should have alternating poles and zeros lying on the real frequency axis. Finally, it is shown that either purely Foster or non-Foster networks can be constructed by combining ideal Foster and non-Foster reactive elements.