Is "ignore" the root of "ignorant" and "ignorance"? I suppose etymologically the three words must be related, maybe having the same Latin root, but in English "ignore" forms an interesting contrast in meaning with "ignorant" and "ignorance". Ignorance of X is lack of knowledge of X, but to ignore X one has to know of X in the first place, that is, one cannot ignore something that one is ignorant of.
However, if one ignores X, one will not be able to know X further than one already does. So, in a way, ignoring X does lead to ignorance of X. On this understanding, knowledge is a matter of degree, and so is ignorance.
The Latin verb is ignoro/ignorare, which means both "to be unfamiliar with/to not know" and "to disregard". The latter (as well as with our English word "to ignore") has an active element to its meaning, expressing a sort of agency or a conscious willing, whereas the former tends to have more of a passivity in its connotation.
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