The first two letters signify a male, the first three letters signify a female, the first four letters signify a great, while the entire word signifies a great woman. What is the word?

A rich man needs it, a poor man has it, if we eat it, we die, and when we die, we bring it with us.

I sound like a dessert or bodily waste depending on how you pronounce me. I am important in geometry. I have a day named in my honor. I have a very famous specific set of numbers.

Used in math but not a number. I can be golden, inverse, or direct. I remain constant when my parts change size.

I ate the one after the next - who was three by three, The count of the rainbow's arcs, The steps on music's marks.