If I say, "je sais rien," does that express "I know nothing" in a satisfactory way?
Or do I need to say, "je ne sais rien."?
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If you write a French text, then make sure to use je ne sais rien as much as possible, because in fact, je sais rien is incorrect. In spoken language, it's not really a problem to use je sais rien.
Same for ne ... pas, ne ... plus and ne ... jamais
Yes, while saying (spoken French) ne may be quite often omitted. So to answer your question yes it is satisfactory.
Ce n'est pas l'usage, mais "je sais rien" devrait être considéré comme correct étymologiquement et signifier "je sais une chose".
On dit bien : "je sais quelques riens"
En effet, "rien" vient du Latin res = chose.