Who were the 'mixed multitude also went up with' the children of Israel in Exodus 12:38?
Were they part of Israel?
What does Judaism, Jewish tradition say about this mixed multitude?
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They were a mixture from other nations that decided to join the Jews at the exodus (Oknkelos, Rashi and pretty much everyone I could find, although some identify them specifically as Egyptians).
Rabbi Gansfried quotes various opinions as to their size, based on the idea that the 600,000 number represents one fifth of the total that left. Whether that represents numbers or a metaphorical statement about influence (or something else) will depend on one's approach to such Aggadah.
Exodus Rabbah 42 says that Moshe accepted them of his own initiative, after initial resistance from G-d. They later become the main instigators of the golden calf incident. (This is a theme, sending the spies was of his own initiative and that turned out badly).