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A siman is a mark, sign or omen. The root is the verb soom, which means to put or place and so also to mark or distinguish. The best known simanim in Judaism are the fruits and vegetables eaten on the first night of Rosh Hashanah. (Some eat them on the second night as well.)
The custom is based on a talmudic teaching of Abaye, who said: "Since you say that omens are significant, you should eat gourds, fenugreek, leeks, beets and dates at the start of the year" (Horayot, 12a).
For each food, something about the name or quality of the food evokes a wish or prayer for the coming year. The most famous is apple dipped in honey, associated with the hope for a sweet New Year. Others include pomegranates, whose many seeds call up the wish that our merits should be correspondingly numerous; and the head of a fish (in some traditions, the head of a sheep) which elicits the hope that we should be the head and not the tail.
Some have created new simanim, based on far-fetched puns: eg eating dates so that the coming year should bring many dates (as in romantic assignations) or the excruciating lettuce, raisins and celery for "let us have a raise in salary".
Rosh Hashanah is known as the womb of the year, containing the next twelve months in embryo. Whatever their efficacy, the simanim help us articulate our hopes and prayers for the year.