How much money is a Machatziz hashekel.
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The Shekel in the Torah was a measure of weight of silver.
Rashi (Exodus 21:32) says the Shekel is "half an ounkia as used in Kolonia"; to translate from Old French, half an ounce as was used in the German city of Cologne. The shekel is generally understood to be about half a troy ounce; the Chazon Ish is quoted as saying the shekel is 0.51 troy ounces. (See this article for many more details). So a half shekel is about a 1/4 of a troy ounce. (That's about the same weight as the new American gold-colored dollar coins.)
Today, silver is worth about $18.70 per troy ounce; taking $18.70 / 4, it looks like right now, a half-shekel's weight of silver is worth about $4.70.
According to the Rambam (Hil. Shekalim 1:2), a shekel was originally the weight of 320 barleycorns of silver, and this was later increased to 384. So half a shekel would be, respectively, 160 or 192 barleycorns. According to measurements given here a barleycorn is around 1/20 of a gram, so half a shekel would be 8 or 9.6 grams. This comes to about 0.25-0.3 troy ounce.
(For purposes of pidyon haben, where five silver shekalim are required, there are customs to use as much as 117 grams of silver. This would make half a shekel, 1/10 of that amount, 11.7 grams or 0.38 troy ounce.)