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Does an index have a currency?
There's no need for an index to have a currency as its purpose is not to act as an asset but rather to signal investors about the performance of a collection of stocks. An index can be price-weighted, meaning that its value equals the (arithmetic) average of the prices of each stock in the index. With no stock splits, the return on this index is the same as the return on a portfolio composed of one share of each stock. If there is a stock split, however instead of dividing by the number of stocks, as you normally would when taking the arithmetic average, you divide it by the number that will make the value of the index pre-stock-split (arithmetic average) equal to the value post stock split. Then use that dividing number for all periods until a new stock split occurs.  An index can be value-weighted, meaning that its changes in value track the percentage changes in total market capitalization of the stocks in the index. Price weighted indexes ignore for "firm size" and percentage changes in price weighted indexes are not robust to stock-splits. Value weighted indexes take "firm size" into account and are robust to stock-splits. DJIA is price-weighted. S&P 500 is value-weighted.