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For example, the integers are made by adding 0 and negative numbers. The rational numbers add fractions, and the real numbers add infinite decimals. Complex numbers add the square root of −1. This chain of extensions canonically embeds the natural numbers in the other number systems. [6] [7] Natural numbers are studied in different areas of math.
An example of the second case is the decidability of the first-order theory of the real numbers, a problem of pure mathematics that was proved true by Alfred Tarski, with an algorithm that is impossible to implement because of a computational complexity that is much too high. [122]
Photographs of caucus math worksheets taken by caucus "captains" showed errors in adding up votes for candidates and in calculating "state delegate equivalents". [10] IDP chair Troy Price said that a recount of votes would be required to correct the miscalculations on the handwritten tally sheets from precincts.