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The JSON specification doesn't support special floating values such as any NaN value or +/-infinity. But many (if not most) Json libraries at least provide a special Flag to allow them. As an example see the flag serializeSpecialFloatingPointValues of the google-Gson library:
It seems that the choice of representation chosen here (NaN for a NaN, Infinity and -Infinity for positive/negative infinity is quite popular and I recommend to use them when this feature is active.