1 Gigasecond is also a Teramilisecond?

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I'm reasoning about some time parsing thing in JavaScript which uses millisecond-precision for Date objects and Unix System time does use seconds.

Putting problems with precision loss for conversion- can one say if

if 10^9 seconds is a Gigasecond, then 10^12 Miliseconds is a Gigasecond a Teramilisecond?

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If $10^9$ seconds is a gigasecond, $10^{12}$ seconds is a terasecond, not a teramillisecond. A teramillisecond would be $10^{12} \cdot 10^{-3} = 10^9$ seconds as well, but only one metric prefix can be used on the same word.