is the two different sayings of isaac newton birth date are december 25 1642 and january 4 1643

Yes, you've described correctly the two different ways Isaac Newton's birth date is often given due to the switch from the Julian calendar to the Gregorian calendar.

Newton was born on December 25, 1642, according to the Julian calendar, which was in use in England at the time. However, the Gregorian calendar was introduced in 1582 by Pope Gregory XIII to correct for inaccuracies in the Julian calendar. The difference between these calendars is accounted for by a 10-day shift.

The switch from the Julian to the Gregorian calendar occurred at different times in different countries due to religious reasons and political considerations. England didn't adopt the Gregorian calendar until later in 1752. To align with the Gregorian calendar, England had to drop 11 days (to correct the drift since the last leap year in the Julian calendar), and also shifted the start of the new year from Lady Day (March 25) to January 1.

So, when you convert Newton's birth date from the Julian to the Gregorian calendar, taking into account the 11-day adjustment and the change from March to January as the start of the year, his birth date becomes January 4, 1643.

Thus, both December 25, 1642, and January 4, 1643, correctly represent Newton's birth date depending on whether you're using the Julian or Gregorian calendar.