The origin of the date actually seems to come from a second or third century tradition tying the date of Jesus' death to the anniversary of his conception. December 25 is 9 months after March 25, which is the date which equated to the date of his death comparing the Hebrew and Roman calendars for that year.
But the Hebrew calendar doesn't correspond exactly with the Roman one from one year to the next (c.f. the changing dates of Passover, etc.), and it's just a generic "+ 9 months" origin, never stated as a precise anniversary.
Other theories have tied the date of Jesus' birth to the Jewish Feast of Tabernacles, largely based on the eschatological associations of "the day of the Lord" with that feast in Zechariah and elsewhere. These theories are more modern in origin, and fit some of the evidence well, though they don't line up with the traditional date. Christians in the first couple centuries didn't seem to pay much attention to the date of Jesus' birth - it was the third century or so before any records start popping up. So an exact date is impossible to determine in any case.
Yes, it's often simplified as "Christmas is Jesus' birthday" but I don't know that many people when pressed would say that they think means December 25 is the exact date he was born, or even interpret the phrase as being a claim of that. It's a holiday that's celebrated near one of the dates that's a possibility for when he could've been born, and that date's as good as any other we have, so we celebrate it as "Jesus' birthday" whether or not he was really born on that date, and aren't terribly concerned with the precision of the estimate.
But the Hebrew calendar doesn't correspond exactly with the Roman one from one year to the next (c.f. the changing dates of Passover, etc.), and it's just a generic "+ 9 months" origin, never stated as a precise anniversary.
Other theories have tied the date of Jesus' birth to the Jewish Feast of Tabernacles, largely based on the eschatological associations of "the day of the Lord" with that feast in Zechariah and elsewhere. These theories are more modern in origin, and fit some of the evidence well, though they don't line up with the traditional date. Christians in the first couple centuries didn't seem to pay much attention to the date of Jesus' birth - it was the third century or so before any records start popping up. So an exact date is impossible to determine in any case.
Yes, it's often simplified as "Christmas is Jesus' birthday" but I don't know that many people when pressed would say that they think means December 25 is the exact date he was born, or even interpret the phrase as being a claim of that. It's a holiday that's celebrated near one of the dates that's a possibility for when he could've been born, and that date's as good as any other we have, so we celebrate it as "Jesus' birthday" whether or not he was really born on that date, and aren't terribly concerned with the precision of the estimate.
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