@DoubleAA pointed out that I should mention that the dates based on the birth of a child do not account for however many months after each father's birthday a child was born. That would add some months to the timeline in each generation. This could add an extra year every two generations, but it would be too complicated to go into. That is why the meforshim who use this method do not go into it.
Someone else pointed to the analogy of the Chinese counting ages from New Years. If that analogy holds, the time is counted using Rosh Hashannah. Thus, the extra months would be dropped rather than being added in as an extra year as occurred for the reigns of the kings. I do not know if this is true or not, but it could be a consideration.
I should also note that if the secular estimate of 586 BCE for the destruction of the first Bais Hamikdash is used, then 3338 would have been 2604 years ago, making the current year 5942. The statement that the current year is 5778 is based on the destruction of the second temple in 68 CE and 3828 AM.
There are those who do not have the zero point at the creation of Adam. Thus, some count year 1 as the time before Adam, and year 2 as the year after Adam was created. This would add two years to the year number but not change the number of years elapsed. Thus, after 3828 years, the second Bais hamikdash would have been destroyed in the year 3830 which would match with the year 70 CE. The conversion factor of 3760 would stay the same.
We should note that the Torah only discusses time passing after the creation of Adam. That is, the age of the universe is only counted from the creation of Adam. Once the Adam was created, we have the timeline that I show as part of Rabbi Leibtag shiurim: The Hebrew Calendar and its Missing Years- Parts 1 - 3. Thus the flood occurred in 1656, Avraham was born in 1948, The Exodus occurred in 2448, The first temple was built in 2928, etc. This gives the current year as 5778. The dates that I mention (from the creation of Adam until the destruction of Bais Hamikdash Harishon) are directly from the pesukim in the Tanach.
The wikipedia article that you quote apparently has certain dates that are either mistaken or based on philosophical reasoning. I would say that the dates shown are mistaken as I show in the timelines below. The article claims that the first temple lasted 430 years, However, it neglects to account for the fact that partial years in the reigns of kings were counted as full years.
In order to calculate the duration of the first Beit HaMikdash, Sefer
Melachim records the length of each king’s reign. Adding up the reigns
of the kings from Shlomo – in whose fourth year as king the Beit
HaMikdash’s existence began – to Tzidkiyahu – in whose reign it was
destroyed – we have a total of 433 years. However, because the
dating system then was focused on the king and not on an absolute,
continuous calendar (as we mentioned above), the final partial year of
a king’s rule was counted as a full year, and the rest of that year
was also considered to be a full year for the next king. Therefore, we
can conclude that there was an extra year of overlap recorded for each
king. Accounting for the 19 rulers, and therefore 19 years of overlap,
our total reduces to 414 years. We also need to remember that
construction began in the fourth year of Shlomo’s reign. We therefore
remove four years to give the final count of 410 years for which the
first Beit HaMikdash stood. Thus, the Beit HaMikdash was destroyed in
The time line that I show is
Name Father’s age at time of birth Year of birth
Adam N/A 0
Sheit 130 130
Enosh 105 235
Keinan 90 325
Mahalaleil 70 395
Yered 65 460
Chanoch 162 622
Metushelach 65 687
Lemech 187 874
Noach 182 1056
The flood occurred when Noach was 600 years old (1656).
Noach 10:10 says
These are the generations of Shem: Shem was one hundred years old, and
he begot Arpachshad, two years after the Flood.
This means in 1658. Adding the 290 years of his descendants - Noach 10:26
And Terah lived seventy years, and he begot Abram, Nahor, and Haran
we get Avraham was born in 1948, Yitzchak was born 100 years later, and the Exodus was 400 years after that (2448). This assumes that Avraham was the eldest born when Terach was 70, or the youngest so that the verse means that when Terach was 70 all three had been born.
Melachim I 6:1
And it was in the four hundred and eightieth year after the departure
of the children of Israel from Egypt, in the fourth year, in the month
Ziv, which (is) the second month of Solomon's reign over Israel, that
he did (begin to) build the house of the Lord.
The first temple was built 480 years after the Exodus or 2928. It lasted 410 years, bringing us to 3338. This was based on the count of the kings in Tanach and Seder Olam. This brings us to the end of the Tanach.
I carefully do not use the common secular date that is written for the destruction of the first Bais Hamikdash as there are disputes in the matter which would affect the actual date that we derive for the current year and the age of the universe.
The meforshim and the gemara tell us that the galus bavel lasted 70 years (from the destruction) and the second bais hamikdash lasted 420 years. See the timelines at the end of the post that I cite for details. Seder Olam tells us that the building of the second temple occurred at the end of the 70 years of galus bavel (3408) and lasted 420 years, being destroyed in 3828.
We are told that this was the equivalent of 68 CE in the modern secular calendar. Thus, the conversion factor is 3760 years. That is, add 3760 to the current secular year to get the age of the universe count. That is, when the secular and Hebrew years are again in sync (after January 1, 2018), we see that our count of the age of the universe is 5778 years.