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1900s ≠ 19th century. Don't forget about the first century!
1900s formally refers to the years 1900-1909 and 19th century refers the years 1801-1900. There is no year zero so 1800 is a part of the 18th century, 1900 a part of the 19th century, and 2000 is part of the 20th century.
There is only one year every century where the informal terminology you are using and the formal terminology coincide.
Even though 1900s is used to refer to years 1900-1909, this is logically impossible because that would mean the first "decade" CE and the last "decade" BCE would be only 9 years each as there is no year zero and even if their was a year zero you would need one year zero each which makes even less sense.
In reality 1900s can only mean the years from 1901-1910.
Let me make tables for you:
Your usage of 1700s, 1800s, and 1900s:
1700s 18th century 1800s 19th century 1900s 20th century
Proper and logical usage of 1700s, 1800s, and 1900s:
1700s 1701-1710 1800s 1801-1810 1900s 1901-1910
All this may sound unnecessarily pedantic, but it's important to point out.
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