The mathematician Augustus DeMorgan lived his entire life during the 1800s. In the last year of his life he announced "Once I was x years old in the year x^2". In which year was he born?

the only perfect square between 1800 and 1900 is 1849 = 43^2

so, he was 43 in 1849, and born in 1806

40^2 = 1600

50^2 = 2500

So his age has to be between 40 and 50

What number between 40 and 50 when squared is a number between 1800 and 1900?

43^2 = 1849

if he was 43 in the year 1849, then he was 0 (the age he was born) in ???

To find out in which year Augustus DeMorgan was born, we need to solve the equation "Once I was x years old in the year x^2."

Let's assume that he was born in the year "y." When he was x years old, it means that the current year is y + x. According to the statement, this year, y + x, must be equal to x^2.

So our equation becomes: y + x = x^2

To solve this equation, we need to isolate y. Rearranging the equation, we get: y = x^2 - x

Now, let's substitute the given information into the equation. Augustus DeMorgan lived his entire life during the 1800s, so the year in which he died would be between 1800 and 1899. If we assume that he announced this in the last year of his life, then the maximum value of x would be 99, as he could not have been older than 99 years old.

Now, we can substitute the values of x into the equation y = x^2 - x and check each possible solution from x = 1 to x = 99 to find the year of his birth.

By checking each value of x, we find that when x = 44, the equation holds true: 44^2 - 44 = 1936 - 44 = 1892.

Therefore, Augustus DeMorgan was born in the year 1892.