Yes, we can write 2.797979... as the decimal expansion of an irrational number.
We can express this number in the form of a repeating decimal, which is of the form p/q, where p is a finite decimal and q is the number of repeating decimals.
Let x = 2.797979...
Multiplying both sides by 100 gives us:
100x = 279.797979...
Subtracting x from 100x, we get:
99x = 277
x = 277/99
This is a rational number because it can be expressed as a fraction of two integers.
However, we can express this number in terms of an irrational number by taking the square root of 3.
Let y = sqrt(3)
We can write y in terms of a continued fraction, which is an expression of the form:
a0 + 1/(a1 + 1/(a2 + 1/(a3 + ...)))
where a0, a1, a2, a3, ... are integers.
The continued fraction for sqrt(3) is:
sqrt(3) = [1; 1, 2, 1, 2, 1, 2, ...]
This means that:
sqrt(3) = 1 + 1/(1 + 1/(2 + 1/(1 + 1/(2 + 1/(1 + 1/(2 + ...))))))
We can approximate this expression to get a decimal expansion of sqrt(3):
sqrt(3) ≈ 1.73205080757...
If we multiply this by 1.61616161... (which is the repeating decimal expansion of 0.797979...), we get:
sqrt(3) * 1.61616161... = 2.7979594554...
This is very close to the original number 2.797979..., but it is not the same. Therefore, 2.797979... is not a decimal expansion of an irrational number.