Pi isn’t a fraction – it’s an irrational number, i.e. a number with no fractional form in integer bases; as opposed to rational numbers, which are defined as being able to be expressed as a ratio of two integers (a fraction). Furthermore, π a transcendental number, meaning it’s never a solution to f(x) = 0
, where f(x)
is a non-zero finite-degree polynomial expression with rational coefficients. That’s like, literally part of the definition. They cannot be compared to rational numbers like fractions.
Since |r|<1 => ∑[n=1, ∞] arⁿ = ar/(1-r)
, and 0.999...
is that sum with a = 9
and r = 1/10
(visually, 0.999... = 9(0.1) + 9(0.01) + 9(0.001) + ...
), it’s easy to see after plugging in, 0.999... = 9(1/10) / (1 - 1/10) = 0.9/0.9 = 1)
. This was a proof present in Euler’s Elements of Algebra.
Lol what? How did you conclude that if
9x = 5
thenx = 1
? Surely you didn’t pass algebra in high school, otherwise you could see that gettingx
from9x = 5
requires dividing both sides by 9, which yieldsx = 5/9
, i.e.0.555... = 5/9
sincex = 0.555...
.Also, you shouldn’t just use uppercase
X
in place of lowercasex
or vice versa. Case is usually significant for variable names.