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.
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, wheref(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), and0.999...is that sum witha = 9andr = 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.