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by dmfdmf 121 days ago
Pi is not a number, it stands for a method (i.e. infinite series) to calculate a number. Conceptually it is the ratio of circumferance to the diameter of a circle which are incommensurate quanties, i.e. can't be represented as a rational number.

It is a subtle distinction but important. We define the exact value based on the context. If I am tiling my circular patio then 3.14 is fine to calculate how many tiles I need. If I going to the moon or mars then I need more decimals or I will miss the target.
1 comments
pavel_lishin 120 days ago
By that logic, the square root of two and i are also not numbers, right? In fact, any non-rational "number" isn't a number by that definition.
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dmfdmf 120 days ago
Setting aside i, math symbols that stand for irrational numbers can be treated as numbers in theorems or derivations or proofs but one needs to be careful that a symbol such as Pi or SQRT[2] actually designates an infinite series to calculate a rational number. This distinction is important when math equations are actually used to calculate or measure something specific which is the whole point of math. All valid measurements can only result in a rational number. It is the distinction between doing math and doing physics or engineering (i.e. applied math) which has to be integrated, they are not separate fields with regards to measurement.

The imaginary number i=SQRT[-1] is the base solution to the polynomial equation -y= x^2. If you read the history it was invented to solve certain types of cubic equations, i.e. as a heuristic of method. So not only is it not a number it is a bare contradiction. While i was useful to solve some subclass of all cubics it did not lead to a general solution to cubic equations. Nevertheless the mathematicians ran with it and added complex numbers to the definition of number so that the number system could solve all possible polynomial equations.

In my opinion imaginary numbers are a kludge and deadend and it is masking the real issues in math, i.e. the ghosts of departed quantities. In math some symbols stand for an infinite series but you can't just choose any arbitrary series for Pi or e or SQRT[2], they all have to be defined as part of a system of measurement with clearly defined and globally defined epsilon/delta (i.e. precision) of measurement to get valid results.

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