If a number like Pi is infinite, how do we know each decimal that is newly calculated is valid?

Question

Not a mathematician here at all so perhaps my question is phrased incorrectly.

Let’s say through thorough testing in reality, we can prove with certainty Pi is correct up until 5 decimal places,

3.14159

The computers that are calculating Pi to an endless degree, how do they validate new values that are calculated as correct and cannot be otherwise?

In: 434

Answer 1

This is NOT how a calculator does it (the formula is different) but ancient mathematicians (before computers) used geometry to calculate approximations of pi.

First, start with the definition of pi: the ratio of the area of a circle to the square of its radius. This gives the geometric definition.

Now draw a circle (declare it a circle with 1 unit radius), therefore the area of that circle is (by definition) pi.

Within that circle draw a square that is fully within the circle and has the vertices on the edge of the circle. By Pythagoras theorem, we know that the side length of the square has to be 2/sqrt(2). and therefore the area is 2.

Now draw another square except the square encloses the circle and the midpoint of the edges of the square just touch the edge of the circle. We know that the square’s side length is 2 (twice the radius of the circle). The area of this square is 4.

Since the area of the circle must be greater than the smaller square and smaller than the larger square, we establish that pi must be between 2 and 4.

The trick is that we don’t have to use squares. By using to regular polygons with more sides (hexagon, heptagon, octagon…) as long as there are methods to calculate the areas (generally very simple formulas), it is always possible to bracket the lower and upper bounds of the value of pi by geometry.

As long as number of sides of the polygon are increased, the estimate for pi becomes more accurate to more decimal places.

Modern calculators use a different method (there is an infinite series expansion to give pi) but conceptually similar. As long as more and more terms of the infinite series are calculated, more and more accurate values of pi can be calculated.