Double posting this.
Ok the observable universe is 8.8*1026 meters in diameter. We'll call this d.
The difference between the actual circumference of the universe and the 39 digit representation would be written as:
πd - π39d
This would be about -2.5*10-12 meters. The width of a hydrogen atom is 2.5*10-11 meters. So you can see the difference between the actual circumference and approximation is smaller than a hydrogen atom.
Now to calculate how long it would take to require an extra decimal place I thought of it like this:
Since the difference between the margin of error in the approximation and the size of a hydrogen atom is almost exactly one order of magnitude, the size of the universe would need to be one order of magnitude larger before the size of the hydrogen atom would eclipse the margin of error. This would be the math involved in that thinking:
8.8*1026π - 8.8*1026π39 = -2.5*10-12 (margin of error in the approximation)
8.8*1027π - 8.8*1027π39 = -2.5*10-11 (margin of error with the universe one order of magnitude larger, now the same size as a hydrogen atom)
Now we need to find out how many meters that order of magnitude is.
8.8*1027 - 8.8*1026 = 7.92*1027 meters
So the universe will need to be 7.92*1027 meters larger before another decimal place is needed. Now how many years will it take for the observable universe to
expand that much?
The observable universe is expanding by 1 light year per year. One light year is 9.46*1015 meters.
All we need now is some division. We have to divide the difference in meters by the amount of meters the observable universe expands per year.
7.92*1027 meters / 9.46*1015 meters per year = 8.37*1011 years
Written normally, that would be 837 billion years.
For reference, the universe is only 13.8 billion years old.
MATH IS FUN
