If we add smaller pieces of pizza infinitely, will we get an infinitely big pizza?

No, not always! Let’s look at this scenario:

Person A gets 1 piece as big as 1/4 or quarter of the pizza.

Person B gets 1 piece 1/16 of the pizza (=a quarter of person A’s pizza)

Person C gets 1 piece 1/64 of the pizza (=a quarter of person B’s pizza)

Person D get 1 piece 1/256 of the pizza (=a quarter of person C’s pizza)

……………………………………………

The pattern here is that the next person will get a quarter of what previous person has.

Continue this pattern infinitely (never ending pattern)

Now if we collect all pieces of pizza from all these people. Do we have an infinitely big pizza??? Let’s see, our sum will be:

1/4+ 1/16 + 1/64 + 1/256 + … = what is this??

Now, we can’t manually add them since we have an infinite number of terms but we can observe that the later the person is in the line, the smaller the pizza he/she will get and they can be super super super tiny that is like nothing.

Let’s turn this analysis graphically and see what we get! Let’s look at the picture below and observe the area of the purple square.

You may notice that:

The first purple square = ½ x ½ = ¼
The 2nd purple square = ¼ x ¼ = 1/16
The 3rd purple square = 1/8 x 1/8 = 1/64
The 4th purple square = 1/16 x 1/16 = 1/256

……….

As you can see, the pattern for the area of the purple squares also resembles an exact pattern for our problem. What is obvious in the graphical representation is that you can see the sum of all the area of the purple square is less than 1 since the area of all the purple square adding with the area of the white square = 1 x 1 = 1. Hence we have just graphically prove that regardless the fact that we are adding the area of the purple square infinitely, the sum is still less than 1 and hence it is a finite!
The above summation pattern is actually called a GEOMETRIC SERIES and it has a very nice property: the summation = (1 – common ratio) / common ratio.

In our case, the area of the current square is always ¼ smaller than the previous square so common ratio in this context is ¼ . If we apply this property, our result will be [1 – ¼] / (1/4) = (3/4) / (1/4) = 3/1 = 3.
Conclusion and implication: if we have an infinite population, 1st person gets a quarter (1/4) of a pizza, 2nd person gets a piece of pizza 4 times smaller than the 1st person, 3rd person gets a piece of pizza 4 times smaller than the 2nd person, and so on…. Our world will have a total of 3 pizzas.