The Indians got to zero in two stages. First they overcame the problem of denoting empty spaces in place-value notation by drawing a circle around the space where there was a "missing" entry. This much the Babylonians had done. The circle gave rise to the present-day symbol 0 for zero. The second step was to regard that extra symbol just like the other nine. This meant developing the rules for doing arithmetic using this additional symbol along with all the others. This second step – changing the underlying conception so that the rules of arithmetic operated not on the numbers themselves but on symbols for the numbers – was the key.
The paragraph given explains how the Indians “got to zero”. The first step was denoting the empty place values in numbers with a circle. This later became the symbol 0. The second step, according to the paragraph, was the key. This involved treating the the extra symbol zero just like the other symbols for numbers such as 1, 2, 3 etc., and developing rules for doing arithmetic using zero as with the other numbers. The penultimate line explains how this involved applying the rules of arithmetic not on the numbers themselves (which would have excluded zero) but on the symbols for the numbers (which included zero).
Now let us look at the options to see which one completes the paragraph best:
Option A. Indeed, our sense of numbers depends on the symbols, and we cannot divorce the symbols from the numbers they represent.
This line talks of “our sense of numbers” and how we think of them in terms of their symbols. This is not the right choice to complete the paragraph given, which is about how the concept of zero evolved.
Option B. Over time, it led to a change in the conception of numbers to a more abstract one that included zero.
The paragraph details the two-step process by which Indians got to zero. The penultimate sentence explains the key idea that changed the way the rules of arithmetic were applied. Option B, which talks of how this “led to a change in the conception of numbers to a more abstract one that included zero” seems to be a good choice to complete the paragraph.
Option C. Everything becomes much clearer when there is a special symbol to mark a space with no value.
The use of a circle to mark empty spaces in place value was discussed as step one towards zero in the paragraph. The paragraph then moves on to the next idea about how the rules of arithmetic were applied to zero. It does not make sense to conclude the paragraph by stating a fact that was already noted earlier. So we rule out option C.
Option D. A remarkable thing about this number system is that using just the ten digits from 0 to 9, we can represent any of the infinitely many positive whole numbers.
This is a new, related idea about the number system that included zero and its remarkable ability to represent infinitely many positive whole numbers. However, the given paragraph is about how the concept of zero evolved. Option D does not not fit in at the end of this paragraph.
So, of the given choices, B is the best one to complete the paragraph.
Correct Answer: Choice (b)