The world of mathematics has had a profound understanding of zero. Even its mere existence had been in question for many years - as its abstraction from the sensory world had been a hard concept to fully grasp for humankind. However, either used in binary, in operations as a placeholder, or as the concept of “nothingness”, zero has been an integral part of our world and the world of mathematics for the last 5000 years - when it was first used in Mesopotamia.  

 

Zero was first utilized as a placeholder in numbers. Consider the number 100, for instance. Although the idea of zero denotes nothingness, it clearly does not mean that it could be ignored. The “0”, in this case, simply reveals that there is nothing in the “tens column”. When the Roman numerals are considered, however, zero’s usage as a placeholder is not encountered due to how the numerals are formatted - for instance in X, otherwise, 10. Thus, zero’s first appearance in the world of mathematics was what initiated our ability to be able to simply add, subtract, and engage in other simple yet immensely integral sorts of operations in mathematics as it introduced the idea of “placeholders”, laying the foundation of humankind’s mathematical understanding of the world surrounding them.

 

After nearly 3500 years, zero was introduced as a number, signifying ‘nothing’ in India, whilst the ancient Mayans introduced zero in their number system as well. It was only then, when it was theorized that if zero was added or subtracted from a number, the number stayed as it was. This simple discovery rapidly spread around the world, to the Middle East followed by Europe, popularizing the current number system we have right now.

 

When we consider the set of zero’s existence in nature—or counting numbers—however, it is important to acknowledge the Peano axioms. Introduced by the Italian mathematician Guiseppe Peano in 1889, the Peano axioms or postulates provide a rigorous ground for Natural Numbers. This set of axioms (or self-evidently true ideas) establishes the foundation of our understanding of the Natural Numbers. The most important axioms are as follows:

 

1. 0 is a natural number.

2. Every natural number has a successor number, which is also a natural number.

3. Zero is not the successor of any natural number.

 

    The first axiom, in a self-explanatory manner, introduces 0 as a natural number. Axioms 2 and 3 open a new concept for natural numbers. In these axioms, zero is used as a set, ground value, which is where all other numbers derive from. The existence of 1, for example, is not self-evident. However, the existence of a successor number to 0 (denoted as S(0) for a successor function S) is self-evident, producing the number 1. Thus, it is not the usefulness of zero that makes it important, but rather how it lays the foundation for our daily understanding of numbers surrounding us.

 

    Whether in Mesopotamia 5000 years ago or in India and Central America 3500 years ago or in today’s world, zero has had a notable impact on humankind’s understanding of basic mathematical concepts. Its existence as a concept of nothingness, while true, also fails to fully explain its integral role in mathematics, as it is the source of the mathematical world surrounding us.