What is Zero?

What is Zero?

Until now, we have not said anything, either in single digit numbers or counting, about a special number which we all know as zero.

Zero is a number which we need to understand, before studying representation of numbers bigger than 9.

Zero as signifying “Nothing”

The idea of zero must have developed as another way of saying that there was “nothing.”

It is just a “mental” concept. There is nothing in this world which is truly “nothing.” Even in the far reaches of the universe in the vacuum there is “space” and a few molecules.

When we use the word “zero” in this chapter, we mean the “idea of zero.” Different cultures gave different names for this idea.

In human cultures the idea of numbers started as a way to quantify collections. Zero cannot be represented with physical materials as the numbers 1 to 9 can be. Hence for a very long time zero was never conceptualised as a number.

Zero & the Greek & Sumerian Civilizations

Greeks primarily focused on Geometry and the logic connecting various topics in it. They did not focus too much on arithmetic.

They did not develop a distinct numeral system. They used their alphabets with a line over them in place of numerals.

Even their study of numbers and their properties was through geometrical shapes.

They were aware of the idea of zero representing "nothing".

They were also aware of the sexagesimal system & the use of a blank space (concept of zero) as a placeholder, developed by the Sumerians.

But several issues prevented them from thinking of zero as a number. Their philosophy stood in the way of conceptualising zero. Aristotle had stated that "Nature abhors a vacuum".

Zero could not be represented geometrically. Their geometry used ratios and zero had no role in ratio representation.

Hence though Greeks developed Geometry to a very complex levels, they missed out on thinking of zero as a number and developing arithmetic.

Zero/ Soonya in the Hindu Culture

It was Hindus who gave a mathematical perspective to the idea of "nothing".

The concept of Zero started in India as "Sunya" in Sanskrit. In Indian & Buddhist philosophies Sunya meant "nothingness" as well as "completeness".

The opening verse of Ishopanishad “Om Purnamadah Purnamidam Purnat Purnamudachyate Purnasya Purnamadaya Purnameva Vashishyate Om Shanti,Shanti, Shanti” talks of taking the infinite from the infinite leaving the infinite.

Indian philosophy talked of "Smaller than the smallest & larger than the largest". The Rig Veda uses at least 3000 different number names of various magnitudes. It also mentions several huge powers of ten.

This cultural mooring possibly helped them to conceive of "nothing" as something! It was easy for Hindus to think of "nothing" as a concept as well as a number.

Zero & Infinity

Archimedes gives us a way to understand the equivalence between zero & infinity.

Imagine we start with a triangle, which has 3 corners. If we keep on increasing the number of sides, the number of corners would also increase. If we imagine a polygon with infinite number of sides, the number of corners would also approach infinity. But then the polygon approaches a circle which has zero corners.

In a sense zero corners has become the same as infinite corners!

Zero & the Hindu Counting Systems

Hindus had the system of chanting a mantra 108 or 1008 times. Many cultures had similar rituals and usually used a string of beads.

Hindus however used the fingers of both hands to keep count. The palm of a hand has 12 finger joints or divisions. Ten of these could be touched sequentially by the thumb, to count from One to Ten. When the thumb reaches the last joint (which is ten), the thumb of the other palm is made to touch the first joint. This is equivalent to showing "ten" on the second palm.

Simultaneously the other palm is opened and shows that there is "nothing" in the palm or it implies "sunya" or "zero". Thus the idea of "zero" had a concrete representation of an open palm.

Zero as a Number

Hindus thought of zero as a number on par with numbers one to nine. They treated zero as meaning that there is nothing to count of a particular thing in a particular context. This is different from the simplistic notion of zero as nothing.

To give an example, the classroom has many boys, girls, windows, fans etc each of which can be represented by a number. But there are no elephants in the classroom! Hence in the context of the classroom, there are “no elephants.” A mathematical way of telling this is to say “there are zero elephants in this classroom!”

Use of zero also lets us inject some humour into the classroom and train children to say that "I have zero Geometry Boxes" instead of saying "I have not brought my Geometry Box"! The humor will help them remember the concept.

In spite of this understanding, in primary schools, even today, students are mostly taught that "Zero is Nothing". Brahma Gupta in the 6th & 7th century was the first to write about zero and operations with zero in his Brahmasputasiddhantha. This was also the first written record of zero being considered a number.

Number One

It may surprise most people that even the idea of "one" as a number took a long time for Greek philosophers to accept. Similar to the idea of "zero as nothing", they thought of one philosophically as the "whole".

It took them some time to conceptualize the idea of one as a number!