Understanding Zero 2

< 6.6 Understanding Zero 1 | Topic Index | 6.8 Zero &amp; Place Value System >

Zero across cultures

The concept of Zero started in India as "Sunya" in Sanskrit. In Indian philosophy 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". So it was easy for Indians to think of "nothing" as a concept as well as a number.

The Rig Veda uses at least 3000 different number names of various magnitudes. It also mentions several huge powers of ten.

Though the inventor of numeral 0 is not known, Brahma Gupta in the 6th &amp; 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.

Arabs in the 10th century translated it as "Sifr" which meant empty. When Italians borrowed the idea from Arabs in the 13th century, they used the Latin term "Zephrium". After a few centuries, it became "Zero" in Italian.

"Sifr" also transformed into another term "Cipher" which was popular in Indian schools a generation ago to indicate 0. Today Cipher indicates a code.

Zero & the Greek Civilization

Greeks primarily focussed 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 sexagesimal system & the concept of zero as a placeholder, developed by the Sumerians. They were aware of the idea of zero representing "nothing".

But several issues prevented them from thinking of zero as a number. Zero could not be represented geometrically. Also Aristotle had stated that "Nature abhors a vacuum". 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 investigating zero & its properties. Greek philosophy stood in the way of conceptualising zero and working on place value systems.

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" acquired a concrete representation of an open palm.

Idea of Whole Numbers

Until the acceptance of Zero as a number, numbers 1 to 9, were always identified with physical objects or events. They were also called Natural or Countable numbers. Zero made mathematicians think of numbers in an abstract manner, as mental objects, unrelated to physical reality. This gave rise to the idea of Whole Numbers which started from 0 and went on to 1, 2, 3 etc.

Whole numbers are numbers thought of as "mental or abstract" ideas. It is one of the initial steps in math ideas getting free from representation using concrete objects.

0 and Integers

0 gave a mathematical base for understanding negative numbers. They were defined as "additive inverses" of whole numbers. The sum of a number and its "additive inverse" was 0.

Its placement at the beginning of the number line enabled negative numbers to be marked on the opposite side of whole numbers.

A Typical Confusion 

The subtle fact that 0, though included in the number system, is not a counting number can cause some confusions in calculating some national days.

Consider that a school is celebrating the Independence Day on 15th August 2015. What is the count of the Independence Day in 2015? If a child was born on 15th August 1948, its age on 15th August 2015 is calculated as 2015-1948 = 67. It is considered that on the day of birth, the child is 0 years old.

With the same logic, many a time, the Independence Day in 2015 is sometimes called the 67th Independence Day. This is wrong because for a country, 15th August 1948 was the 1st Independence Day and that on 2015 is actually the 68th.

Zero in Algebra

In the 18th century, a new method was found for solving algebraic equations by shifting all the terms to one side of an equation leaving the other side equal to 0.

An expression like x2 + 4x = 3 was re-written as x2 + 4x - 3 = 0.

This method had far reaching consequences in the solution of algebraic equations of various types. The solutions of an equation also were referred to as "zeros" of the equation!

This method also provides a visual idea of the solutions to the polynomial. When the polynomial is plotted on a XY plane, the solutions are values of x where the curve intersects the X axis (where y=0)

Symbol "0" for Zero

Even though, Hindus discovered the place value system & the place of zero in it, it was a long time before a separate numeral form was given to it. One reason could be that, in those days, tradition was to write out its mathematics in verse form, using various synonyms in place of numbers. We have already seen how the Roman Number system or word representation of numbers does not require a word representing zero! If tens nines can be represented by the word ninety, there is no need for 0.

The Bakshali Manuscript

The Bodelian Library at Oxford houses a birch bark manuscript of about 70 pages, which was discovered in 1881 in the village of Bakhshali, near Peshawar. It was in Sanskrit and seemed to contain exercises in practical arithmetic. It is called the Bakhshali Manuscript.

It seemed to be a training manual for merchants on commercial transactions. It provides techniques and rules for solving arithmetic, geometry & algebra problems. It also provides a formula for calculating square roots. The manuscript uses a "dot" as a symbol for zero.

This was a conclusive evidence of the use of a numeral for zero. But this manuscript was dated around the 10th to 12th centuries. Hence there were questions as to whether this was the "first ever" use of a numeral for zero!

Recently however, radiocarbon dating, has revealed that the manuscript dates to 2nd or 3rd century AD. Hence it is clear that by 2nd or 3rd century AD, a numeral for zero was common enough in the society, to appear in arithmetic examples used for commercial purposes. This is also evidence that a numeral for zero, appeared first in India.

Possibly by the time Brahmagupta codified the operations involving zero, the numeral form had reached its present shape "0". This must have been the form which reached Europe through Arabs.

Zero in the Physical World

We have been talking till now of zero as a mathematical concept. But advancements in science have pointed to the existence of a zero in the physical world - the Absolute Zero temperature.

All materials are made of molecules which have a certain energy which is the energy of their motion. Lord Kelvin showed that as we cool the material, the motion of the molecules and hence their energy reduces. There is a point at which all motion ceases.

This temperature can never be reached but has been calculated as -273 degrees centigrade. This temperature is the "absolute zero" of the Kelvin temperature scale. According to the Kelvin scale the freezing point of water is 273 degrees Kelvin.