Zero as a Number

Once zero was considered a number, solutions to several related issues were also worked out.

A Numeral Shape for Zero

As the first step of considering zero as a number, a numeral shape had to be found for it. It took a long time before a separate, acceptable numeral shape was conceptualised.

One reason could be that, in those days, tradition was to write out mathematics in verse form, using various synonyms in place of numbers. This is known today as “rhetoric math” as against the modern “symbolic math.”

In rhetoric math, zero could be written in text as “zero or soonya” and there was no need to use a numeral symbol. Hindus gave zero a numeral shape. It has gone through several shapes, including a dot, and finally became "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 seems 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 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 confirming 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 Spreads across Cultures

The decimal place value system developed by the Hindus, using a zero, spread to Europe through the Arabs.

Arabs in the 10th century translated the “idea of zero” 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.

Operations with Zero

Once the Hindus accepted zero as a number, they worked out the rules of arithmetic operations with 0.

The Indian mathematician Brahmagupta (7th century) was the first to give these rules.

Some of these rules were intuitive. Some others needed deeper thinking.

Additions involving 0

These rules were intuitive as 0 can be imagined as a basket with “no pencils”.

“Putting together” the contents of 2 baskets, one containing 3 pencils and the other not containing any pencils will result in a total of 3 pencils.

Hence 3 + 0 will be 3.

We can extend the rule to say “any number” + 0 = “the same number”.

By the same logic 0 + 0 will also be 0.

Subtractions involving 0

These rules can also be derived intuitively using the “take away” metaphor.

“Taking away” “no pencils” from a basket with 3 pencils will result in the basket having 3 pencils.

Hence 3 – 0 = 0

We can extend the rule to say “any number” – 0 = “the same number”.

By the same logic 0 – 0 will also be 0.

Multiplications involving 0

We can use the metaphor of “addition of equal quantities” to derive the results.

We can imagine 3 X 0 as 3 baskets each having “no pencils.” Hence the total of pencils in all the 3 baskets would be 0.

Hence 3 X 0 = 0.

We can extend the rule to say “any number” X 0 = 0.

By the same logic 0 X 0 will also be 0.

Division by zero

The first attempt to define the division by zero was done by Brahmagupta around 628 CE. After that, Bhaskaracarya (c. 1150), while discussing the mathematics of zero in Bijaganita, explains that infinity (ananta-rasi) which results when some number is divided by zero is called khahara.

Let us assume that 3 ÷ 0 = n where “n” is some number.

Then by the property of division & multiplication we can write n X 0 = 3

This goes against the property that n X 0 = 0.

Since dividing by 0 gives contradictory results, division by 0 is accepted to be “not defined”.

The current understanding in math is that division by 0 is not defined.

There is an unauthenticated story about Ramanujan & zero.

While his teacher was explaining the idea that any number divided by itself would always be 1, Ramanujan is supposed to have asked if that is true of 0 ÷ 0 also! We do not know how the teacher responded.

Number Zero as the Place Holder

An understanding that any number multiplied by 0 remains 0, made Hindus realise that 0 can be used as a place holder in the place value system, in place of the blank space.

Take the problem of representing four hundred five with numerals.

In a previous section we had mentioned that Sumerians wrote the number as 4 5, with a space between 4 & 5 to convey that the number had no numeral in the “ten’s” place. We also dealt with how the blank space could confuse the readers.

Hindus proposed that this number could now be written as 405.

The value of this number is 4*10^2 + 0*10^1 + 5*1 which is four hundred five because the middle term 0*10^1 will be 0 since it is a number multiplied by 0!

By considering 0 as a number (without value), Hindus removed the confusion which existed in the representations with the place value system.

Advanced Operations involving 0

With the development of math, many other operations involving 0 were also defined.

Factorial of 0: 0!

The normal definition of the factorial of a (whole) number is the product of all numbers preceding that number. It is represented with a “!” sign placed after the number.

Hence 4! = 1 X 2 X 3X 4 = 24, 6! = 1 X 2 X 3 X 4 X 5 X 6 = 120

But to satisfy laws of arithmetic 0! needs to be 1.

Factorial operation emerged out of the study of "permutations", the number of ways a set of things can be rearranged.

There is only 1 way you can permute "nothing", which is by simply doing nothing. In this sense, permute 0 objects and 1 object is the same. In both scenarios, you cannot change anything, therefore in the combinatorial sense, the two are equivalent.

So, 0! =1 & 1!=1

0 as an Exponent

By applying the rules of operations with exponents any number raised to 0 has to be 1.

So, 3^0 = 1 & 5^0=1. In general n^0=1 where n is any number.

This property is useful in making the place value system an algebraic one in structure. We will see details in a later chapter.