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It has been said that the invention of zero was a great leap forward, not only in abstract understanding, but in the ability to introduce place value notation and do computations; computing using Roman numerals was notably clumsy.

However, the ancient Chinese did not really have place value notation either, and did not have a symbol for zero. Yet that culture discovered many pretty advanced arithmetic results (for example, the Chinese Remainder Theorem).

So what exactly was it about the invention of zero in Western culture that was so useful?

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  • $\begingroup$ In my understanding, it was the fact that it was avoided. For example the fundamental theorem of algebra would have been stated way earlier if negative square roots were accepted smoother. $\endgroup$ – Ali Caglayan Oct 28 '14 at 21:40
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It has been said that the invention of zero was a great leap forward, not only in abstract understanding, but in the ability to introduce place value notation and do computations; computing using Roman numerals was notably clumsy.

Zero is not necessary in order to have a place-value system. Nor is zero necessary (or sufficient) in order to develop sophisticated mathematics. The Mesopotamians had a place value system by, at the latest, 1800 BCE, but had no symbol for zero at that time. Theirs was a sexagesimal (base-60) system. Where we would put a zero, they left a blank space. (But only between digits, and not as a trailing digit, which had the implication that a string of digits defined a number only up to a power of 60 (which could, effectively, be negative). The magnitude of a number had to be inferred from context.) By around 200 BCE, the Mesopotamians had introduced a place-holder zero, but there is no evidence the zero was treated as a stand-alone number.

The Greeks did not use a place value system (although Archimedes devised a special purpose place value system for dealing with very large numbers in The Sand Reckoner) and did not recognize zero as a number. Nevertheless, they accomplished great things in mathematics, including developing a sophisticated axiomatic geometry, proving the irrationality of the square root of 2, proving the infinitude of the primes, developing a sophisticated theory of ratios of incommensurable magnitudes, developing the method of exhaustion and using it to compute the volume of the sphere and the area of a parabolic segment, developing the theory of conic sections, and so on.

The Maya, by 1000 CE (and probably well before), had a place value system that did use zero. Theirs was a vigesimal (or base-20) system. Furthermore, zero was treated as a number: counting started from zero rather than one in their calendar. Their calendar was quite sophisticated and they also had a well-developed astronomy. Nevertheless, it does not appear that their mathematics advanced very far (although it is hard to say for sure, since the majority of their artifacts were destroyed). For example, there is no evidence that they had a decent multiplication algorithm.

The example of China has been addressed in the other answer. The rod numeral/counting board system certainly was a place value system. In that system, an empty place was indicated by leaving a gap rather than by an explicit zero. Many of the problem solutions in classic Chinese mathematical texts were formulated as steps for carrying out a computation on the counting board. To develop something like the Chinese remainder theorem doesn't require a zero. The notion "leaves remainder zero" can always be formulated as "divides evenly". (A similar formulation would have been used in Greek mathematics.) As already noted, the counting board technology had an implicit place-holder zero, but not a full-fledged zero. The counting board technology was sufficiently powerful to enable many great achievements, such as a very accurate estimate of pi.

So what exactly was it about the invention of zero in Western culture that was so useful?

The modern zero was, in fact, invented in India. Knowledge of the Indian system diffused, first to the Islamic world, and later to Europe. Indians knew how to perform arithmetic with zero, and introduced the notion that $1/0$ produces an infinite result.

I think it is clear that efficient calculation systems can be developed without the recognition of an explicit zero. That is not to say that recognition of zero as a number was not extremely important. To give but one example, the theory of roots of polynomials becomes much less cumbersome when zero (and negative numbers) are recognized. Before that happened, the theory involved a complicated case-by-case analysis. The algebraic solution of the cubic and quartic, discovered by del Ferro, Fontana, Cardano, and Ferrari during the Renaissance, as well as earlier work by Omar Khayyam, saw the equations $x^3 = a$, $x^3 + bx = a$, $x^3 = bx + a$, $x^3 + bx^2 + cx = d$, and so on, as different equations, each requiring an individual treatment. The ability to treat such sets of related equations in a uniform manner certainly accelerated progress.

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The successful numbering system did not have 0, even as a place holder, due to them not needing it, according to the University of St. Andrew's website Chinese numerals, due to them developing a system that had symbols for the larger values, i.e.

enter image description here

However

a second form of Chinese numerals began to be used from the 4th century BC when counting boards came into use. A counting board consisted of a checker board with rows and columns.

enter image description here

(Both images are from the link above).

To represent a '0' place holder, a gap was left in its place.

Xiahou Yang's Xiahou Yang suanjing (Xiahou Yang's Mathematical Manual) written in the 5th century AD notes that to multiply a number by 10, 100, 1000, or 10000 all that needs to be done is that the rods on the counting board are moved to the left by 1, 2, 3, or 4 squares. Similarly to divide by 10, 100, 1000, or 10000 the rods are moved to the right by 1, 2, 3, or 4 squares. What is significant here is that Xiahou Yang seems to understand not only positive powers of 10 but also decimal fractions as negative powers of 10.

The significance to Western ideas, according to the YaleGlobal web article The History of Zero (Wallin, 2002), originally, zero was a placeholder (in the sense of being used for tens, hundreds, thousands etc). According to the article, it was in India where the notion of zero started to have meaning by itself in about the 7th century.

The significance of zero in Western Culture when it arrived in Europe in the mid-12th century through Spain and via Fibonacci in the early 13th century, gaining merchants a more efficient way to balance their books, with

Fibonacci's developments quickly gained notice by Italian merchants and German bankers, especially the use of zero. Accountants knew their books were balanced when the positive and negative amounts of their assets and liabilities equaled zero.

This actually got the number outlawed, so was named 'cipher' in merchant based encrypted messages to overcome the ban.

Rene Descarte was able to progress the use of zero in what we know as Cartesian coordinate system. With the origin of the coordinate system occurring at (0,0).

From there, arguably the greatest innovation of the notion of zero was from the dilemma of dividing by zero, which (from the article):

In the 1600's, Newton and Leibniz solved this problem independently and opened the world to tremendous possibilities. By working with numbers as they approach zero, calculus was born without which we wouldn't have physics, engineering, and many aspects of economics and finance.

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  • $\begingroup$ Does that book discuss the advances that the Chinese achieved without either zero or place notation? $\endgroup$ – rogerl Oct 28 '14 at 23:48
  • $\begingroup$ No, they were focused on your main question regarding the impact on Western Culture - the Chinese innovations could almost be a separate question. $\endgroup$ – user22 Oct 28 '14 at 23:51
  • $\begingroup$ Hmmm...then I posed my question poorly. Yes, I was interested in why zero was such an advance...but what I was trying to ask really was "why could the Chinese do as much as they did without a zero". Perhaps answering that question sheds light on why the invention of zero was (or was not) a watershed moment. $\endgroup$ – rogerl Oct 28 '14 at 23:53
  • $\begingroup$ This question, as it is, is very valid - as the introduction of the concept of '0' into Europe did have a monumental effect (as stated in my answer). The Chinese arithmetic would be a great additional question - I believe is a very different aspect. $\endgroup$ – user22 Oct 29 '14 at 0:03
  • $\begingroup$ @rogerl I have found a great resource for what the Chinese used - I will edit it into the answer $\endgroup$ – user22 Oct 29 '14 at 0:14
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The value of a digital system of recording numbers is that it allows for a more concise and understandable method of doing arithmetical computations. Before Hindu numerals were used, the use of composite systems (like Roman numerals) were usual. So, for example, 83 is LXXXIII. The modern digital system was originated by Fibonacci in the Liber Abaci, but it took literally hundreds of years before it became used systematically. For a long time there was a fight between the "abacists and algorithmists" over which system was better. There were even contests and tournaments where the two different schools would compete head to head in speed adding matches.

Eventually it became clear that adding and multiplying digital numbers was easier than working with fractional or composite values. This occurred with the publication of Napier's Mirifici Logarithmorum, as edited and improved by Edward Wright in 1616. At first Napier's book was just noticed by calculators, but after John Wallis adopted its methods around 1660 the use of digital notation and decimals became universal and decisive.

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  • $\begingroup$ They were already common in Surya Siddhanta, at least 4th century, probably as old as Rg Veda.Astronomy used concepts like Yuga and Kalpa and used number of orbits in each $\endgroup$ – Partha Shakkottai Jul 21 at 20:33
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Abstraction. The value of abstraction is that it enables us to work formally, without regard for meaning. The invention of zero was important because it increases the level of the abstraction, and hence decreases the need to keep track of meaning.

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0 and the place value notion is required for good arithmetic. Without it modern Science, Engineering and Astronomy and an accurate Calendar and Navigational Charts would not have developed. An example is shown below.

The importance of 0 is for accurate practical utility and not merely notional advantages of easy descriptions.

The Bakhsh1ali Formula (from 4th century manuscript)

Suppose we wish to find sqrt(N), where N > 0. We express N as N = A^2 + b, where |b| is small in comparison with A (as small as possible within the limits of easy guesswork

By taking A=3 and b=1,the fraction 3 and 1/6 = 3.167; computation, using only rational operations, i.e., +, -, * , /). Using the linear approximation sqrt(1+x)= 1+x/2 ,

for x ~ 0 (this is the tangent approximation) we get: sqrt(A^2+b)= A+ b/2A. This formula, which was known to the Babylonians, yields fairly good approximations; e.g., for SQRT(10) it yields, A=3 and b=1,the fraction 3 1⁄6 = 3.167 compared to 3:162.

But the Bakhsh1ali formula goes further, by inserting an extra term. It states, in effect: In the case of a non-square number, subtract the nearest square number, divide the remainder by twice this nearest square; half the square of this is divided by the sum of the approximate root and the fraction. This is subtracted and will give the corrected root. After one has deciphered this recipe, it turns to be equivalent to the following formula: 
 (A^2 +b)^1/2 = A +b/2A - (b/2A)^2 / 2(A +b/2A).

For example, if N = 11 we may take A = 3 and b = 2. The formula then gives: 
3 + 1/3 - 1/9 /2(3+1/3) =3+1/3-1/60 = 3.31667

Compare this with the true value: (11)^1/2 = 3.31662. We see that though b is far from being `small' in comparison with A, we have still got four decimal place accuracy.

Choosing A=3.3 and b = 11- 3.3^2 the accuracy jumps up to 79201/23880 which is 3.316624790, accurate to 9 decimals. from

https://www.ias.ac.in/article/fulltext/reso/017/09/0884-0894

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