The Invention of Zero: Thanks for Nothing!
In this month's issue, I want to delve into something we all encounter daily, yet rarely fully grasp its profound significance: the concept of “numbers”. More than just symbols, numbers are the bedrock of our understanding and interaction with the world.
The Indispensable Role of Quantification
At their most fundamental, numbers are essential for describing things quantitatively. Without them, our ability to compare and differentiate between sets of objects would be limited,to situations when the visual difference between groupings is exceptionally obvious. Imagine trying to distinguish between two large groups of items without the ability to count them!
This limitation becomes even more apparent when dealing with particularly vast or minuscule quantities. Our natural senses simply aren't equipped to discern subtle differences at these extremes. For instance, we can't accurately measure or compare the sizes of two oceans or the weight of two microbes using only our eyes or other senses. Numbers provide the precision necessary to analyze and comprehend these otherwise imperceptible realities. They allow us to scientifically examine the world around us, revealing intricate patterns like multiples and fractions that would otherwise remain hidden and incomprehensible.
Numbers: The Engine of Civilization
Beyond simple measurement, the very introduction of numbers as a concept has had a fundamental and transformative impact on human society. They paved the way for the development of currency, which in turn spurred the flourishing of trade and commerce, ultimately leading to the sophisticated banking systems we rely on today.
Before the advent of numbers and, consequently, currency, economic exchange was limited to bartering. Numbers liberated us from this constraint, enabling us to communicate precise quantities — like the size, length, or volume of an object — to a remote audience who couldn't visually verify the information themselves. Suddenly, a buyer didn’t have to be in front of a field of barley, or see a warehouse full of sacks of grain, in order to understand how much he needed to provide in exchange. When banks came into the picture, concepts like promissory notes followed them. The oldest bank still in operation is Banca Monte dei Paschi di Siena, which was founded in 1472 in Siena, Italy. Thanks to these new systems, sales could happen without two merchants ever needing to meet in person, making trade more affordable and accessible, as well as less dangerous and time consuming.
Banking systems overall completely revolutionized economic activity. Loans became feasible, empowering individuals to transform innovative ideas into thriving businesses, even without an immediate access to funds. This financial fluidity has undeniably contributed to the welfare and growth of nations with more robust economic structures. Furthermore, numbers today stand as the only truly international language. Once we collectively agreed on a standard, numbers transcended different populations, cultures, and beliefs. Today, we may not necessarily speak our client's language, but we can effortlessly communicate the cost and quantity of goods we wish to purchase or sell.
The field of engineering has also reaped immense benefits from numbers. Precise computations became possible, and the detailed transcription of project specifications onto paper allowed for seamless collaboration and sharing of information among diverse teams and across various locations.
The Evolution of Numerical Systems: From Ancient Practices to Modern Standards
As the common idiom goes, "numbers don't lie!"
Historically, however, that adage wasn't always true. Different cultures employed diverse representations of quantities, leading to significant challenges in cross-cultural communication and understanding. It's only in the last few centuries that humanity has reached common ground on the standardization of numbers. While we still grapple with some discrepancies, such as the ongoing debate between the metric and imperial systems—my soul still hurts when I think about the Mars Climate Orbiter (MCO) that crashed years ago due to a metric / imperial conversion error—the progress towards standardization has been monumental.
Ultimately, numbers are a convention; we, as a collective, have agreed to associate specific meanings to particular symbols. While various cultures and populations in the past used different methods to represent numbers, even with the existence of numerical symbols, humans still found basic mathematics surprisingly difficult for a long time.
Simple addition or subtraction was far from straightforward, and if the resulting total was too large, there might not even be a symbol to represent it. Moreover, rules were often ignored or standards unenforced, as evidenced by some parts of the Ancient Roman Empire using "IIII" instead of "IV" for the number four, as can be seen in historical and archeological records.
| Numeral in English | Hindi (Devanagari Script) | Arabic Numerals | Burmese | Ancient Roman | Hebrew | ||
|---|---|---|---|---|---|---|---|
| Numeral | Word | Western | Eastern | ||||
| zero | ० | शून्य | 0 | ٠ | ၀ | - | |
| one | १ | एक | 1 | ١ | ၁ | I | א |
| two | २ | दो | 2 | ٢ | ၂ | II | ב |
| three | ३ | तीन | 3 | ٣ | ၃ | III | ג |
| four | ४ | चार | 4 | ٤ | ၄ | IV | ד |
| five | ५ | पांच | 5 | ٥ | ၅ | V | ה |
| six | ६ | छह | 6 | ٦ | ၆ | VI | و |
| seven | ७ | सात | 7 | ٧ | ၇ | VII | ז |
| eight | ८ | आठ | 8 | ٨ | ၈ | VIII | ח |
| nine | ९ | नौ | 9 | ٩ | ၉ | IX | ט |
| ten | दस | X | י | ||||
The Revolutionary Concept of Zero and Positional Value
A pivotal breakthrough occurred with the introduction of the concept of “Zero” and the mathematical principles behind it.
This groundbreaking idea, in the form we use today, originated in India around the 5th century CE and was later introduced to the Western world by Arabic scholars. Its development wasn't instantaneous and began via the practical need to represent empty spaces in numerical representations, slowly evolving into the concept of zero as a proper number.
Thanks to this invention and the concept of positional value (more on that later), we now possess the incredible ability to represent any number, no matter how large or small, using a limited set of symbols. Before this innovation, new symbols had to be constantly added to each society’s written language systems as quantities grew.
The number of symbols used to represent numbers in any numerical system is known as its "Base." The numeric system we use daily in Western civilization, for example, is called Base 10 because it's composed of ten distinct symbols (0, 1, 2, 3, 4, 5, 6, 7, 8, 9).
The earliest trace of a concept similar to “zero” was found in Babylon (around 400-300 BCE), though it wasn't introduced to Western culture until the 13th century by the Italian mathematician Leonardo Fibonacci, of Fibonacci Sequence fame, (c. 1170—c. 1240–50) who encountered it during his travels in the Arabian peninsula. However, even for the Babylonians, it was a later addition to their system. Interestingly, more recent discoveries revealed that the Mayans in modern-day Latin America had independently developed the concept of “zero” within their own culture around 100-200 CE.
Earlier, I mentioned something called “positional value”. Let’s dig into that.
Zero is what enables us to assign a positional value to each symbol or digit in any numeric base. Without the concept of positional value, we wouldn't be able to distinguish whether the symbol "6" represents 6, 60, or 600. The strategic addition of zeros allows us to place a digit in the correct position to represent its intended value. Thanks to zero, we can now represent an infinite range of numbers—from infinitesimally small to infinitely large—using the same predefined set of digits. This eliminated the previous necessity of having to constantly come up with new symbols to represent larger numbers, as the ancient Romans did.
Now on to positional value.
Consider the digit "2."
It signifies different magnitudes depending on its position in a line of text: at the rightmost position, it simply represents 2.
If you shift it one position to the left, it becomes 20.
Similarly, the numbers 47 and 74 use the same two symbols (4 and 7) but, due to their different positions, they represent two entirely distinct quantities.
The Romans, lacking the concept of zero, had to introduce new symbols like "L" for fifty, "C" for one hundred, "D" for five hundred, and "M" for one thousand, and so on.
With the advent of “zero”, all mathematical operations became significantly easier. The fundamental algorithm for counting became a much more simplified system of rotating digits.
Starting from "0," we progress through each symbol until we reach the last element ("9"). We then return to the beginning of the set ("0") and increment the symbol to the left by one unit:
And so on and so forth.
By convention, any number can be considered to have an infinite sequence of zeros to its left, though we typically omit them (e.g., 1 is understood as ...000001). This positional value of symbols is precisely what allows us to write "12" and "21" and understand them as representing two distinct quantities.
In our widely adopted Base 10 system (0, 1, 2, 3, 4, 5, 6, 7, 8 9) , the position of a digit represents a power of 10 multiplier:
If we were to use a Base 8 system (0, 1, 2, 3, 4, 5, 6, 7), the calculation would differ:
Our society collectively opted for the Base 10 numeral system, likely because it feels intuitive to humans with ten fingers. However, in the past, populations like the Babylonians utilized a sexagesimal system (Base 60). Even without an original concept of zero, this system is believed to be the origin of our current measurements of time (60 minutes in an hour, 60 seconds in a minute) as well as angles in geometry (360 degrees in a circle).
Numbers in the Digital Age: Beyond Base 10
While Base 10 dominates our everyday lives, it's not the only system actively in use today.
In computer science, at least three other numeric systems are prevalent: Binary (Base 2), Octal (Base 8), and Hexadecimal (Base 16).
These systems are favored in computing because they more closely represent how computers internally handle numbers due to their hardware characteristics:
Binary (2): Uses only 0 and 1, directly corresponding to the two logical states understood by computer circuitry (on/off or power / no power).
Octal (8): A byte, the atomic memory element a computer can address, is an 8-bit cell, making octal a convenient representation.
Hexadecimal (16): Provides a concise representation for groups of bytes (16, 32, or 64 bits long––all multiples of 8)
Here are the standardized symbols used by these systems:
Binary: 0, 1
Octal: 0, 1, 2, 3, 4, 5, 6, 7
Hexadecimal: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F
The Broad Impact: Science and Society
As I've already mentioned, numbers have been a fundamental concept in the evolution of our civilization. Beyond their financial impact (currency, banking, etc.), their scientific influence cannot be overstated.
In 1620, Francis Bacon published his treatise, Novum Organum (The New Organ), which broadly laid out what we now recognize as The Scientific Method. This method, which forms the bedrock of all modern science, would have been impossible without numbers. It demands that every new discovery undergo rigorous verification and reproduction before being announced and recognized. Only by describing the process with precise numerical details can different teams proceed with such verification and validation. Without numbers, there would be no data sets to compare.
In space exploration, extreme precision is paramount when calculating the orbit of a satellite or a spaceship. Consider the extraordinary achievement of sending the DART probe in 2021 to an asteroid (Dimorphos), which is orbiting approximately 6.8 million miles (11 million kilometers) from Earth.
The probe took about 14 months to reach this moving target, yet it struck with incredible precision, successfully altering its orbit. The immense volume of numbers and computations required to achieve such a feat is truly staggering.
Numbers are also vital for demographic studies. The Census is perhaps the most well-known example. We rely on numbers to monitor population growth, understand average age and age distribution within a country, all data which subsequently informs state and global policies.
Numbers: A Tool for Truth and Deception
Earlier, I said, "numbers don't lie."
Well, that was a partial truth. Numbers, in reality, are unbiased; when derived from a sound measurement process, they simply report an amount. However, the associated meaning or implication of that amount can indeed be subject to manipulation.
We're all familiar with the classic statistical report stating that "everyone, on average, eats three chickens every month." The underlying truth, often obscured by having the data presented in this manner, might be that some individuals are starving while others are eating more than their fair share. Numbers can’t tell the difference within that context, so an average is an average.
Similarly, an accusation about 3 million dollars of waste discovered in a company budget might initially sound scandalous. However, if you scale things up, you may realize the amount only represents 0.01% of the entire company’s budget. Suddenly, that huge number becomes a mere "drop in the bucket." The number itself hasn’t changed. Its tangible, numerical value remains constant. However, the meaning of that number in our eyes, and its impact, has.
Numbers themselves don't lie, but they must be accompanied by accurate communication to provide the right context and convey their true significance within the larger system. Otherwise, it is all too easy to get to the wrong conclusions if faced with just a number with no context. After all, as I mentioned at the start of this article, humans are not built to be able to accurately perceive or understand numbers once they get to be over, or under, certain limits.
A useful suggestion, if you are ever faced with a number you wish to question, is to attempt to re-do a computation using a different approach and verify if the final result remains consistent, as it should. This can be done via actually redoing the math yourself, if that’s within your skill set, or simply looking for additional sources that can confirm or deny your results. Additionally, always seek to understand the margin of error in any statistical estimate. This crucial piece of information reveals how confident you should be in the figures presented to you.
Yes, Numbers are Important
From the simple act of counting to the complexities of global finance and space exploration, numbers underpin nearly every facet of our existence. They are the silent architects of our modern world, enabling precision, fostering commerce, and driving scientific advancement.
While we often take them for granted, their evolution—from basic representations to the revolutionary concept of zero and positional value—highlights humanity's incredible journey of understanding and mastery.
So, the next time a number is presented to you, don't stop there. Pause to consider not just its face value, but the intricate web of meaning and utility it represents. Always strive to understand its context and actual impact.
Because while numbers don’t lie, people definitely do.
*If you found these topics interesting and want a deeper dive, you can find two additional appendixes here with more information:
More About Number Systems & Sets
How Numbers are Represented in a Computer