Nothing to Something: Why Top Mathematicians Hated the Number Zero
What is zero? Most people today would say “nothing.” It’s the absence of quantity, the empty space on the number line. It seems simple, obvious even. But would you believe that for centuries, some of the smartest people in the world fiercely resisted the idea of zero as a real number?
The journey of zero from a humble placeholder to a cornerstone of modern mathematics is a mind-bending story. It involves ancient empires, clever inventions, philosophical debates, and a conceptual leap that fundamentally changed how humans understood quantity itself. It’s a tale that reveals how even the most basic mathematical ideas had to fight for acceptance and how cultural perspectives can shape our understanding of numbers.
Get ready to rethink “nothing.” We’ll trace zero’s path from a mere gap-filler to a number with its own unique powers, exploring why it caused so much controversy and how it eventually became indispensable. This story might even make you look at numbers, and the idea of “nothing,” in a completely new light.
Keeping the Place: Zero’s Humble Beginnings
The story starts not with zero the number, but with zero the idea – the idea of needing to show that a place was empty. This need arose in ancient Mesopotamia, the land often called the “Cradle of Civilization,” sometime before 1600 B.C.
The Babylonians, as we learned from clay tablets they left behind, had a sophisticated system for writing numbers. Unlike the Egyptians or Romans who simply added up symbol values, the Babylonians used a place-value system, similar to ours. The key difference? Our system is based on powers of ten (ones place, tens place, hundreds place, etc.). Their system was based on powers of sixty! This is called a sexagesimal system. We still see echoes of it today in how we measure time (60 seconds in a minute, 60 minutes in an hour) and angles (360 degrees in a circle).
They used simple wedge-shaped marks made in clay: a vertical wedge (▼) for ‘one’ and a sideways wedge (◄) for ‘ten’. They combined these to write numbers from 1 to 59. For example, 23 might be ◄◄▼▼▼.
Now, how did they write a number like 72? That’s one group of sixty plus twelve left over (1 x 60 + 12). They would write the symbol for ‘one’ (▼), leave a space, and then write the symbol for ‘twelve’ (◄▼▼). The position of the ▼ told you it meant ‘sixty’, not just ‘one’.
The Problem of the Empty Space:
This worked well, mostly. But what about a number like 3612? That’s one ‘sixty squared’ (1 x 3600) and twelve ‘ones’ (12 x 1), with no sixties in between. How do you show that the sixties place is empty?
Initially, they didn’t! They just left a bigger gap. They might write ▼ followed by a large space, then ◄▼▼. You had to guess from the context whether ▼ ◄▼▼ meant 1 x 60 + 12 (which is 72) or 1 x 3600 + 12 (which is 3612). Imagine the confusion! Trying to read ancient Babylonian accounting records must have been a headache.
The Placeholder is Born:
Sometime much later, between 700 and 300 B.C., the Babylonians came up with a solution. They started using a special symbol – often two small slanted wedges that looked a bit like a dot (we’ll use • here) – to explicitly mark an empty place.
So, 72 was still ▼ ◄▼▼.
But 3612 became ▼ • ◄▼▼.
This little dot wasn’t a number itself. It didn’t represent a quantity. It was just a placeholder, a signal saying, “Skip this place value.” This was the birth of zero, not as a number, but as a crucial part of writing numbers clearly in a place-value system.
A Circle of Emptiness: Zero Travels East
While the Babylonians used their dot, a different symbol for emptiness arose independently, much later, in India. Sometime before 600 A.D., mathematicians in India developed the decimal place-value system we use today, based on powers of ten. They had unique symbols for the digits 1 through 9. Crucially, they also invented a symbol to represent an empty place – initially a dot, but later evolving into a small circle (0).
This system, combining place value with a symbol for an empty place (our zero), was incredibly efficient for writing numbers and, importantly, for doing arithmetic calculations directly with the written numbers.
The Indian Leap: Zero Becomes a Number:
Here’s where things get really interesting. The thinkers of India didn’t just stop at using zero as a placeholder. They took a giant conceptual leap, one that seems to have eluded many other cultures for centuries. They began to treat zero as a number in its own right.
What does this mean? It means they started asking questions like:
- What happens when you add zero to another number?
- What happens when you subtract zero?
- What happens when you multiply by zero?
- What happens when you divide by zero?
Think about that last one. Division by zero still trips up students today! The fact that Indian mathematicians were even asking this question shows they were treating zero not just as emptiness, but as an actual mathematical entity you could operate with.
Around 850 A.D., the mathematician Mahavira stated rules like:
- A number multiplied by zero is zero.
- A number minus zero remains the same number.
Around 1100 A.D., another mathematician, Bhaskara II, went further. He declared that a number divided by zero resulted in an infinite quantity. While we view division by zero as undefined today, Bhaskara’s statement shows the depth of their engagement with zero as a number with unique properties.
This transformation of zero from a mere placeholder (“emptiness”) into a number (“the quantity of nothing”) was revolutionary. It required a significant level of abstraction – moving from numbers tied directly to counting objects (like 3 goats or 5 coins) to thinking of numbers as concepts in themselves, including the concept of “nothing.”
The Greek Resistance: Why Zero Didn’t Fit
Why was this leap so hard for others, especially the ancient Greeks, who were such sophisticated mathematicians and philosophers? The Greeks had a very different idea about what numbers were.
For thinkers like Pythagoras, Plato, and Aristotle, numbers were fundamentally linked to magnitudes – lengths, areas, volumes. They represented properties of things. You could have the length of a line segment or the area of a square. You couldn’t really have a “length of nothing.” The idea of zero as an actual quantity, representing the absence of magnitude, simply didn’t fit their philosophical framework.
Their geometry was incredibly advanced, but their arithmetic was somewhat limited by this view. They didn’t embrace the abstract concept of number in the same way the Indians did. They knew about ratios and proportions, but the idea of zero operating like other numbers seemed illogical, perhaps even paradoxical, within their system. This philosophical block meant that zero, as a number, had no place in Greek mathematics.
Zero Enters Europe (Slowly and Suspiciously)
The Indian system, including the concept of zero as both placeholder and potential number, traveled westward primarily through the work of Arab scholars like Al-Khwarizmi in the 9th century.
Al-Khwarizmi’s Ambivalence:
When Al-Khwarizmi explained the Indian number system (which we now call Hindu-Arabic numerals), he described it as having “nine symbols” (1 through 9). He explained the use of the zero symbol (which Arab scholars often wrote as a dot) as a placeholder to keep columns aligned, but he didn’t really treat it as a number on par with the others in his explanations of arithmetic. The conceptual leap made in India hadn’t fully transferred, or perhaps Al-Khwarizmi himself was cautious about it.
Translation and Transformation:
When Al-Khwarizmi’s books were translated into Latin starting in the 12th century, European scholars began to learn the new numeral system. The Indian word for zero, sunya (meaning “empty” or “void”), had become the Arabic sifr. This, in turn, was Latinized as zephirum or cifra. These words eventually gave rise to both our zero and cipher. (Think about “cipher” meaning a secret code – zero was once seen as mysterious!)
Learning the new system forced Europeans to grapple with the role of zero, at least as a placeholder. Doing calculations like long addition or multiplication is almost impossible without it.
Lingering Doubts:
Yet, the idea of zero as a number was still met with deep suspicion for centuries. Many prominent mathematicians of the 16th and 17th centuries, figures active during the Scientific Revolution, refused to accept zero as a legitimate solution (or “root”) to an equation. If solving a problem led to an answer of zero, it often meant, to them, that the problem had no real solution. It felt like an absurdity. How could the answer be “nothing”?
Zero Finds Its Power: Algebra Needs Nothing
What finally forced mathematicians to fully embrace zero as a number? Ironically, it was the development of algebra itself.
Harriot’s Principle:
In the early 17th century, the English mathematician Thomas Harriot (who also surveyed the Virginia colony) proposed a simple but powerful technique for solving equations. His idea was: move everything to one side of the equals sign, so the equation looks like:
= 0
This might seem obvious today, but it was a revolutionary shift. Why? It harnesses a unique property of zero. If you have two numbers multiplied together, say A times B, and the result is zero (A x B = 0), then you know for sure that either A must be zero, or B must be zero (or both). No other number has this property!
Consider solving x² + 2 = 3x.
Harriot’s approach: Rewrite it as x² – 3x + 2 = 0.
The left side can be factored into (x – 1)(x – 2).
So, we have (x – 1)(x – 2) = 0.
Using zero’s unique property, this means either (x – 1) = 0 or (x – 2) = 0.
Solving these simple equations gives x = 1 or x = 2.
This technique, popularized by René Descartes, made solving many equations much easier. But it only works if you accept zero as a legitimate number that can stand alone on one side of an equation and has unique multiplicative properties. Algebra needed zero to unlock its full potential.
Zero on the Graph:
When Descartes and Fermat linked algebra to geometry through coordinate systems (see Sketch 16), zero gained even more importance. The point where the axes cross is the origin (0, 0). Finding where a graph crosses the x-axis means finding the value of x where the function’s value y is zero. Graphing functions to solve equations of the form f(x) = 0 became a powerful tool, again relying on zero as a key reference point.
Zero as the Center:
By the 18th century, zero was generally accepted as a number, though some philosophical unease remained. It sat at the center of the number line, the crucial point separating the positive numbers from the newly accepted (but still slightly weird) negative numbers (see Sketch 5).
Nothing Becomes Everything: Zero in Modern Math
The story doesn’t end there. As mathematics became more abstract in the 19th and 20th centuries, zero took on even more fundamental roles.
In modern algebra, mathematicians study abstract structures like rings and fields. These are sets of elements (not necessarily numbers) where you can define operations that behave like addition and multiplication. In these systems, there’s often a special element, called the additive identity, which acts just like zero does with regular numbers: when you add it to any other element, that element stays unchanged. This abstract “zero element” is critical to the structure.
Furthermore, Harriot’s Principle lives on. Systems where the product of two non-zero elements is never zero are called integral domains. This property, inspired by our number zero, turns out to be fundamental to understanding the structure of many algebraic systems.
From a simple placeholder dot in ancient Babylon, zero journeyed through India, the Arab world, and a skeptical Europe. It battled philosophical objections and deep-seated intuitions about what numbers should be. Yet, its practical utility in writing numbers and its unique properties in algebraic manipulation proved irresistible. It evolved from representing “nothing” to becoming a number in its own right, and finally, to being a fundamental concept defining the structure of modern abstract mathematics. Not bad for a number that started out as nothing at all!
Zero’s Hidden Dimensions: A Numerological Reflection
The journey of zero is fascinating mathematically, but it also resonates with deeper, more spiritual ideas found in numerology.
Zero as Potential: In numerology, zero is often seen not just as nothing, but as the representation of pure potential, the void from which all things emerge. It’s the circle, the All, the Infinite before it takes form. Before the number 1 (representing initiation, the individual), there is 0 – the unlimited possibility. This resonates with zero’s mathematical journey. It started as an empty space, a potential place for a digit, before becoming a number embodying the concept of “nothingness” itself – a concept essential for the potential to solve equations in new ways (like Harriot’s Principle).
The Void and Creation: Many spiritual traditions speak of the Void or the Great Mystery as the source of creation. Mathematically, zero is the origin on the number line, the point from which positive and negative numbers extend. It’s the center point, the source. Its acceptance allowed the number system to become complete, embracing negatives and paving the way for complex numbers. This mirrors the idea of the creative void.
Cipher and Mystery: The word “cipher” originally meaning zero, now often means a secret code or something mysterious. Zero was mysterious for centuries. Its properties (especially division by zero) seemed paradoxical. This aligns with the numerological view of zero as representing the unknown, the unseen potential, the mystery that underlies manifest reality. It holds possibilities that aren’t immediately obvious.
A Bridge Between Worlds? The resistance to zero, especially by the Greeks who tied numbers to physical magnitudes, highlights a tension between the concrete and the abstract. Zero forced a move towards abstraction. In numerology, numbers bridge the physical and the metaphysical. Zero, being both “nothing” and a powerful mathematical concept, acts as a unique bridge. It represents the intangible source from which tangible quantities (other numbers) arise. Its acceptance in math required a leap beyond purely physical intuition, embracing a more abstract, perhaps more spiritual, understanding of quantity.
The Power of the Placeholder: Even in its earliest role as a placeholder, zero had a subtle power. It brought order and clarity to the Babylonian and Hindu-Arabic systems. It allowed numbers to be written efficiently and unambiguously. In life, creating “space” – emptiness – is often necessary for clarity and order to emerge. Zero, as the empty space, allows the other digits to reveal their true value based on position. It structures the entire system.
The story of zero is more than just math history. It’s about how humans grapple with fundamental concepts like emptiness, potential, and the abstract. Its journey from a controversial “nothing” to a foundational mathematical and symbolic element reminds us that sometimes, the most powerful ideas are the ones that represent the very space from which everything else can grow.