The Invention of Zero: From Ancient Mesopotamia to Modern Math

If the ancient Arab world had refused foreign ideas, our modern understanding of medicine, astronomy, and mathematics would not exist. Central to our grasp of the universe is the concept of zero. This idea originated in ancient Mesopotamia and sparked a profound shift in human thought. It was first developed in pre-Arab Sumer, located in modern-day Iraq. Later, ancient India provided it with a symbolic form. This fusion of concept and symbol shaped mathematics, which now underpins our best scientific models of reality. Zero also permeated human culture. Shakespeare referenced it in King Lear as "an O without a figure." Today, the binary bit in computers, using 1s and 0s, enables everything from typing to complex computations.

Mathematician Robert Kaplan chronicles this revolutionary story in his book The Nothing That Is: A Natural History of Zero. It is a story of scientific discovery, where an abstract concept drawn from nature is named and symbolized. It is also a cross-cultural tale that celebrates human reason across time and space.

Kaplan writes: "If you look at zero you see nothing; but look through it and you will see the world. For zero brings into focus the great, organic sprawl of mathematics, and mathematics in turn the complex nature of things. From counting to calculating, from estimating the odds to knowing exactly when the tides in our affairs will crest, the shining tools of mathematics let us follow the tacking course everything takes through everything else – and all of their parts swing on the smallest of pivots, zero... As we follow the meanderings of zero's symbols and meanings we'll see along with it the making and doing of mathematics — by humans, for humans. No god gave it to us."

Kaplan also considers a fundamental philosophical question: is mathematics discovered or invented? Is zero something inherent in the world, or is it a human creation? This leads to a deeper inquiry about whether we invent or discover the fundamental patterns of reality.

Like all transformative inventions, zero began with necessity — the necessity for counting without getting bemired in the inelegance of increasingly large numbers. Kaplan explains: "Zero began its career as two wedges pressed into a wet lump of clay, in the days when a superb piece of mental engineering gave us the art of counting."

The story starts roughly 5,000 years ago with the Sumerians of Mesopotamia, now Iraq. The Sumerians counted by 1s and 10s, but also by 60s. While this may seem unusual, we retain this system today: 60 seconds in a minute, 60 minutes in an hour, and 360 degrees in a circle. Other counting systems, like 12 months or 24 hours, also have historical origins in custom and compromise. The Sumerian base-60 system likely emerged from trade with neighboring cultures that used different weights and currencies.

Mixing decimal and sexagesimal (base-60) systems created confusion. The Sumerians wrote on wet clay tablets using a three-sided reed stylus, which produced triangular cuneiform marks to represent numbers and concepts. By 2000 BCE, their numerical system was highly complex.

This cumbersome system persisted for millennia. Then, between the sixth and third centuries BCE, a solution emerged. Scribes began using a mark to separate columns in their accounts, symbolizing "nothing in this column." The concept of zero was born, though it was not yet a standardized symbol. Kaplan points to a tablet from around 700 BCE found at Kish. The scribe wrote his zeroes with three hooks, not two wedges. Another scribe used a single hook, which resembled the symbol for ten. This variety indicates these were early, experimental uses of a separation sign that would evolve into zero.

Zero nearly vanished with the civilization that conceived it. The narrative then shifts to ancient Greece, where the need for zero resurfaced. The Greek polymath Archimedes worked on naming enormous numbers. The term "myriad" meant 10,000. With his system for orders of large numbers, Archimedes nearly invented the concept of mathematical powers. More crucially, he demonstrated how to think concretely about vast scales by building up to them in manageable steps.

This concept of the infinite underscored the need for its opposite: nothingness. (Negative numbers were still a long way away.) Yet the Greeks had no word for zero, even as they sensed its conceptual presence. Kaplan observes: "Haven't we all an ancient sense that for something to exist it must have a name?" Children often believe numbers must end when names run out. For them, a number like a googol (1 followed by 100 zeroes) is a real, immense entity. Archimedes did not use zero; he named his "myriad myriads" and orders of magnitude. This approach gave constructive vitality to vastness, making it more comprehensible.

Typically, names are given to things that exist. Zero presents a paradox: it is a name for no-thing. Kaplan reflects: "Names belong to things, but zero belongs to nothing. It counts the totality of what isn't there... So, what does the name zero mean?" It remained an unnamed idea, continuing its journey.

After Babylon and Greece, zero arrived in India. The first surviving written symbol for zero appeared on a stone tablet dated to 876 AD. It recorded garden measurements: 270 by 50, written as "27°" and "5°." Kaplan notes that similar zeros appear on copper plates from three centuries earlier, though forgeries from the eleventh century cast some doubt on their authenticity.

Why search for zero in India before 876 AD? Because this period represents a crucial junction in the story—the intellectual bridge between the ancient Mediterranean world and ancient India.

If zero had a high priest in ancient India, it would be the mathematician and astronomer Āryabhata. His identity, like Shakespeare's, is somewhat mysterious, but his legacy is central to zero's development.

Kaplan details Āryabhata's work. Seeking a concise way to record large numbers without modern positional notation (where an 8 in 9,871 means 800), Āryabhata devised a coded system using nonsense words. Syllables in these words stood for digits in specific places. Digits were represented by consonants, and the nine vowels of Sanskrit indicated the places. To write 386 (which he ordered as 6,8,3), one would combine specific consonants and vowels.

This system allowed for only 9 places. As an astronomer, Āryabhata needed more. His complex solution was to double the system to 18 places, using the same nine vowels twice and dividing consonants into groups for odd and even places. Thus, 386 might be coded as CASAGI. There is no zero in this system. However, Āryabhata explained it using the word "kha" for "place." This word "kha" later became a common Indian term for zero. Kaplan describes this as a "slow-motion picture of an idea evolving"—a shift from a named notation to a purely positional one, moving from an empty placeholder to "the empty number" itself.

Kaplan considers the multicultural origins of zero. Possessing a symbol for zero is significant, but possessing the underlying notion is more profound. Did this notion travel directly from Babylon, or did it pass through Greece? The critical development in India concerned the character of the idea itself. Would it remain merely the idea of the absence of any number, or would it become a number representing that absence? Is it the mark of emptiness, or is it the empty mark? The first keeps it separate from numbers, a part of their landscape. The second grants it equal standing among numbers.

In his book, Kaplan further explores how other cultures, including the Mayans and Romans, contributed to zero's story. He examines its deep influence on philosophy, literature, and mathematics. The journey of zero reveals how human thought connected across civilizations to grasp a fundamental concept: nothingness. Ultimately, the transition from a mere placeholder to a mathematical entity allowed humanity to unlock the universe's deepest secrets, transforming zero from a simple gap into the cornerstone of modern science.