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Book Summary: Zero – The Biography of a Dangerous Idea

Zero (2000) is the fascinating story of a number banned by the ancient Greeks and worshipped by ancient Indians. Zero – as well as its twin, infinity – is a number that’s been at the heart of both mathematics and philosophy over the centuries.

Introduction: Discover the history of a heretical number.

Zero is a curious sort of number. It’s not much like 4, or 32, or 83.

When you add zero to other numbers, nothing happens. When you multiply other numbers by zero, you always get zero back. And when you divide by zero, all hell breaks loose.

It’s such a strange number, in fact, that back in the ancient world many great mathematicians denied its existence altogether. And, in modern times, even the philosopher René Descartes claimed it wasn’t real.

But since its eventual acceptance, it’s been found at the heart of pretty much every breakthrough in math or physics. How so? You’re about to find out.

In these summaries, you’ll learn

  • why the Babylonians invented zero;
  • why Aristotle banned it; and
  • why infinity is zero’s twin.

Book Summary: Zero - The Biography of a Dangerous Idea

Zero didn’t exist in the earliest days of math; it first emerged in ancient Babylonia.

Can you imagine a world with zero numbers?

Back in the Stone Age, that was how things were – until some enterprising cavepeople started carving notches onto a wolf bone.

What were they counting? We don’t know. But it must have been something practical, like animals or spearheads. Because prehistoric math was strictly functional, there was no need for the concept of zero. They didn’t need a special word for “zero” deer; there just . . . weren’t any deer.

But, over time, math advanced, and people developed complex counting systems. And the ancient Babylonians eventually realized that something – or rather, nothing – was missing.

This is the key message: Zero didn’t exist in the earliest days of math; it first emerged in ancient Babylonia.

To understand why zero first appeared, you’re going to need to know how the ancient Babylonian counting system worked. So, here’s a quick breakdown.

You probably know that our modern counting system is decimal, or in base 10: we group things into 1s, 10s and 100s. But back in ancient Babylonia, the system was sexagesimal – it was in base 60. And get this: it had just two symbols.

Those two symbols represented “1” and “10.” The Babylonians just repeated those symbols however many times they needed – much like in the later, better known Roman system. Fifty, for instance, would be five times the “10” symbol; fifty-one would be the same plus a “1” symbol – and so on, until you got to 60.

Here’s the confusing bit: at 60, they’d just start again with the “1” symbol. Sixty and 1 were represented by the same symbol. And so was 60 times 60, or 3,600.

If you’re thinking that sounds ambiguous, you’re right. But it was especially ambiguous when it came to numbers like 61 and 3,601. Those were both represented simply by two “1” symbols, side by side. So how could you tell the difference between them?

Eventually, the Babylonians found a solution: zero. To write 3,601, they wrote a totally new symbol in between the two “1” symbols; this made clear that the first number wasn’t 60, but a degree higher up. This was the birth of zero.

But this still wasn’t quite our modern-day zero. Really, it was just a placeholder denoting an absence. It was only later on that the strange, mystical properties of zero would become fully apparent – to the amazement, and horror, of the ancient Greeks.

Philosophically minded ancient Greeks rejected zero despite its usefulness.

For many ancient civilizations, numbers were merely tools for counting and dividing up land. For the ancient Greeks, however, numbers were a whole philosophy. Mathematician-philosophers like Pythagoras saw a harmony of numbers within every shape.

But the ancient Greeks were not on board with zero – not at all. In fact, Aristotle declared that it simply didn’t exist; it was merely a product of man’s imagination. And his views on this subject, as on so much else, resounded through the centuries – to the detriment of math in the Western world.

The key message here is: Philosophically minded ancient Greeks rejected zero despite its usefulness.

Conventional wisdom in ancient Greece was that zero didn’t exist. But one philosopher, Zeno, devised a paradox that questioned this accepted belief.

Imagine the great athlete Achilles is racing against a tortoise. The tortoise gets a head start of one foot. Can Achilles overtake the tortoise and win?

Well, Achilles makes up the tortoise’s one-foot advantage in, say, a second. But by that time, the tortoise has moved slightly further on. OK, so Achilles gets to the tortoise’s new point in a fraction of a second – but the tortoise, of course, has already inched ahead again. And so on, and so on, ad infinitum.

Every time Achilles catches up with where the tortoise was, it’s already moved ahead. The gap between them becomes smaller and smaller . . . but Achilles can never quite get there. Right?

We all know that, in reality, Achilles would simply overtake the tortoise. That’s because the gap between Achilles and the tortoise has a limit: zero. Sure, it takes an infinite number of increasingly tiny stages to close the gap, but it does eventually happen.

But the Greek mathematical system couldn’t account for Zeno’s paradox – because it banished zero.

According to Aristotle, zero and the infinite simply didn’t exist; everything was finite. The universe had an outer sphere and then abruptly stopped. Time was finite too; at some point in the distant past, it had just begun. That was the bedrock of their belief system.

But what happened before time began? The answer is either nothing at all – zero – or there was no starting point – infinity. It doesn’t make sense to deny the existence of zero and infinity. Yet that’s what happened in ancient Greece, and through the Middle Ages in the West, because of Aristotle’s huge influence.

Further east, however, Aristotle wasn’t so influential.

Ancient Indian and Arabic mathematicians embraced zero and made huge mathematical strides.

In ancient India, there was nothing to fear about infinity or zero. While Aristotle rejected these concepts altogether, they were a key part of the Indian belief system. Ancient Indians believed the universe was created from a void of nothingness, and that it was infinite – but that because the world came from nothing, it would one day return to nothingness.

Ancient Indian mathematicians realized that zero deserved a place among the numbers. And that realization opened all sorts of doors.

Here’s the key message: Ancient Indian and Arabic mathematicians embraced zero and made huge mathematical strides.

Another vital difference between ancient Greece and India concerned geometry. For the Greeks, geometry was at the heart of math; numbers essentially represented proportions and shapes. But in India, mathematicians thought of numbers in abstract terms.

Here’s an example of the difference that makes. What’s 2 minus 3?

To someone in ancient Greece, that question doesn’t even make sense. If you have a field that’s two acres large, you can’t subtract three acres from it. But if the numbers don’t represent anything in particular, you can just solve the equation. And, of course, you get -1.

Along with negative numbers, the ancient Indians were happy to include zero in their number system; it fit neatly between the positives and the negatives. But they still thought it was sort of strange.

Multiply anything by zero, and you get zero again. As for division, well, that’s chaos. How many 0s are there in 1? The twelfth-century Indian mathematician Bhaskara realized the answer was infinity. Incidentally, infinity had strange properties too; you could add or subtract any number, and it stayed exactly the same.

When these mathematical advances reached Muslim, Jewish, and Christian thinkers, it was a big deal – and not just because of zero and infinity’s weird properties. The three religions were heavily influenced by Aristotle, so these new concepts were challenging to their worldview. Eventually, though, all three adopted them.

The Christians were the last to accept zero – and, in the end, it was commercial pressure that caused them to embrace it. Italian merchants realized that our modern system of counting with ten digits, known as the Arabic system, was a lot simpler to use than the Roman system that the church still required. But by then, that Arabic system included a digit for zero.

So, in the Middle Ages, zero crept into the Western system of numbers. But it was still treated with suspicion – even by some of the greatest mathematical minds.

Embracing zero in the West was theologically tricky, but it yielded a mathematical revolution: calculus.

René Descartes was born in 1596. Like so many great thinkers before him, including Pythagoras, he was both a mathematician and a philosopher. But even he didn’t totally embrace zero.

Descartes lends his name to the Cartesian coordinate system – the x and y axes we learn about in high school. Those two axes have a zero in the bottom left corner by necessity. If you started with 1, you’d soon end up with errors.

This powerful new coordinate system ushered in a whole new universe of mathematical advances. Yet Descartes would always insist that zero itself did not actually exist. Having been brought up on Aristotle’s teachings, zero was just a step too far for him.

Later mathematicians, though, were less squeamish – with spectacular results.

This is the key message: Embracing zero in the West was theologically tricky, but it yielded a mathematical revolution: calculus.

You probably remember a little calculus from school, but you might not appreciate just how closely bound up it is with zero and infinity. So let’s go over some fundamentals.

Say you draw a curve on a Cartesian grid. How do you calculate the area underneath it?

You might start off by drawing a rectangle under the curve – one that covers as much of the area as possible. That’s a good starting point, but it’s not very accurate.

To get closer, draw two smaller rectangles instead; that way, you can cover a greater amount of the area. Three rectangles gets you closer still – and so on. But to get the actual area under the curve, you need infinite rectangles, each with an area that is infinitely small – that is to say, an area of zero.

That might sound nonsensical, but the weird thing is, it works. The mathematicians Isaac Newton and Gottfried Leibniz realized this almost simultaneously and developed systems of calculus. This allowed them to calculate the area mathematically, even though it involved some bizarre math around zero and infinity.

People found this theoretically troubling. The Irish bishop George Berkeley noted that, unlike other branches of math, which had been fully proved, calculus was faith-based – nobody truly understood what was happening with all those zeros.

The mathematician who solved this problem was Jean Le Rond d’Alembert. Just like with Zeno’s paradox, d’Alembert explained the answer using limits. A sequence might stretch toward infinity – but it may still approach a finite limit.

Mathematicians soon discovered that zero and its opposite, infinity, had a complex but enthralling relationship.

Quadratic equations are a mainstay of high school math classes. But that doesn’t mean they’re straightforward. Far from it.

Take a seemingly simple one, for instance: x2 + 1 = 0. What’s x?

As you might remember, quadratic equations usually have two solutions, one positive and one negative. And the two answers to that equation sound a bit peculiar: the square root of -1 and the negative square root of -1.

These numbers don’t exist, so mathematicians say that they’re imaginary. As such, they call them i and -i.

What does this have to do with zero? Nothing – and everything.

The key message here is: Mathematicians soon discovered that zero and its opposite, infinity, had a complex but enthralling relationship.

Once you allow the existence of imaginary numbers, you can combine them together and get numbers like i + 2, or 2i – 4. To visualize numbers like that – known as complex numbers – it’s helpful to map them onto a Cartesian grid, with the x axis for the real part of the number (say, -4) and the y axis for the imaginary part (say, 2i).

But that grid doesn’t work quite like a normal grid. Say you plot the point i one square up on the y axis, and at zero on the x axis. What happens when you square that number? i2, by definition, is -1. So in fact, that point rotates 90 degrees to the left. The same is true for any complex number: multiplying it by itself causes it to rotate around the grid.

If we stay on a two-dimensional grid, things become pretty complicated at this point – so the mathematician Bernhard Riemann realized that it made more sense to visualize things on a sphere. Imagine a sphere with i at one point on it, and -i directly across from it. Perpendicular to those two points are 1 and -1. And what goes at the top and bottom points of the sphere? Zero and infinity.

The strange logic of math with complex numbers reveals that zero and infinity are equal and opposite poles, just like 1 and -1.

The Riemann sphere makes some previously problematic equations easier to understand. Take y = 1/x, for example. In two dimensions, it looks messy: the curve shoots out of the picture toward infinity as x approaches zero. But on the sphere, it makes perfect sense: the curve simply reaches the topmost point.

This might all sound a bit theoretical. But if you’re wondering what any of this has to do with the actual, real world, stay tuned – because zero and infinity are mainstays not just of math, but of physics too!

Zero and infinity aren’t just mathematical concepts – they abound in physics too.

We’ve just been talking about imaginary numbers – but zero and infinity are so real that they make themselves felt in the real world in all manner of important ways. In fact, they underpin many of the advances made in physics over the past hundred or so years.

One example goes back even before that. In the 1850s, physicist Lord Kelvin discovered that it was literally impossible to cool an object below around -273 degrees Celsius. In other words, he discovered absolute zero.

The key message here? Zero and infinity aren’t just mathematical concepts – they abound in physics too.

It’s actually impossible to ever reach the temperature of absolute zero, because absolute zero is the state a gas reaches when it has zero energy at all. This is unattainable; there are always particles around giving off energy and warming stuff up again. But absolute zero really is there, as a limit, in the natural world.

Another zero in physics was uncovered by the work of Albert Einstein: the black hole. Einstein’s theories helped account for a curious and concerning phenomenon that happens in deep space. When a massive star dies, its gravitational pull is so strong that it collapses on itself, getting ever smaller, until it eventually takes up zero space. But despite taking up zero space, it still has mass. And that incongruous combination causes a curve in space-time itself, sucking in anything that approaches it.

Other advances in physics have different relationships with zero altogether. String theory, for instance, takes the curious step of more or less banning it – but not quite in the same way as Aristotle all those millennia ago.

According to string theory, the universe exists in ten or possibly eleven dimensions, so what seems like zero to us may not really be zero when all the other dimensions are factored in. This theory helps to explain certain enigmatic features about the universe – but some argue that this is more a philosophy than true science because it’s not provable through experimentation.

Then again, zero and philosophy have always gone hand in hand. Right from the beginning of time – the big bang itself, another zero, of course – through to the eventual end of the universe, zero has always exerted a mysterious, vacuous power.

Nothing can be created from nothing, the poet and philosopher Lucretius once said. But that nothing has strange, mystical properties. And we’re still uncovering them today.


The key message in these summaries is that:

Zero didn’t exist in the earliest days of math – and in Babylonia, where it was first invented, it was merely a placeholder. Despite the ancient Greeks’ prowess in math, zero was banned by Aristotle. This meant that it wasn’t fully appreciated in the Western world for many centuries. In places like India, however, it was embraced – and math progressed immensely. Zero has since gained its rightful place in the system of numbers, along with its twin, infinity. And it’s proved to be a vital yet mysterious component of every new concept in math or physics, from calculus to relativity.

About the author

Charles Seife is the author of five previous books, including Proofiness and Virtual Unreality. He has written for a wide variety of publications, including The New York Times, Wired, New Scientist, Science, Scientific American, and The Economist. He is a professor of journalism at New York University and lives in New York City.

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