Summary of the book "Zero" - By Charles Seife

Key Insights in this book:

1. Zero didn't exist in the beginnings of mathematics; it was invented in ancient Babylonia.
2. Despite its usefulness, ancient Greek philosophers rejected zero.
3. Ancient Indian and Arabic mathematicians adopted zero and made significant advances in mathematics.
4. In the West, accepting zero was theologically difficult, but it resulted in a mathematical revolution: calculus.
5. The link between zero and its polar opposite, infinity, was quickly discovered by mathematicians to be complex but fascinating.
6. Infinity and zero aren't simply mathematical ideas; they're also prevalent in physics.

Who can benefit the most from this book:

• Popular science enthusiasts.
• History buffs are curious about how concepts have evolved over time.
• Philosophers interested in everything . . . and nothing.

What am I getting out of it? Learn about the origins of a heretical number.

The number zero is a strange one. It's not the same as 4, 32, or 83.

Nothing happens when you add zero to other numbers. When you multiply other numbers by zero, the result is always zero. And when you divide by zero, everything goes to hell.

It's such a bizarre number, in fact, that many brilliant mathematicians in the ancient world denied it ever existed. Even the philosopher René Descartes claimed it wasn't real in modern times.

But, since its eventual adoption, it's been at the centre of almost every major mathematical or scientific advance. What do you mean by that? You're about to learn.

• You'll learn why the Babylonians originated zero.
• Why Aristotle forbade it.
• And why infinity is zero's twin in this summary.

1. Zero didn't exist in the beginnings of mathematics; it was invented in ancient Babylonia.

Can you envision a world where there are no numbers at all?

That was how things were back in the Stone Age – until some enterprising cavepeople started carving grooves into a wolf bone.

What were they keeping track of? We have no idea. However, it had to be something practical, such as animals or spearheads. There was no need for the idea of zero in prehistoric math because it was solely utilitarian. There was no need for a separate phrase to describe "zero" deer; there simply weren't any.

However, as math progressed, individuals invented more complicated counting systems. And the ancient Babylonians soon discovered that something was missing – or rather, that nothing was missing.

This is the most important message: Zero didn't exist in the beginnings of mathematics; it was invented in ancient Babylonia.

You'll need to know how the ancient Babylonian numbering system functioned to understand why zero first appeared. So, here's a quick rundown of what's going on.

You're probably aware that our modern counting system is decimal, or in base 10, in which we divide things into groups of one, ten, and hundred. However, the system in ancient Babylonia was sexagesimal - it was in base 60. What's more, it just had two symbols.

"1" and "10" were represented by those two symbols. Like the later, more well-known Roman system, the Babylonians simply repeated those symbols as many times as they wanted. For example, fifty would be five times the "10" symbol, fifty-one would be the same plus a "1" symbol, and so on until you reached sixty.

Here's where it gets confusing: at 60, they'd simply start over with the "1" symbol. The symbols for sixty and one were the same. 60 times 60, or 3,600, was also a factor.

You're right if you believe that sounds unclear. However, when it came to numbers like 61 and 3,601, it was especially unclear. Both were symbolized by two "1" symbols placed side by side. So, how would you be able to tell them apart?

The Babylonians eventually came up with a solution: zero. They used a completely new symbol in between the two "1" symbols to write 3,601; this made it clear that the first number wasn't 60, but rather a degree higher up. This was the beginning of zero's existence.

This wasn't, however, our modern-day zero. It was really only a placeholder for an absence. The weird, mysterious powers of zero would not become completely apparent until much later, much to the shock, and fear, of the ancient Greeks.

2. Despite its usefulness, ancient Greek philosophers rejected zero.

Numbers were essential instruments for counting and dividing the land for many ancient civilizations. Numbers, on the other hand, were philosophy in and of itself for the ancient Greeks. Pythagoras, a mathematician-philosopher, perceived the harmony of numbers in every shape.

The ancient Greeks, on the other hand, were not fans of zero. In fact, Aristotle argued that it didn't exist at all; it was just a figment of man's imagination. And, as with so much else, his ideas on this issue resounded down the centuries, to the damage of math in the Western world.

The main point here is that, despite its utility, ancient Greek philosophers rejected zero.

In ancient Greece, conventional thinking held that zero did not exist. However, Zeno, a philosopher, constructed a paradox that cast doubt on this widely held notion.

Imagine Achilles, the great athlete, competing against a tortoise. The turtle gets a one-foot head start. Is it possible for Achilles to beat the tortoise and win?

Achilles, on the other hand, makes up for the tortoise's one-foot advantage in a fraction of a second. However, at that time, the tortoise had progressed a little further. Okay, so Achilles arrives at the tortoise's new location in a fraction of a second – but the tortoise has already advanced. And so on, ad nauseam, ad nauseam, ad nauseam, ad nauseam, ad nauseam, ad nauseam

The tortoise has already gone ahead by the time Achilles catches up with it. The distance between them narrows... yet Achilles never quite manages to close it. Right?

Achilles would just surpass the turtle in reality, as we all know. This is due to the fact that the distance between Achilles and the tortoise has a limit: zero. Sure, closing the gap takes an infinite number of increasingly small steps, but it does happen eventually.

The Greek mathematical theory, however, was unable to account for Zeno's paradox since it excluded zero.

Zero and the infinite, according to Aristotle, did not exist; everything was finite. The universe had an outer sphere before abruptly coming to a halt. Time was finite as well; it had only just begun at some point in the distant past. That was the foundation of their faith.

But what happened before the beginning of time? Either there was no starting point – infinity – or there was nothing at all – zero. Denying the existence of zero and infinity is illogical. Because of Aristotle's enormous influence, this is exactly what transpired in ancient Greece and throughout the Middle Ages in the West.

Aristotle, on the other hand, was less prominent in the east.

3. Ancient Indian and Arabic mathematicians adopted zero and made significant advances in mathematics.

There was no such thing as infinity or zero in ancient India. Despite Aristotle's rejection of these ideas, they were an important element of Indian belief. The ancient Indians thought that the cosmos was created from a void of emptiness, that it was limitless, and that it would one day revert to nothingness because it came from nothing.

Zero earned a position among the numbers, according to ancient Indian mathematicians. And that discovery led to a slew of new opportunities.

Here's the main point: Ancient Indian and Arabic mathematicians adopted zero and made significant advances in mathematics.

Another significant distinction between ancient Greece and India was in terms of geometry. Geometry was at the heart of math for the Greeks, as numbers fundamentally expressed proportions and shapes. Mathematicians in India, on the other hand, conceived about numbers in abstract terms.

Here's an illustration of how that makes a difference. What is the difference between 2 and 3?

That question makes no sense to someone in ancient Greece. You can't take three acres away from a field that is two acres in size. However, if the numbers don't represent anything, you can just answer the equation. You receive a -1, of course.

The ancient Indians were content to have zero in their number system with negative numbers since it fit perfectly between the positives and negatives. Nonetheless, they felt it was unusual.

When you multiply anything by zero, you get back to zero. Division, on the other hand, is chaos. How many 0s are there in a single number? Bhaskara, an Indian mathematician from the 12th century, discovered the answer was infinite. Infinity, by the way, has peculiar features as well: you could add or subtract any number and it would remain the same.

It was a significant issue when these mathematical breakthroughs reached Muslim, Jewish, and Christian intellectuals — and not simply because of zero and infinity's strange qualities. Aristotle had a strong effect on the three religions, therefore these new ideas were a challenge to their worldview. However, they were eventually adopted by all three.

The Christians were the last to adopt zero, and it was ultimately due to commercial pressure that they did so. Italian merchants understood that our contemporary ten-digit counting system, known as the Arabic system, was far more user-friendly than the Roman system that the church still needed. However, by that time, the Arabic system featured a zero digit.

As a result, zero crept into the Western numeral system during the Middle Ages. But it was nevertheless viewed with mistrust, even by some of the world's most brilliant mathematicians.

4. In the West, accepting zero was theologically difficult, but it resulted in a mathematical revolution: calculus.

In the year 1596, René Descartes was born. He was a mathematician and a philosopher, like so many great philosophers before him, including Pythagoras. Even he, though, did not fully embrace zero.

The Cartesian coordinate system — the x and y axes we study in high school – is named after Descartes. By necessity, the bottom left corner of those two axes has a zero. If you started with 1, you'd quickly run into problems.

This revolutionary new coordinate system ushered in a new era of mathematical progress. Descartes, on the other hand, was adamant that zero did not exist. Zero was a step too far for him, having been raised on Aristotle's ideas.

Later mathematicians, on the other hand, were less apprehensive, and the results were magnificent.

This is the most important message: In the West, accepting zero was theologically difficult, but it resulted in a mathematical revolution: calculus.

You certainly remember a little calculus from school, but you may not realize how tightly zero and infinite are linked. So, let's go over some basics.

Let's say you're using a Cartesian grid to design a curve. How do you figure out how much space is beneath it?

You may begin by drawing a rectangle beneath the curve that covers as much of the region as feasible. That's a fine place to start, although it's not particularly precise.

To get closer, draw two smaller rectangles instead; this will allow you to cover a larger area. Three rectangles bring you further closer – and so on. To acquire the actual area under the curve, though, you'll need an endless number of rectangles, each with an infinitely small area - that is, an area of zero.

That may appear absurd, yet the strange thing is that it works. Isaac Newton and Gottfried Leibniz, both mathematicians, realized this almost simultaneously and built calculus systems. This allowed them to determine the area numerically, despite some strange math involving zero and infinite.

This was regarded as theoretically worrisome by some. Calculus, unlike other fields of math that had been fully proven, was faith-based, according to Irish bishop George Berkeley, because no one actually grasped what was going on with all those zeros.

Jean Le Rond d'Alembert was the mathematician who solved the problem. Like Zeno's paradox, d'Alembert used limits to describe the solution. Even if a series stretches to infinity, it will eventually reach a finite limit.

5. The link between zero and its polar opposite, infinity, was quickly discovered by mathematicians to be complex but fascinating.

Quadratic equations are a common topic in high school math. However, this does not imply that they are straightforward. Not at all.

Consider this example, which appears to be straightforward: x2 + 1 equals 0. What is the value of x?

Quadratic equations normally have two solutions, one positive and one negative, as you may recall. The square root of -1 and the negative square root of -1 are the two answers to the equation, and they sound strange.

Mathematicians call these numbers imaginary since they don't exist. As a result, they're referred to as I and -i.

What exactly does this have to do with the number zero? Nothing - yet everything at the same time.

The main point is this: The link between zero and its polar opposite, infinity, was quickly discovered by mathematicians to be complex but fascinating.

You can mix imaginary numbers to obtain numbers like I + 2 or 2i - 4 if you accept their existence. It's helpful to map complex integers onto a Cartesian grid, with the x-axis representing the real component of the number (say, -4) and the y axis representing the imaginary part (say, 2i).

However, its grid isn't quite like a regular grid. Assume you plot the point I on the y axis one square up from zero on the x-axis. When you square a number, what happens? i2 is -1 by definition. That point, in fact, rotates 90 degrees to the left. Any complex number that is multiplied by itself rotates around the grid in the same way.

Things get extremely confusing if we continue on a two-dimensional grid at this point, so mathematician Bernhard Riemann realized that seeing things on a sphere made more sense. Consider a sphere with the letter I at one end and the letter -i on the other. 1 and -1 are perpendicular to the two places. What happens at the sphere's top and bottom points? Infinity and zero.

The peculiar logic of complex numbers reveals that zero and infinity, like 1 and -1, are equal and opposite poles.

The Riemann sphere makes it easy to understand some previously difficult equations. Take, for example, y = 1/x. In two dimensions, it appears jumbled: as x approaches zero, the curve shoots out of the picture toward infinity. On the sphere, though, it makes complete sense: the curve just reaches the highest point.

This may all sound a little academic. But, if you're wondering what any of this has to do with the real world, keep reading because zero and infinite are fundamental concepts in both math and physics.

6. Infinity and zero aren't simply mathematical ideas; they're also prevalent in physics.

We've been discussing imaginary numbers, yet zero and infinity are so real that they manifest themselves in the actual world in a variety of ways. Indeed, they are at the heart of many of the physics breakthroughs of the last century or so.

One example dates from considerably earlier. Lord Kelvin, a physicist, determined in the 1850s that it was physically impossible to chill an object below -273 degrees Celsius. He discovered absolute zero, in other terms.

What is the main point here? Infinity and zero aren't simply mathematical ideas; they're also prevalent in physics.

Absolute zero is the state a gas achieves when it has no energy at all, therefore it's actually impossible to reach. This is impossible to achieve since there are always particles around emitting energy and warming things up again. However, in the natural world, absolute zero exists as a limit.

Albert Einstein's work revealed yet another zero in physics: the black hole. Einstein's theories contributed to the explanation of a strange and disturbing event that occurs in outer space. When a large star dies, its gravitational pull is so strong that it collapses in on itself, shrinking until it occupies no space at all. Even though it takes up no space, it has mass. And this strange combination generates a bend in space-time, sucking in anything that comes close.

Other physics breakthroughs have very different correlations with zero. String theory, for example, takes the unusual step of effectively prohibiting it — though not in the same manner that Aristotle did millennia ago.

According to string theory, the universe has ten or eleven dimensions, thus what appears to be zero to us may not be zero when all the other dimensions are taken into account. This theory explains several puzzling aspects of the cosmos, but some claim that it's more of a philosophy than actual science because it can't be proven through experimentation.

However, zero and philosophy have always been linked. Zero has always had a mysterious, empty force, from the beginning of time – the great bang itself, of course, another zero – until the eventual end of the universe.

Lucretius, the poet and philosopher, famously remarked that nothing can be made from nothing. That nothing, on the other hand, has odd, supernatural powers. We're still finding new ones today.

The main message in these blinks is that zero didn't exist in the early days of math - and that it was only a placeholder in Babylonia when it was first invented. Despite the ancient Greeks' mathematical prowess, Aristotle prohibited the use of zero. For many centuries, this meant that it was not completely recognized in the Western world. However, it was adopted in areas like India, and math improved tremendously. Along with its twin, infinity, zero has since reclaimed its due place in the number system. From calculus to relativity, it's shown to be a crucial yet enigmatic component of every new notion in math or physics.