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It seems unnatural.
When we are dealing with the counting numbers—1, 2, 3, and so on—it is easy to rank them in order.
We don’t have to worry about mixing up the value of the number—its cardinality—with the order in which it arrives—its ordinality—since
0, 1, 2, 3: zero came first, one was second in line, and two was in third place. No longer were cardinality and ordinality interchangable.
The first hour of the day starts at zero seconds past midnight; the second hour starts at 1 AM, and the third hour starts at 2 AM. Though we count with the ordinals (first, second, third), we mark time with the cardinals (0, 1, 2).
It’s hard to blame the monks for their ignorance. The world of Dionysius Exiguus, Boethius, and Bede was dark indeed.
As these medieval thinkers imported the philosophy and science of the ancients, they inherited the ancient prejudices: a fear of the infinite and a horror of the void.
The fear of the void was so great that Christian scholars tried to fix the Bible to match Aristotle rather than vice versa.
When we last saw zero, it was simply a placeholder. It was a blank spot in the Babylonian system of numeration. Zero was useful but was not truly a number on its own—it
India, all this changed.
Greece, India never had a fear of the infinite or of the void. Indeed, it embraced them.
Like many Eastern religions, Hinduism was steeped in the symbolism of duality.
In the earliest age of the gods, existence was born from non-existence.
Indian mathematicians did more than simply accept zero. They transformed it, changing its role from mere placeholder to number. This reincarnation was what gave zero its power.
Like the Egyptians, the Indians had rope stretchers to survey fields and lay out temples.
sophisticated system of astronomy; like the Greeks,
An important difference between the new Indian number system and the Babylonian style was that Indian numbers were base-10 instead of base-60.
by rights they should be called Indian numerals rather than Arabic ones
Nobody knows when the Indians made the switch to a Babylonian-style plac...
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They apparently didn’t have a deep interest in the plane figures that the Greeks loved so much. They never worried about whether the diagonal of a square was rational or irrational, nor did they investigate the conic sections as Archimedes had. But they did learn how to play with numbers.
Indian system of numbering allowed them to use fancy tricks to add, subtract, multiply, and divide numbers without using an abacus to help them. Thanks to their place-number system, they could add and subtract large numbers in roughly the same way we do today.
Contests between the abacists and the so-called algorists who used Indian numerals were the medieval equivalents ...
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Unlike the Greeks, the Indians did not see squares in square numbers or the areas of rectangles when they multiplied two different values.
This was the birth of what we now know as algebra.
After all, if you are thinking in geometric terms, what is a negative area? It simply didn’t make any sense to the Greeks.
Brahmagupta, an Indian mathematician of the seventh century, gave rules for dividing numbers by each other, and he included the negatives. “Positive divided by positive, or negative by negative, is affirmative,” he wrote. “Positive divided by negative is negative. Negative divided by affirmative is negative.” These are the rules that we recognize today:
Brahmagupta’s mistake did not last for very long. In time the Indians realized that 1 ÷ 0 was infinite.
Does man forget that We created him out of the void?
Mohammed, a thirty-year-old native of Mecca, fell into a trance on Mount Hira. According to legend, the angel Gabriel told him, “Recite!” Mohammed did, and his divine revelations started a wildfire.
Caliph al-Mamun founded a great library: the House of Wisdom at Baghdad. It was to become the center of learning in the Eastern world—and one of its first scholars was the mathematician Mohammed ibn-Musa al-Khowarizmi.
like Al-jabr wa’l muqabala, a treatise on how to solve elementary equations;
He also wrote a book about the Hindu numeral system, which allowed the new style of numbers to spread quickly through the Arab world—along with algorithms,
Though the Arabs took the notation from India, the rest of the world would dub the new system Arabic numerals.
The Indian name for zero was sunya, meaning “empty,”
which the Arabs turned into sifr. When some Western scholars described the new number to their colleagues, they turned sifr into a Latin-sounding word, yielding zephirus, which is the root of our word zero.
However, as Indian mathematicians had made quite clear, zero was the embodiment of the void. Thus, if the Muslims were to accept zero, they had to reject Aristotle. That was precisely what they did.
a profound and important idea which appears so simple to us now that we ignore its true merit.
The man who reintroduced zero to the West was Leonardo of Pisa.
There the young man—better known as Fibonacci—learned
Pythagoras had noticed that nature seemed to be governed by the golden ratio.
Fibonacci discovered the sequence that is responsible.
Fibonacci had learned his mathematics from the Muslims, so he knew about Arabic numerals, including zero. He included the new system in Liber Abaci, finally introducing Europe to zero.
The Germans called the counting board a Rechenbank, which is why we call moneylenders banks.
A contemporary of Brunelleschi, a German cardinal named Nicholas of Cusa, looked at infinity and promptly declared, “Terra non est centra mundi”: the earth is not the center of the universe. The church didn’t yet realize how dangerous, how revolutionary, that idea was.
When Tempier declared that the omnipotent God could create a vacuum if he so desired, Tempier insisted that God could break any Aristotelian law.
Nicholas was sure that God had, indeed, created an infinite number of other worlds. Earth was no longer at the center of the universe.
the meantime another Nicholas turned Cusa’s philosophy into a scientific theory. Nicolaus Copernicus showed that Earth is not the center of the universe.
The power of Copernicus’s idea was in its simplicity. Instead of placing Earth at the center of the universe filled with epicycle-filled clockworks, Copernicus imagined that the sun was at the center instead, and the planets moved in simple circles.
Nicholas of Cusa and Nicolaus Copernicus cracked open the nutshell universe of Aristotle and Ptolemy.
Copernicus published his magnum opus on his deathbed—in 1543, just before the church started clamping down on new ideas. Copernicus’s book, De Revolutionibus, was even dedicated to Pope Paul III. However, the church was under attack. As a result, the new ideas—the questioning of Aristotle—could no longer be tolerated.
