From
placeholder to the driver of calculus, zero has crossed the greatest minds and
most diverse borders since it was born many centuries ago. Today, zero is
perhaps the most pervasive global symbol known. In the story of zero, something
can be made out of nothing.
Zero, zip, zilch - how often has a
question been answered by one of these words? Countless, no doubt. Yet behind
this seemingly simple answer conveying nothing lays the story of an idea that
took many centuries to develop, many countries to cross, and many minds to
comprehend. Understanding and working with zero is the basis of our world
today; without zero we would lack calculus, financial accounting, the ability
to make arithmetic computations quickly, and, especially in today's connected
world, computers. The story of zero is the story of an idea that has aroused
the imagination of great minds across the globe.
When anyone thinks of one hundred, two
hundred, or seven thousand the image in his or her mind is of a digit followed
by a few zeros. The zero functions as a placeholder; that is, three zeroes
denotes that there are seven thousands, rather than only seven hundreds. If we
were missing one zero, that would drastically change the amount. Just imagine
having one zero erased (or added) to your salary! Yet, the number system we use
today - Arabic, though it in fact came originally from India - is relatively
new. For centuries people marked quantities with a variety of symbols and
figures, although it was awkward to perform the simplest arithmetic calculations
with these number systems.
The Sumerians were the first to develop
a counting system to keep an account of their stock of goods - cattle, horses,
and donkeys, for example. The Sumerian system was positional; that is, the
placement of a particular symbol relative to others denoted its value. The
Sumerian system was handed down to the Akkadians around 2500 BC and then to the
Babylonians in 2000 BC. It was the Babylonians who first conceived of a mark to
signify that a number was absent from a column; just as 0 in 1025 signifies
that there are no hundreds in that number. Although zero's Babylonian ancestor
was a good start, it would still be centuries before the symbol as we know it
appeared.
The renowned mathematicians among the
Ancient Greeks, who learned the fundamentals of their math from the Egyptians,
did not have a name for zero, nor did their system feature a placeholder as did
the Babylonian. They may have pondered it, but there is no conclusive evidence
to say the symbol even existed in their language. It was the Indians who began
to understand zero both as a symbol and as an idea.
Brahmagupta, around 650 AD, was the
first to formalize arithmetic operations using zero. He used dots underneath
numbers to indicate a zero. These dots were alternately referred to as 'sunya',
which means empty, or 'kha', which means place. Brahmagupta wrote standard
rules for reaching zero through addition and subtraction as well as the results
of operations with zero. The only error in his rules was division by zero, which
would have to wait for Isaac Newton and G.W. Leibniz to tackle.
But it would still be a few centuries
before zero reached Europe. First, the great Arabian voyagers would bring the
texts of Brahmagupta and his colleagues back from India along with spices and
other exotic items. Zero reached Baghdad by 773 AD and would be developed in
the Middle East by Arabian mathematicians who would base their numbers on the
Indian system. In the ninth century, Mohammed ibn-Musa al-Khowarizmi was the
first to work on equations that equaled zero, or algebra as it has come to be
known. He also developed quick methods for multiplying and dividing numbers
known as algorithms (a corruption of his name). Al-Khowarizmi called zero
'sifr', from which our cipher is derived. By 879 AD, zero was written almost as
we now know it, an oval - but in this case smaller than the other numbers. And
thanks to the conquest of Spain by the Moors, zero finally reached Europe; by
the middle of the twelfth century, translations of Al-Khowarizmi's work had
weaved their way to England.
The Italian mathematician, Fibonacci,
built on Al-Khowarizmi's work with algorithms in his book Liber Abaci, or
"Abacus book," in 1202. Until that time, the abacus had been the most
prevalent tool to perform arithmetic operations. Fibonacci's developments
quickly gained notice by Italian merchants and German bankers, especially the
use of zero. Accountants knew their books were balanced when the positive and
negative amounts of their assets and liabilities equaled zero. But governments
were still suspicious of Arabic numerals because of the ease in which it was
possible to change one symbol into another. Though outlawed, merchants
continued to use zero in encrypted messages, thus the derivation of the word
cipher, meaning code, from the Arabic sifr.
The next great mathematician to use
zero was Rene Descartes, the founder of the Cartesian coordinate system. As
anyone who has had to graph a triangle or a parabola knows, Descartes' origin
is (0,0). Although zero was now becoming more common, the developers of
calculus, Newton and Lebiniz, would make the final step in understanding zero.
Adding, subtracting, and multiplying by
zero are relatively simple operations. But division by zero has confused even
great minds. How many times does zero go into ten? Or, how many non-existent
apples go into two apples? The answer is indeterminate, but working with this
concept is the key to calculus. For example, when one drives to the store, the
speed of the car is never constant - stoplights, traffic jams, and different
speed limits all cause the car to speed up or slow down. But how would one find
the speed of the car at one particular instant? This is where zero and calculus
enter the picture.
If you wanted to know your speed at a
particular instant, you would have to measure the change in speed that occurs
over a set period of time. By making that set period smaller and smaller, you
could reasonably estimate the speed at that instant. In effect, as you make the
change in time approach zero, the ratio of the change in speed to the change in
time becomes similar to some number over zero - the same problem that stumped
Brahmagupta.
In the 1600's, Newton and Leibniz
solved this problem independently and opened the world to tremendous possibilities.
By working with numbers as they approach zero, calculus was born without which
we wouldn't have physics, engineering, and many aspects of economics and
finance.
In the twenty-first century zero is so
familiar that to talk about it seems like much ado about nothing. But it is
precisely understanding and working with this nothing that has allowed
civilization to progress. The development of zero across continents, centuries,
and minds has made it one of the greatest accomplishments of human society. Because
math is a global language, and calculus its crowning achievement, zero exists
and is used everywhere.
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