As you see below, we owe more than the "Hindu-Decimal" numeration, although that alone would be an enormous debt. (Would you like to balance your checkbook or report to the IRS in Roman numerals?)
In 527 B. C. Justinian, Roman Emperor of the West wing of the Roman Empire, became also Emperor of the East. He decided that the "pagan" learning of the Academy founded in Plato's name and other philosophical schools threatened orthodox Christianity, so, in 529, all these schools were closed, effectively ending speculative research in Europe.
The challenge was taken up by Islamic scholars who not only preserved the knowledge of the ancients but advanced it in "forbidden areas".
Moslem armies, sweeping the seventh and eighth century Arabian peninsula, annexed territories from Spain to Persia, absorbing the works of Plato, Aristotle, Democritus, Pythagoras, Archimedes, Hippocrates and other Greek thinkers. Hellenistic culture had been spread eastward by the armies of Alexander the Great and by religious minorities, including various Christian sects.
The mostly illiterate Muslim conquerors needed local intellectuals to help govern, who absorbed Greek learning not yet effectively transmitted to the West, or even translated into Latin. "The West had a thin version of Greek knowledge," said Arabic historian, Dr. Lindberg. "The East had it all."
And Dr. David King, a historian of science at Johann Wolfgang Goethe University in Frankfurt, pointed out in his book "Astronomy in the Service of Islam" (1993), that Islam is one of the few religions in history which requires some scientific procedures for religious ritual. Arabs had always been knowledgeable about the stars and used them to navigate the desert, but Islamic religion increased the use of astronomy. The requirement that Muslims face toward Mecca to pray, for example, involved knowledge of the size and shape of the Earth. The best Muslim astronomers produced tables or diagrams by which the qibla (sacred directions) could be found from any point in the Islamic world. Their efforts developed precision far beyond peasant use, noted Dr. King. Astronomers at Samarkand observatory, founded about 1420 by the ruler Ulugh Beg, measured star positions to a fraction of a degree, said Dr. El-Baz.
In the 13th and 14th centuries Islamic astronomy pushed against the limits of the Ptolemaic world view that had ruled for a millennium. In this model, celestial bodies moved in circles at uniform speeds. The reality of very ununiform motions of planets and the Sun as seen from Earth was correctionted by orbits within orbits, known as epicycles, with geometrical modifications.
13th century astronomer, Al-Tusi, restored most of the symmetry to Ptolemy's model by adjoining pairs of cleverly designed epicycles to each orbit. And 14th-century Ala al-Din Abul-Hasan ibn al-Shatir constructed a completely symmetrical model.
Copernicus, who overturned the Ptolemaic universe in 1530 by proposing that the planets revolved around the Sun, expressed ideas similar to the Muslim astronomers in his early writings. This has led some historians to suggest that there is a previously unknown link between Copernicus and the Islamic astronomers, even though neither ibn al- Shatir's nor al-Tusi's work is known to have ever been translated into Latin, and therefore was presumably unknown in the West.
From the 10th to the 13th century Europeans, especially in Spain, were translating Arabic works into Hebrew and Latin "as fast as they could," said Dr. King. The result was a rebirth of learning that ultimately transformed Western civilization.
One of their greatest precursors was astronomer Mohammed ibn Musa al-Khowarizmi (circa 825 A.S.) 
whose name inspired the label "algorithm" for a "sure-fire" method for problem-solving. And the title of his book on Arithmetic inspired our word for "algebra".
The most important Islamic advance was to arithmetize motion and challenge the Aristotlean "Law of Falling Bodies". Peculiarly, Islamic arguments for arithmetizing motion derived from analogies in pharmacy and optics. It appeared possible to change magnitudes in medicines and in light intensities, hence, it was extrapolated to motion.
The Koran bade them to seek knowledge and signs of the Creator in nature. Discovering riches of ancient Greek learning (little known in Europe), Muslim society in the Middle Ages became the scientific center of the world. The Arabic language disseminated a golden age of learning and science for 500 hundred years. Their universities prepared the way for modern universities. They created and named the vast mathematical field of algebra. They named some of the most famous of our stars, such as Aldebaran, Arcturus and Betelgeuse. (In the film, "Ben Hur", the hero's chariot was driven by the beautiful white horses of an Arabic sheik, all given the Arabic names for stars.) Islamic scholar developed the notion of science as empirical inquiry.
"Nothing in Europe could hold a candle to what was going on in the Islamic world until about 1600," said Dr. Jamil Ragep, a professor of the history of science at the University of Oklahoma. Infusion of this knowledge into Western Europe instigated fueled the Renaissance and the scientific revolution.
In this "Medieval-Renaissance" period, Arabic artists excelled in the geometric art of figrative adornment. The finest example of this is The Alhambra in Grenada (Spain), a fortess and palace. Mathematicians recognize 128 distinct patterns of "the wallpaper-tile group". All of these appear in the decorations of The Alhambra (as in an Alhambra portal or an Alhambra relief or an Alhambra mosaic).
As you may read in The Medieval Machine: The Industrial Revolution of the Middle Ages, by Jean Gimpel, 1977, in the 12th century (A.D.) began what should be called "The Mechanical Industrial Revolution". Monks -- desiring to raise their food and other needs, yet have time for prayer and meditation -- were given special permission by the Pope to read pagan texts showing how to build watermills and windmills. In this century and the next, these mills spread across Europe. (The Domesday Book of 1086 lists 5624 water-wheel driven mills in England south of Trent, or about one mill for each 400 persons.) Clearly, further extension of these mills would have eliminated much of the slavepower then existing, but "the lesson" was still not obvious enough.
In 1202, Fibonacci (a.k.a. Leonardo of Pisa) published Liber abaci ("book of the abacus"), the first European books to use decimal notation in arithmetic (which Fibonacci learned in his native North Africa).
The spread of decimal arithmetic made possible the great "Age of Navigation" and "Age of Trading", which followed -- although historians choose to ignore this.
Recently, some scholars have found evidence that European mathematicians, particularly Italian ones, may have copied, and sometimes built upon, work of Islamic and Jewish mathematicians working before the Renaissance. In fact, a few of their writings actually credit such contributions, but mainstream mathematicians have chosen, up until now, to ignore this.
To assess our debt to Islamic scholars, consider the standard precollege mathematical curriculum:
- arithmetic, in a form mainly originating with the ancient Greeks, but using Hindu-Arabic decimal notation;
- geometry, originating with ancient Greeks, with Hindu and Islamic contributions;
- algebra, mainly an Islamic contribution;
- trigonometry, originating in work of the mathematician and astronomer, Hipparchus (c. 440 BC), computing chords of a circle, extended by Ptolemy (2nd century AD), but taking the more modern trigonometric functional form in the work of Islamic mathematicians;
- analytic geometry, originated by Descartes (1596-1650) and Fermat(1601-1665) in the 17th century;
- differential and integral calculus, originated by Newton (1643-1727), Leibnitz (1646-1716) and Fermat in the 17th century.
Hence, the Pre-Renaissance "roots" of our precollege curriculum are about evenly divided among contributions of Greek and Islamic scholars.
Yet historians still know little about this golden age, since few of its major scientific works have been translated from Arabic, leaving thousands of manuscripts unread by modern scholars. Dr. Sabra says the history of Islamic science as a field that "hasn't even begun yet."
Why didn't Eastern science fare as well? "Nobody has answered that question satisfactorily," said Dr. Sabra of Harvard, although historians offer many reasons. For one, the Islamic empire began to be taken over in the 13th century by Crusaders from the West and Mongols from the East.
Christians reconquered Spain, with its magnificent libraries in Córdoba and Toledo, rich in Arabic learning. So, Islamic centers of learning began to lose touch with one another and with the West, and lost financial support.
In the West, science paid off in such new technology asthe steam engine and it attraced financing from industry. But in the East, science still depended on the patronage and curiosity of sultans and caliphs. Furthermore, the Ottomans, who took over the Arabic lands in the 16th century, were builders and conquerors, not thinkers, said Dr. El- Baz of Boston University, and support waned. "You cannot expect the science to be excellent while the society is not," he said.
Yet others argue that Islamic science seems to decline only in Western, secular eyes. "It's possible to live without an industrial revolution if you have enough camels and food," Dr. King said. "Modern science doesn't claim to address the purpose of life; that is outside the domain. In the Islamic world, purpose is integral, part of that life."
Islamics can decide what is best for them. But it should be clear -- considering what we westerners have already benefited from the fraction of Islamic scholarship available to us -- that more, perhaps vastly more, could accrue from translations of those thousands of manuscripts from "The Gold Age of Islamic Science".