 In 662, Bishop Sauris Seibacht, living in Kineshra on the Euphrates River, now in Syria, chided his colleagues for being too enamored of the accomplishments of the Greeks. He pointed to the discoveries made by Indian astronomers using their nine signs for numerals. This is one of the earliest references to what we now call the Hindu-Arabic numerals. So here's a quick review. In additive notation, the value of a symbol is fixed and the value of a number is the sum of the symbols present. Egyptian hieroglyphic, Greek numerals, and Roman numerals are additive. In a positional notation, the value of a symbol depends on where it is written. Mesopotamian notation is positional. A symbol might represent 1 or 60. One problem with positional numeration occurs when an order of magnitude is absent. The Mesopotamian symbol for 1 could mean 1 or 60. The problem was solved in various ways. The Mesopotamians didn't bother. The value of a number should always be read in context. The Chinese rod numerals altered the orientation. So 333 will be written with the 3s oriented differently to indicate they were indifferent but adjacent orders of magnitude. And the Chinese written numerals explicitly indicated the orders of magnitude 300, 310s, and 3. And these all represented one solution to the problem of magnitude. But sometime in the 8th century, maybe, someone in India, probably, invented a symbol representing the absence of an order of magnitude, the zero symbol. The symbols underwent a long evolution. The earliest forms of 1, 2, and 3, which we see in India, in China, and many other places, were 1, 2, and 3 marks, which could also be written vertically. However, if you're not careful to separate the lines or if your pen drips ink, you form ligatures that adjoin adjacent symbols. And so the vertical marks of 2 and 3 began to look a lot like our modern symbols. And here's a bit of irony. The new numerals entered Europe mostly through Spain and southern Italy. This meant they were introduced to Europe by those in the western half of the Islamic world. But this area was the cultural backwater of the Islamic world, and the cultural center was in the east. And the numerals had a different form in the eastern half of the Islamic world. Which meant that, by and large, the Arabs weren't using Arabic numerals. Now the open circle is probably the most natural way to represent an absence. It's a whole or a void, sunya, in Sanskrit. When the Arabs got ahold of this, they called it their own word for empty, sifer. This became transliterated by Europeans as cipher, then progressively mangled as words like sifer, sifer, and eventually zero. Now an interesting thing happens when the Hindu-Arabic numeral system gets translated into Europe. The Indians wrote the numbers starting with the lowest order of magnitude. So 3,257 would be 7 ones, 5 tens, 2 hundreds, 3 thousands. Islamic mathematicians followed the same pattern. But since Arabic is written from right to left, this put the smallest order of magnitude on the right. So this 7 ones, 5 tens, 2 hundreds, 3 thousand would be written this way. European translators read this as a single symbol, so they retained the order. However, since European languages are written left to right, this meant we see the highest order of magnitude first as we read our numbers. Now, Algarizmi wrote a book on Hindu reckoning. Unfortunately, it's been lost and we only have translations made centuries after he wrote it, so we don't know exactly how he described these Hindu numbers. However, his work proved influential enough so that the steps for computing with the new numerals came to be known as algorithms, after Algarizmi, and books describing the procedures were called algorithms. And at this point, European translators penchant for fanciful stories got the better of them. At least one European writer claimed that the name came from Algor, a king of India. No, not him. Now, because the Hindu Arabic numerals are how we write numbers today, and how we compute with them, you might expect that these algorithms for working with these numbers resemble pretty much our own algorithms. And that's true in the most general sense of the word true, while we can find our algorithms in these more ancient works. It's worth pointing out that in the early stages of any technology, there's a number of different solutions to the same problem. And it's only over time that one preferred solution emerges. So let's take a look at some of those algorithms.