Algebra – the finished product Recall that last week we were studying the “art of solving equations,” or algebra This name comes from a very influential book by a Persian mathematician named Muhammad Ibn Musa Al Khowarizmi (about 780 – 850 C.E.) called “Kitab Al Jabr wa-l- muqabala” (“Complete book on calculation by completion and balancing”) (Al Khowarizmi's name is also the source of our word algorithm)

Al Khowarizmi's Work A major part of Al Khowarizmi's book dealt with techniques for quadratic equations leading to the quadratic formula that we discussed briefly (and that you used on the first problem set), if you express them in our modern symbolic form. [derivation]

Higher degree equations? Are there similar formulas for solving higher degree equations? Chances are you have probably never seen one! They do exist for equations of degrees n = 3 (cubics) and degree 4 (quartics) The cubic formula was found by the Italian Renaissance mathematician Nicolo Tartaglia (1499 – 1557)

Higher degree equations, cont. First published by Girolamo Cardano (1501 – 1576) in his “Ars Magna” (essentially stolen from Tartaglia!) The quartic formula was found by a student of Cardano named Ludovico Ferrari (1522- 1565). Both the cubic and quartic formulas are so complicated that they are rarely used, though! [“Cardano's formulas” for cubics]

No formulas for degrees > 4 In the early 1800's, through work of Abel, Ruffini, Galois, it was proved that there are no analogous formulas for general polynomial equations of degrees 5, 6, … That is: no formulas using just the four arithmetic operations, and taking square, cube, fourth, … roots.

Historical Orientation We are now ready to begin our historical study of where the mathematics you have learned (and that we have reviewed the past few days) originally “came from.” “The past is a foreign country; they do things differently there” L.P. Hartley, The Go- Between Today, we'll start that with a bit of orientation (in location and time) for the first two of the ancient civilizations we will look at.

Ancient Mesopotamia the ``land between the rivers'' – Tigris and Euphrates – mostly contained in current countries of Iraq, Iran, Syria.

A very long history ~5500 BCE -- First village settlements in the South ~3500 - 2800 BCE -- Sumerian city-state period, first pictographic texts ~3300 - 3100 BCE -- first cuneiform writing created with a reed stylus on a wet clay tablet, then sometimes baked in an oven to set combined with a pretty dry climate, these records are very durable!

A tablet with cuneiform writing Note the limited collection of forms you can make with a wedge-shaped stylus:

Cuneiform writing Different combinations of up-down and sideways wedges were used to represent syllables Was used to represent many different spoken languages over a long period – 1000 years + We'll see the way numbers were represented in this system in a few days

Concentrate on southern area ~2800 - 2320 BCE -- Early Dynastic Period, Old Sumerian literature ~2320 - 2180 BCE -- Akkadian (Sumerian) empire, first real centralized government ~2000 BCE -- collapse of remnant of Sumerian empire ~2000 - 1600 BCE -- Ammorite kingdom "Old Babylonian Period"-- Hammurabi Code, mathematics texts, including “Plimpton 322,” editing of Sumerian Epic of Gilgamesh

Plimpton 322 The most famous Old Babylonian mathematical text:

Later history This part of the world has been fought over and conquered repeatedly – most recently, of course, in the two Iraq wars of the 1990's and 2000's CE – a very complicated story! Also figures in Biblical history (“Babylonian captivity” of Jewish people) 612 - 539 BCE -- “New Babylonian” period (Nebuchadnezzar) height of Babylonian astronomy Baghdad a world center of learning during “dark ages” in Europe

Geography of Egypt The “gift of the Nile”

Egypt 3200 - 2700 BCE -- predynastic period ~3300 - 3100 BCE first hieroglyphic writing

Egyptian hieroglyphics A very rich system with phonetic signs for single sounds, combinations of sounds, plus some ideographs (signs representing ideas) Pretty much the antithesis of cuneiform in terms of the variety of signs! Some of the most recognizable symbols are the names of kings and queens given in the oval signs called “cartouches”

Tutankhamen's Cartouches Each king had a pair of names (plus several others besides) – note how hieroglyphs can also function as decoration:

Other Egyptian writing Hieroglyphics were the “formal” Egyptian written language, used mostly for temple or tomb inscriptions carved in or painted on stone, grave goods (coffins, etc.) meant to last. The Egyptians also used a paper-like writing medium called papyrus manufactured from plant material grown along the Nile for “everyday” writing – scrolls with stories, business records, school exercises, … Hieratic and demotic writing forms as well

An Egyptian mathematical papyrus A portion of the Rhind papyrus:

A very stable civilization ~2650 - 2134 BCE -- Old Kingdom (pyramid- building period) ~2134 - 2040 BCE -- First intermediate period ~2040 - 1640 BCE -- Middle Kingdom (Moscow mathematical papyrus) 1640 - 1550 BCE -- Second intermediate period (Hyksos) Rhind (Ahmes) mathematical papyrus (possibly copying an older work from Middle Kingdom)

Egyptian timeline, continued 1550 - 1070 BCE -- New Kingdom (Akhenaten, Tutankhamen, Ramses II) after 1070 BCE -- Third intermediate period then Egypt ruled by Nubians, Assyrians, Persians, Ptolemaic Greek dynasty (until Cleopatra), Romans, Byzantines, Islamic caliphate, …

Egyptian number symbols The Egyptians, like us, used a base 10 representation for numbers, with hieroglyphic symbols like this for powers of 10:

Egyptian numbers The Egyptians did not really have the idea of positional notation in this system, though. To represent a number like 4037 (base 10) in hieroglyphics, the Egyptians would just group the corresponding number of symbols for each power of 10 together – four lotus flowers, 3 bolts of cloth, 7 strokes (something like a simpler version of Roman Numerals). There were separate and more involved number systems used in hieratic writing.

Egyptian arithmetic Even though the Egyptians used a base 10 representation of numbers, interestingly enough, they essentially used base 2 to multiply (!) Called multiplication by successive doubling Example: Say we want to multiply 47 x 26

“The Egyptian way” Successively double: 26 x 1 = 26 26 x 2 = 52 26 x 4 = 104 26 x 8 = 208 26 x 16 = 416 26 x 32 = 832 (stop here since 32 x 2 = 64 > 47)

The calculation concluded Then to get the product 47 x 26, we just need to add together multiples to get 47 x 26: 47 = 32 + 8 + 4 + 2 + 1, so 47 x 26 = 32 x 26 + 8 x 26 + 4 x 26 + 2 x 26 + 1 x 26 = 832 + 208 + 104 + 52 + 26 = 1222 Note: this essentially uses 47 (base 10) = 101111 (base 2)!

“Egyptian fractions” Probably the most distinctive feature of the way the Egyptians dealt with numerical calculations was the way they handled fractions. They had a strong preference for fractions with unit numerator, and they tried to express every fraction that way, for example to work with the fraction 7/8, they would “split it up” as: 7/8 = ½ + ¼ + 1/8. More on this next time!