n Writing n Mathematics
Table of contents n Writing n Mathematics 1 n Writing 2 n Mathematics 3
Outline n Writing n Mathematics The Era and the Sources Cuneiform Writing 1 n Writing 2 n Mathematics 3
n Writing n Mathematics Location and Timeline The Era and the Sources Cuneiform Writing
The Era n Writing n Mathematics The Era and the Sources Cuneiform Writing 4000 BCE to 3000 BCE was an incredible millenium of cultural development Metal (beginning of Bronze Age) and widespread use of the wheel Egypts first dynasty began around 3100 BCE Sumerians united the mesopotamian region and built incredible canal systems for irrigation to control the erratic flooding of the Tigris and Euphrates rivers Not predictable like that of the Nile in Egypt Began a widespread use of writing called cuneiform
n Writing n Mathematics Fertile Crescent Valley and Babylon The Era and the Sources Cuneiform Writing The Fertile Crescent Valley, unlike Egypt, has no natural barriers from invasion Semitic Akkadians took over the valley in 2276 BCE under Sargon the Great and began unifying the regions disparate, Sumerian culture Of importance was the adoption of native cuneiform writing Reign ended in 2221 BCE Other invaders dominated the region until about 1900 BCE The era from 1894 BCE to 539 BCE is referred to as Babylonian In 539 BCE, Babylon fell to Cryus of Persia. The city remained, but the empire ended Its mathematics persisted through to the start of the common era
n Writing n Mathematics The Era and the Sources Cuneiform Writing Sources Soft clay tablets baked in the hot sun or in an oven Thousands of tablets contained information like laws, tax accounts, school lessons, etc. Tablets, unlike Egyptian papyri, stand the test of time quite well There exists a much larger body of mathematical knowledge From one site in Nippur, there are approx. 50,000 tablets Interestingly, Egyptian hieroglyphics were still translated first Took until early 1800s when a German philologist F.W. Grotefend started translating Late 1800s finally saw a full translation of n mathematics
Cuneiform Writing n Writing n Mathematics The Era and the Sources Cuneiform Writing Earliest vestiges occur in tablets from Uruk circa 3000 BCE Based in picture-writing In 3000 BCE, approx. 2000 symbols were in use. By 2300 BCE (Sargon invasion), only 700 were in use. Instead of pictures, it now used combinations of wedges Originally, cuneiform was written vertically, left-to-right. By 2300 BCE, it switched to horizontal right-to-left The early stylus was a truncated cone with thin strokes for small values and think strokes for larger values Around 2000 BCE, the numbering system simplified greatly to a single width stylus
n Writing n Mathematics Babylonian Numerals The Era and the Sources Cuneiform Writing By 1800s, the numeral system simplified greatly: Remained this way for millenia
n Writing n Mathematics The Era and the Sources Cuneiform Writing Babylonian Numerals The n number system operated on a Base 60 Theories as to why vary, but most agree it was due to metrology 60 has 2, 3, 4, 5, 6, 10, 12, 15, 20, and 30 as divisors While no culture uses Base 60 anymore, quantities like time and angle measures still use it
Positional Notation n Writing n Mathematics The Era and the Sources Cuneiform Writing The Babylonians utilized positional notation. Small whole numbers < 60 = 1, = 10. Used repetition to reach desired number Similar to Egyptian hieroglyphics Whole numbers 60 Differed greatly from hieroglyphics The same symbol represented a different value depending on where it was relative to the other symbols! This positional notation is much like todays: 222 uses the cipher 2 three times each with a different meaning Ex: = 2(60 2 ) + 2(60) + 2 = 7322 Invented approx. 2000 BCE
n Writing n Mathematics The Era and the Sources Cuneiform Writing The Empty Position Our knowledge of n mathematics comes from two main time frames: Old Babylonian Age (c. 1800 BCE) Seleucid Period (c. 300 BCE) The Old Babylonian Age had a wealth of mathematical contribution, but the notion of an empty position didn t appear in sources until the Seleucid Period. The empty position is essentially what we use 0 for in todays numeration Before 300 BCE, it was impossible to tell whether meant 2(60) + 2 or 2(60 2 ) + 2. Unfortunately, this ambiguity was settled by appealing to context.
n Writing n Mathematics The Era and the Sources Cuneiform Writing The Empty Position By around 300 BCE, when Alexander the Great died and the Seleucid period began, a cipher was finally introduced for the empty position: // // = 2(60 2 ) + 0(60) + 2 = 2(60) + 2 Unfortunately, there was still ambiguity. The Babylonian positional system was relative, not absolute. Thus it was impossible to distinguish between 2(60) + 2 2(60 2 ) + 2(60) 2(60 3 ) + 2(60 2 ) Nevertheless, an incredible improvement over Egyptian hieroglyphics!
n Writing n Mathematics The Era and the Sources Cuneiform Writing Sexagesimal Fractions Incredibly, the Babylonians seamlessly extended the positional system to fractions Ex:, which we saw could have been equal to 2(60) + 2, could also have been equal to: 2 + 2(60 1 ) 2(60 1 ) + 2(60 2 ) While still suffering from ambiguity, it was a definitive improvement over the Egyptian unit fractions This treatment mimics the modern decimal system An advantage of the modern decimal system and our positional notation is that 23.45 9.876 is no more difficult to compute than 2, 345 9, 876 The Babylonian system behaved similarly and they utilized this freedom often.
Outline n Writing n Mathematics 1 n Writing 2 n Mathematics 3
Approximations n Writing n Mathematics Modern discussions of sexagesimal numbers writes: 2, 17; 8, 50 = 2(60) + 17 + 8(60 1 ) + 50(60 2 ). A tablet currently in the Yale collection contains an algorithmic approximation to 2. It computes the value to 2 sexagesimal places: 1; 24, 51, 10 = 1 + 24(60 1 ) + 51(60 2 ) + 10(60 3 ) = 1.414222 Actual value: 2 = 1.41421...
n Writing n Mathematics The Square Root Algorithm Let x = a be the desired root. Let a 1 be a first guess at x. Let b 1 = a a 1 be another guess at x. Either a 1 < x < b 1 or b 1 < x < a 1. Therefore a 2 = a1+b1 2 is a better approximation. Consider b 2 = a a 2. Again, either a 2 < x < b 2 or b 2 < x < a 2. Choose a 3 = a2+b2 2, continue. On the Yale tablet, a 1 = 1; 30. Their final approximation was a 3 = 1; 24, 51, 10. Sometimes this algorithm is attributed to other peoples and cultures Greek scholar Archytas (c. 400 BCE) Heron of Alexandria (c. 100 BCE) Newtons Algorithm An early instance of an infinite procedure.
Tables n Writing n Mathematics A large portion of the Babylonian cuneiform mathematical tablets are table texts. Multiplication tables, reciprocal tables, squares, cubes, square roots, cube roots, etc. 2 30 3 20 Observe: The product of each line is 60. 4 15 5 12 This is a table of reciprocals Ex: 6 10 (unit fractions in Egyptian) 8 7,30 1 9 6,40 8 = 7(60 1 ) + 30(60 2 ) 10 6 12 5
Reciprocal Tables n Writing n Mathematics Observe that 1 7 and 1 are missing. 11 7 and 11 were considered irregular since their sexagesimal expansions are non-terminating Similar to the numbers 1 3, 1 7, and 1 9 in our decimal system Again, an opportunity to investigate the infinite, but the Babylonians seeming skirted around it At one point, however, a scribe noted that 0; 8, 4, 16, 59 1 7 0; 8, 34, 18.
Arithmetic n Writing n Mathematics Multiplication was carried out easily be the n cultures Division was far more sophisticated than Egyptian duplation Babylonians divided two numbers by multiplying the dividend by the reciprocal of the divisor (using the tables) Ex: Much like today, we can compute: 34 5 = 34 1 = 34 0.2 = 34 2, then move one decimal place over. 5 They computed: 34 5 = 34 1 = 34 12, then move one sexagesimal place over 5 = 6; 48 = 6 48 60.
ic Equations n Writing n Mathematics developed in far more than in Egypt Had working understanding of combining like terms, multiplying both sides of an equation to clear fractions, and instances of adding quantities like 4ab to (a b) 2 to yield (a + b) 2 Factoring posted no great challenge Note: They did not use letters to represent unknowns since the alphabet didn t yet exist Used words like length, breadth, area, and volume Interestingly, they routinely added area and volume, suggesting that these words were abstract quantities and not concrete representations
Linear Equations n Writing n Mathematics Egyptian algebra capped out at linear equations, Babylonian texts show that these were handled easily A tablet from the Old Babylonian Age (c. 1800 BCE) encounters a system of linear equations: { 1 4 width + length = 7 hands width + length = 10 hands The scribe provided two solutions. The first was essentially a solution by inspection. The second, however, first multiplied the first equation by 4, then subtracted the two equations to yield 3 width = 18, or width = 6 and therefore length = 4 An incredible instance of what we call the method of elimination.
Quadratic Equations n Writing n Mathematics Babylonians not only handled linear equations easily, but seem to have tackled quadratic equations as well. Something Egypt didn t do. A tablet from the Old Babylonian Age asks: What is the side of a square if the area less the side is 14, 30? The scribe writes the solution: Take half of 1, which is 0; 30, and multiply 0; 30 by 0; 30, which is 0; 15; add this to 14, 30 to get 14, 30; 15. This is the square of 29; 30. Now add 0; 30 to 29; 30, and the result is 30, the side of the square.
Quadratic Equations n Writing n Mathematics In modern notation, the problem asked for the solution to x 2 x = 870. Applying the quadratic formula to x 2 bx = c, (the) solution should be x = (b 2 ) 2 + c + b 2. where b = 1 and c = 14, 30 = 870. The scribe wrote: Take half of 1, which is 0; 30, and multiply 0; 30 by 0; 30, which is 0; 15; add this to 14, 30 to get 14, 30; 15. This is the square of 29; 30. Now add 0; 30 to 29; 30, and the result is 30, the side of the square. This is exactly the correct procedure!
Quadratic Equations n Writing n Mathematics Historically, since complex numbers didn t exist until the late 1500s, quadratic equations of the form x 2 + bx + c = 0 with b, c > 0 were never considered. Quadratic equations came in three types: x 2 + px = q, x 2 = px + q, and x 2 + q = px. In n tablets dating to 2000 BCE, examples of each type appear and are solved. Of interest are instances of classic questions regarding pairs of numbers whose sum and product are fixed. The solutions involve solving systems of equations that reduce to a quadratic equation. We still see these types of problems in texts today.
Cubic Equations n Writing n Mathematics Unlike the Egyptians, Babylonians considered cubic equations Cubics of the form x 3 = a were solved via a table of cubes For non-integer solutions, they used linear interpolation to approximate Similarly, cubics of the form x 3 + x 2 = a were solved by referencing an n 3 + n 2 table. For general ax 3 + bx 2 = c, they reduced to the previous case by multiplying through by a2 b to convert it into 3 ( ax ) 3 ( ax ) + = ca2 b b b 3. This substitution is an incredible observation when one remembers that they didn t have variables Unfortunately, there are no sources indicating a solution to the general cubic ax 3 + bx 2 + cx = d.
Pythagorean Triples n Writing n Mathematics A tablet in the Plimpton collection at Columbia (c. 1800 BCE)
Plimpton Tablet n Writing n Mathematics The tablet translates into the following table: 1,59,0,15 1,59 2,49 1 1,56,56,58,14,50,6,15 56,7 1,20,25 2 1,55,7,41,15,33,45 1,16,41 1,50,49 3 1,53,10,29,32,52,16 3,31,49 5,9,1 4 1,48,54,1,40 1,5 1,37 5 1,47,6,41,40 5,19 8,1 6 1,43,11,56,28,26,40 38,11 59,1 7 1,41,33,59,3,45 13,19 20,49 8 1,38,33,36,36 8,1 12,49 9 1,35,10,2,28,27,24,26,40 1,22,41 2,16,1 10 1,33,45 45,0 1,15,0 11 1,29,21,54,2,15 27,59 48,49 12 1,27,0,3,45 2,41 4,49 13 1,25,48,51,35,6,40 29,31 53,49 14 1,23,13,46,40 56 1,46 15 Analysis suggests the table is a proto-trigonometry table. Column 2 is a, Column 3 is c, and Column 1 is c b = sec2 (θ). θ c b a
Plimpton Tablet n Writing n Mathematics While neither the Egyptians nor the Babylonians had any modern sense of an angle, they wrestled with geometric relations Of interest, the first line is decreasing First row = 1, 59, 0, 15 sec 2 (45 ) Last row = 1, 23, 13, 14, 40 sec 2 (31 ) The choice of lengths is theorized to have been chosen w.r.t the following: Given 2 regular sexagesimal integers p and q with p > q, form p 2 q 2, 2pq, and p 2 + q 2. Observe that these form Pythagorean triples. Restricting to values p < 60 and a < b, there are only 38 possible pairs of p and q, the first 15 of which are on the tablet.
Plimpton Tablet n Writing n Mathematics The tablet appears to have been broken at some point, it is theorized that it would have contained the remaining 23 pairs as well as columns for p, q, 2pq, and tan 2 (θ). A question arises: Why did the Babylonians record this data? Some historians argue that this was an exercise in elementary number theory and done for solely intellectual reasons (pure) Others argue that it was conducted to aid in the computation of real problems involving the areas of squares whose sides are shared with a right triangle. (applied)
Geometry n Writing n Mathematics It is clear that the Babylonians outclassed the Egyptians in algebra, but what about geometry? Recall that the Egyptians had incredible, accurate algorithms for volumes of frustums of pyramids Interestingly, the Babylonians also had formulae for these calculations! They used V = h [ (a + b 2 ) 2 + 1 3 ( ) ] 2 a b where a and b are the side lengths of the top and bottom squares and h is the height. It is not too hard to show that this formula reduces to the one the Egyptians used. 2
n Writing n Mathematics Pythagorean Theorem The Pythagorean Theorem cannot be found in any Egyptian sources, but it was definitely known to A tablet in the Yale collection (c. 1800 BCE) contains a diagram of a square with a diagonal running through it: 42;25,35 1;24,51,10 30
n Writing n Mathematics Pythagorean Theorem 42;25,35 1;24,51,10 30 The tablet says, starting with a square of side length 30, the diagonal is approximately 42; 23, 35. Moreover, it provides the ratio of an arbitrary squares diagonal w.r.t. its side as 1; 24, 51, 10 This is, of course, an approximation to 2. This shows clear knowledge of the Pythagorean Theorem This is an example of a very general observation in Babylonian mathematics
n Writing n Mathematics Other Geometric Facts Babylonians (unlike the Egyptians), as early as 1500 BCE, were aware of the following fact: Theorem An angle inscribed in a semicircle is always a right angle. This theorem is usually referred to as the Theorem of Thales, despite Thales living in 600 BCE!
Outline n Writing n Mathematics Influence and 1 n Writing 2 n Mathematics 3
Influence n Writing n Mathematics Influence and The center of mathematical development began moving from the n Valley to Greece around 800 BCE with the dawn of the Iron Age. A few questions arise: How much mathematics did Greece know? Hellenic period of Greece began with the death of Alexander the Great in 323 BCE. Unfortunately, there are very little sources available on pre-hellenic, Greek mathematics. We are unsure what they knew and what they didn t.
Influence n Writing n Mathematics Influence and How much knowledge transfered between Greece, Egypt, and? n clay tablets stand the test of time far better than Egyptian papyrus and Greek parchment, but did these other civilizations get their hands on the tablets? It would seem Egypt did not have much communication with the n River Valley since their mathematics remained inferior. A plausible explanation is given by geography. While only the Mediterranean Sea lies between Greece and Egypt, was separated from both by large swathes of desert.
n Writing n Mathematics Influence and s Babylonian mathematics seems superior to that of Egypt Babylonians didn t seem to have any concept of the modern interpretation of proof, but they did check their work as proof that a particular division worked. They do not seem to have anything in the way of logical principles that the Greeks would introduce. Their algebra was incredibly sophisticated. Even if they used concrete words such as length and width, these were clearly placeholders for abstract quantities. Much of the n mathematics was of a practical nature with only a few exceptions given to some recreational problems.
n Writing n Mathematics Influence and The Babylonian and Assyrian civilizations have perished; Hammurabi, Sargon and Nebuchadnezzar are empty names; yet Babylonian mathematics is still interesting, and the Babylonian scale of 60 is still used in Astronomy. - G.H. Hardy, A Mathematicians Apology