The Babylonians lived in Mesopotamia, a fertile plain between the Tigris and Euphrates rivers.



Here is a map of the region where the civilisation flourished.

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The region had been the centre of the Sumerian civilisation which flourished beforeBC. This was an advanced civilisation building cities and supporting the people with irrigation systems, a legal system, administration, and even a postal service. Writing developed and counting was based on a sexagesimal system, that is to say base. AroundBC the Akkadians invaded the area and for some time the more backward culture of the Akkadians mixed with the more advanced culture of the Sumerians. The Akkadians invented the abacus as a tool for counting and they developed somewhat clumsy methods of arithmetic with addition, subtraction, multiplication and division all playing a part. The Sumerians, however, revolted against Akkadian rule and byBC they were back in control.However the Babylonian civilisation, whose mathematics is the subject of this article, replaced that of the Sumerians from aroundBC The Babylonians were a Semitic people who invaded Mesopotamia defeating the Sumerians and by aboutBC establishing their capital at Babylon.The Sumerians had developed an abstract form of writing based on cuneiformi.e. wedge-shapedsymbols. Their symbols were written on wet clay tablets which were baked in the hot sun and many thousands of these tablets have survived to this day. It was the use of a stylus on a clay medium that led to the use of cuneiform symbols since curved lines could not be drawn. The later Babylonians adopted the same style of cuneiform writing on clay tablets.



Here is one of their tablets

It was an important task for the rulers of Mesopotamia to dig canals and to maintain them, because canals were not only necessary for irrigation but also useful for the transport of goods and armies. The rulers or high government officials must have ordered Babylonian mathematicians to calculate the number of workers and days necessary for the building of a canal, and to calculate the total expenses of wages of the workers.



There are several Old Babylonian mathematical texts in which various quantities concerning the digging of a canal are asked for. They are YBC 4666 , 7164 , and VAT 7528 , all of which are written in Sumerian ..., and YBC 9874 and BM 85196 , No. 15 , which are written in Akkadian ... . From the mathematical point of view these problems are comparatively simple ...

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4000

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5 25 60 30 3600 5 \large\frac{25}{60}

ormalsize \large\frac{30}{3600}

ormalsize 5 6 0 2 5 ​ 3 6 0 0 3 0 ​

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5

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5 4 10 2 100 5 1000 5 \large\frac{4}{10}

ormalsize \large\frac{2}{100}

ormalsize \large\frac{5}{1000}

ormalsize 5 1 0 4 ​ 1 0 0 2 ​ 1 0 0 0 5 ​

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425

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8 2 = 1 , 4 8^{2} = 1,4 8 2 = 1 , 4

8 2 = 1 , 4 = 1 × 60 + 4 = 64 8^{2} = 1, 4 = 1 \times 60 + 4 = 64 8 2 = 1 , 4 = 1 × 6 0 + 4 = 6 4

5 9 2 = 58 , 1 ( = 58 × 60 + 1 = 3481 ) 59^{2} = 58, 1 (= 58 \times 60 +1 = 3481) 5 9 2 = 5 8 , 1 ( = 5 8 × 6 0 + 1 = 3 4 8 1 )

a b = ( a + b ) 2 − a 2 − b 2 2 ab = \Large \frac {(a + b)^{2} - a^{2} - b^{2}} 2 a b = 2 ( a + b ) 2 − a 2 − b 2 ​

a b = ( a + b ) 2 − ( a − b ) 2 4 ab = \Large \frac {(a + b)^{2} - (a - b)^{2}} 4 a b = 4 ( a + b ) 2 − ( a − b ) 2 ​

a b = a × 1 b \Large \frac a b

ormalsize = a \times \Large \frac 1 b b a ​ = a × b 1 ​

2 0; 30 3 0; 20 4 0; 15 5 0; 12 6 0; 10 8 0; 7, 30 9 0; 6, 40 10 0; 6 12 0; 5 15 0; 4 16 0; 3, 45 18 0; 3, 20 20 0; 3 24 0; 2, 30 25 0; 2, 24 27 0; 2, 13, 20

1 7 , 1 11 , 1 13 \large\frac{1}{7}

ormalsize , \large\frac{1}{11}

ormalsize , \large\frac{1}{13}

ormalsize 7 1 ​ , 1 1 1 ​ , 1 3 1 ​

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1 13 \large\frac{1}{13}

ormalsize 1 3 1 ​

1 13 = 7 91 = 7 × 1 91 ≈ 7 × 1 90 \large\frac{1}{13}

ormalsize = \large\frac{7}{91}

ormalsize = 7 \times \large\frac{1}{91}

ormalsize \approx 7 \times \large\frac{1}{90}

ormalsize 1 3 1 ​ = 9 1 7 ​ = 7 × 9 1 1 ​ ≈ 7 × 9 0 1 ​

1 90 \large\frac{1}{90}

ormalsize 9 0 1 ​

7

1 7 \large\frac{1}{7}

ormalsize 7 1 ​

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... an approximation is given since 7 does not divide.

n 3 + n 2 n^{3} + n^{2} n 3 + n 2

a x 3 + b x 2 = c ax^{3} + bx^{2} = c a x 3 + b x 2 = c .

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a 2 a^{2} a 2

b 3 b^{3} b 3

( a b ) 3 + ( a x b ) 2 = c a 2 b 3 \Large (\frac {a}{b})

ormalsize ^{3} + \Large (\frac {ax}{b})

ormalsize^{2} = \Large \frac {ca^2}{b^3} ( b a ​ ) 3 + ( b a x ​ ) 2 = b 3 c a 2 ​ .

y = a x b y = \Large \frac {ax} b y = b a x ​

y 3 + y 2 = c a 2 b 3 y^{3} + y^{2} = \Large \frac {ca^{2}}{b^{3}} y 3 + y 2 = b 3 c a 2 ​

n 3 + n 2 n^{3} + n^{2} n 3 + n 2

n n n

n 3 + n 2 = c a 2 b 3 n^{3} + n^{2} = \Large \frac {ca^{2}}{b^{3}} n 3 + n 2 = b 3 c a 2 ​

y y y

x x x

x = b y a x = \Large \frac {by}{a} x = a b y ​

a x = b ax = b a x = b

1 n \large\frac{1}{n}

ormalsize n 1 ​

1 a \large\frac{1}{a}

ormalsize a 1 ​

b b b

2 3 \large\frac{2}{3}

ormalsize 3 2 ​

2 3 \large\frac{2}{3}

ormalsize 3 2 ​

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2 3 × 2 3 x + 100 = x \large\frac{2}{3}

ormalsize \times \large\frac{2}{3}

ormalsize x + 100 = x 3 2 ​ × 3 2 ​ x + 1 0 0 = x

( 1 − 4 9 ) x = 100 (1 - \large\frac{4}{9}

ormalsize )x = 100 ( 1 − 9 4 ​ ) x = 1 0 0

2 3 × 2 3 \large\frac{2}{3}

ormalsize \times \large\frac{2}{3}

ormalsize 3 2 ​ × 3 2 ​

1

( 1 − 4 9 ) (1 - \large\frac{4}{9}

ormalsize ) ( 1 − 9 4 ​ )

1 / ( 1 − 4 9 ) 1/(1 - \large\frac{4}{9}

ormalsize ) 1 / ( 1 − 9 4 ​ )

x x x

1 / ( 1 − 4 9 ) 1/(1 - \large\frac{4}{9}

ormalsize ) 1 / ( 1 − 9 4 ​ )

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x 2 + b x = c x^{2} + bx = c x 2 + b x = c and x 2 − b x = c x^{2} - bx = c x 2 − b x = c

b , c b, c b , c

x = ( b 2 ) 2 + c − b 2 x = \sqrt{(\frac b 2)^{2} + c} - \large\frac b 2 x = ( 2 b ​ ) 2 + c ​ − 2 b ​ and x = ( b 2 ) 2 + c + b 2 x = \sqrt{(\frac b 2)^{2} + c} + \large\frac b 2 x = ( 2 b ​ ) 2 + c ​ + 2 b ​ .

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x 2 + 7 x = 1 , 0 x^{2} + 7x = 1, 0 x 2 + 7 x = 1 , 0

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x 2 + b x = c x^{2} + bx = c x 2 + b x = c

x = ( b 2 ) 2 + c − b 2 x = \sqrt{(\frac b 2)^{2} + c} - \large\frac b 2 x = ( 2 b ​ ) 2 + c ​ − 2 b ​

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85200

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Many of the tablets concern topics which, although not containing deep mathematics, nevertheless are fascinating. For example we mentioned above the irrigation systems of the early civilisations in Mesopotamia. These are discussed inwhere Muroi writes:-The Babylonians had an advanced number system, in some ways more advanced than our present systems. It was a positional system with a base ofrather than the system with basein widespread use today. For more details of the Babylonian numerals, and also a discussion as to the theories why they used base, see our article on Babylonian numerals The Babylonians divided the day intohours, each hour intominutes, each minute intoseconds. This form of counting has survived foryears. To write", i.e.hours,minutes,seconds, is just to write the sexagesimal fraction,. We adopt the notationfor this sexagesimal number, for more details regarding this notation see our article on Babylonian numerals . As a basefraction the sexagesimal numberiswhich is written asin decimal notation.Perhaps the most amazing aspect of the Babylonian's calculating skills was their construction of tables to aid calculation. Two tablets found at Senkerah on the Euphrates indate fromBC. They give squares of the numbers up toand cubes of the numbers up to. The table giveswhich stands forand so on up toThe Babylonians used the formulato make multiplication easier. Even better is their formulawhich shows that a table of squares is all that is necessary to multiply numbers, simply taking the difference of the two squares that were looked up in the table then taking a quarter of the answer.Division is a harder process. The Babylonians did not have an algorithm for long division. Instead they based their method on the fact thatso all that was necessary was a table of reciprocals. We still have their reciprocal tables going up to the reciprocals of numbers up to several billion. Of course these tables are written in their numerals, but using the sexagesimal notation we introduced above, the beginning of one of their tables would look like:Now the table had gaps in it since, etc. are not finite basefractions. This did not mean that the Babylonians could not compute, say. They would writeand these values, for example, were given in their tables. In fact there are fascinating glimpses of the Babylonians coming to terms with the fact that division bywould lead to an infinite sexagesimal fraction. A scribe would give a number close toand then write statements such assee for example:-Babylonian mathematics went far beyond arithmetical calculations. In our article on Pythagoras's theorem in Babylonian mathematics we examine some of their geometrical ideas and also some basic ideas in number theory. In this article we now examine some algebra which the Babylonians developed, particularly problems which led to equations and their solution.We noted above that the Babylonians were famed as constructors of tables. Now these could be used to solve equations. For example they constructed tables forthen, with the aid of these tables, certain cubic equations could be solved. For example, consider the equationLet us stress at once that we are using modern notation and nothing like a symbolic representation existed in Babylonian times. Nevertheless the Babylonians could handle numerical examples of such equations by using rules which indicate that they did have the concept of a typical problem of a given type and a typical method to solve it. For example in the above case they wouldin our notationmultiply the equation byand divide it byto getPuttingthis gives the equationwhich could now be solved by looking up thetable for the value ofsatisfying. When a solution was found forthenwas found by. We stress again that all this was done without algebraic notation and showed a remarkable depth of understanding.Again a table would have been looked up to solve the linear equation. They would consult thetable to findand then multiply the sexagesimal number given in the table by. An example of a problem of this type is the following.Suppose, writes a scribe,ofof a certain quantity of barley is taken,units of barley are added and the original quantity recovered. The problem posed by the scribe is to find the quantity of barley. The solution given by the scribe is to computetimesto get. Subtract this fromto get. Look up the reciprocal ofin a table to get. Multiplybyto get the answerIt is not that easy to understand these calculations by the scribe unless we translate them into modern algebraic notation. We have to solvewhich is, as the scribe knew, equivalent to solving. This is why the scribe computedsubtracted the answer fromto get, then looked upand sowas found frommultiplied bygivingwhich istimesto getin sexagesimalTo solve a quadratic equation the Babylonians essentially used the standard formula. They considered two types of quadratic equation, namelywhere herewere positive but not necessarily integers. The form that their solutions took was, respectivelyNotice that in each case this is the positive root from the two roots of the quadratic and the one which will make sense in solving "real" problems. For example problems which led the Babylonians to equations of this type often concerned the area of a rectangle. For example if the area is given and the amount by which the length exceeds the breadth is given, then the breadth satisfies a quadratic equation and then they would apply the first version of the formula above.A problem on a tablet from Old Babylonian times states that the area of a rectangle isand its length exceeds its breadth by. The equationis, of course, not given by the scribe who finds the answer as follows. Compute half of, namely, square it to get. To this the scribe addsto get. Take its square rootfrom a table of squaresto get. From this subtractto give the answerfor the breadth of the triangle. Notice that the scribe has effectively solved an equation of the typeby usingInBerriman givestypical examples of problems leading to quadratic equations taken from Old Babylonian tablets.If problems involving the area of rectangles lead to quadratic equations, then problems involving the volume of rectangular excavationa "cellar"lead to cubic equations. The clay tablet BM+ containingproblems of this type, is the earliest known attempt to set up and solve cubic equations. Hoyrup discusses this fascinating tablet in. Of course the Babylonians did not reach a general formula for solving cubics. This would not be found for well over three thousand years.