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Babylonians

Calculation from 2000 BC The Babylonians used the formula
ab = (a + b)²/4 - (a - b)²/4
to make multiplication easier.
This shows that a table of squares is all that is necessary to multiply numbers, simply taking the difference of two numbers that were looked up in the table.

Test your Babylonian maths.


x =


n²/4 rounded down to nearest integer
1 0 21 110 41 420 61 930 81 1640
2 1 22 121 42 441 62 961 82 1681
3 2 23 132 43 462 63 992 83 1722
4 4 24 144 44 484 64 1024 84 1764
5 6 25 156 45 506 65 1056 85 1806
6 9 26 169 46 529 66 1089 86 1849
7 12 27 182 47 552 67 1122 87 1892
8 16 28 196 48 576 68 1156 88 1936
9 20 29 210 49 600 69 1190 89 1980
10 25 30 225 50 625 70 1225 90 2025
11 30 31 240 51 650 71 1260 91 2070
12 36 32 256 52 676 72 1296 92 2116
13 42 33 272 53 702 73 1332 93 2162
14 49 34 289 54 729 74 1369 94 2209
15 56 35 306 55 756 75 1406 95 2256
16 64 36 324 56 784 76 1444 96 2304
17 72 37 342 57 812 77 1482 97 2352
18 81 38 361 58 841 78 1521 98 2401
19 90 39 380 59 870 79 1560 99 2450
20 100 40 400 60 900 80 1600 100 2500

For example
To work out 23 x 24:

23+24=47, 24-23=1
Look up 47 Þ 552
Look up 1 Þ 0
Subtract 552-0 = 552
To work out 37²:

37+37=74, 37-37=0
Look up 74 Þ 1369
Look up 0 Þ 0
Subtract 1369-0 = 1369
  Thus it is easy to work the answers to multiplications of two number that differ by 1 or to square a number.



Other early mathematics The Babylonians had an advanced number system, in some ways more advanced than our present system. It was a positional system with base 60 rather than the base 10 of our present system. Now 10 has only two proper divisors, 2 and 5. However 60 has 10 proper divisors so many more numbers have a finite form. The Babylonians divided the day into 24 hours, each hour into 60 minutes, each minute into 60 seconds. This form of counting has survived for 4000 years. To write 5h 25' 30", i.e. 5 hours, 25 minutes, 30 seconds is just to write the base 60 fraction, 5 25/60 30/3600 or as a base 10 fraction 5 4/10 2/100 5/1000 which we write as 5.425 in decimal notation.
Plimpton 322
(1900 - 1600 BC)		 plimpton 322
One of the Babylonian tablets (Plimpton 322) which is dated from between 1900 and 1600 BC contains answers to a problem containing Pythagorean triples, i.e. numbers a, b, c with a² + b² = c².
It is said to be the oldest number theory document in existence.
(Note Pythagoras lived from about 569 BC to about 475 BC)


According to tradition a brick-arch bridge was built about 1800 BC in Babylon.

Links:
Babylonian mathematics
Babylonian square roots

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