Rhind Mathematical Papyrus (2 of 8)

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Multiplying unit fraction series

How much is 184 times 2 '6 '7 times 5 '3 '5 ?

184 times 2 '6 '7 equals 368 31 26 = 425

425 times 5 '3 '5 equals 2125 142 85 = 2352 *

184 times 5 '3 '5 equals 920 61 37 = 1018

1018 times 2 '6 '7 equals 2036 170 145 = 2351 *

average result 2351 '2 (mistake less than '10)

How much is 317 times 3 '3 '11 times 4 '5 '13?

317 times 3 '3 '11 equals 951 106 29 = 1086

1086 times 4 '5 '13 equals 4344 217 84 = 4645

317 times 4 '5 '13 equals 1268 63 24 = 1355

1355 times 3 '3 '11 equals 4065 452 123 = 4640

average result 4642 '2 (mistake less than '30)

How much is 5 '3 '7 '11 by 5 '3 '7 '11? This time we multiply the product by a factor of 100 and then divide the result by 100 again:

100 times 5 '3 '7 '11 equals 500 33 14 9 = 556

556 times 5 '3 '7 '11 equals 2780 185 79 51 = 3095

3095 / 100 = practically 31 (mistake less than '135)

A calculation of interest

Vizier Milo (a former incarnation of Milo Gardner, whom I shall mention later :-) saved 68,954 Egyptian Dollars and brings the money to the First City Bank of Amarna, which offers interest at '101 '202 '303 '606 (2/101, a little more than 2 percent). How will Milo's fortune rise in the coming years?

Year 1 fortune 68,954 interest 683 341 228 114 = 1,366

year 2 fortune 70,320 interest 696 348 232 116 = 1,392

year 3 fortune 71,712 interest 710 355 237 118 = 1'420

year 4 fortune 73,132 interest 724 362 241 121 = 1,448

year 5 fortune 74,580 interest 738 369 246 123 = 1,476

year 6 fortune 76,056 interest 753 377 251 126 = 1,507

year 7 fortune 77,563 interest 768 384 256 128 = 1,536

year 8 fortune 79,099 interest 783 392 261 131 = 1,567

year 9 fortune 80,666 interest 799 399 266 133 = 1,597

year 10 fortune 82,263 interest 814 407 271 136 = 1,628

year 11 fortune 83,891 interest 831 415 277 138 = 1,661

year 12 fortune 85,552 interest 847 424 282 141 = 1,694

year 13 fortune 87,246 interest 864 432 288 144 = 1,728

year 14 fortune 88,974 interest 881 440 294 147 = 1,762

YEAR 15 FORTUNE 90,736 DOLLARS

And the exact value?

68954 x (1 + 2/101) exp 14 = 68954 x 1.3158971... = 90736.365...

Although we have rounded all numbers, the margin of error is not even 40 cents. And if we begin with 6,895,400 cents instead of 68,954 dollars, the mistake would be less than 4 (four) cents.

Playing with beans

o o o o o o o

o o o o o o o o o o o o o o

o o o o o o o o o o o o o o o oooooooooooo

3 + 4 + 5 = 3 x 4 = 2 x 6 = 1 x 12

oooooooooooo 12 or 1

oooooo oooooo 6+6 or 1/2 + 1/2

oooooo oo oooo 6+2+4 or 1/2 + 1/6 + 1/3

oooooo oo o ooo 6+2+1+3 or 1/2 + 1/6 + 1/12 + 1/4

1/1 = '1 1/2 = '2 1/3 = '3 1/4 = '4 1/6 = '6 1/12 = '12

1 = '1 1 = '1

1 = '2 '2 1 = '1x2 '2

1 = '2 '6 '3 1 = '1x2 '2x3 '3

1 = '2 '6 '12 '4 1 = '1x2 '2x3 '3x4 '4

RHIND MATHEMATICAL PAPYRUS, duplations and conversions

The famous Rhind Mathematical Papyrus was written around 1650 BC, and represents a copy of a lost scroll dated around 1850 BC. At the begin of the RMP are found divisions of 2 by the odd numbers from 5 up to 101. Examples:

"5 = '3 '15 (2/5 = 1/3 + 1/15)

"7 = '4 '28 (2/7 = 1/4 + 1/28)

"9 = '6 '18 (2/9 = 1/6 + 1/18)

"11 = '6 '66

"13 = '8 '52 '104

"19 = '12 '76 '114

"35 = '30 '42

"43 = '42 '86 '129 '301

"67 = '40 '268 '670

"91 = '70 '130

"95 = '60 '380 '570

"99 = '66 '198

"101 = '101 '202 '303 '606

The calculations are carried out as follows. If you wish to divide 2 by any number a, note the numbers 1 and a. Divide them by handy numbers until you get a number b that is smaller than 2. Now subtract number b from 2 and complete your series. Examples:

1 9 (number a)

'3 3

'6 1 '2 (number b)

2 minus 1'2 equals '2

'18 '2

"9 = '6 '18 (since 1'2 '2 equals 2)

1 19

'3 6 '3 (divided by 3)

'6 3 '6 (divided by 2)

'12 1 '2 '12 (divided by 2)

2 minus 1'2'12 equals ??

24 minus 12+6+1 equals 5 or 3 + 2 (multiplied by 12)

2 minus 1'2'12 equals '4 '6 (divided by 12)

'76 '4

'114 '6

"19 = '12 '76 '114 (since 1'2'12 '4 '6 equals 2)

1 35

'5 7

'15 2 '3

'30 1 '6

2 minus 1'6 equals '2 '3

35 divided by 2+3 equals 7 / remainder '2x3x7 equals '42

"35 = '30 '42

1 43

'2 21 '2 (divided by 2)

'6 7 '6 (divided by 3)

'42 1 '42 (divided by 7)

2 minus 1'42 equals '2 '3 '7

'86 '2

'129 '3

'301 '7

"43 = '42 '86 '129 '301 (since 1'42 '2 '3 '7 equals 2)

1 91

'7 13

'14 6 '2

'70 1 '5 '10

2 minus 1'5'10 equals '2 '5

91 divided by 2+5 equals 13 / remainder '2x5x13 or '130

"91 = '70 '130

1 93

'31 3

'62 1'2

2 minus 1'2 equals '2

'186 '2

"93 = '62 '186 (since 1'2 '2 equals 2)

1 101

'101 1

2 minus 1 equals '2 '3 '6

'202 '2 '303 '3 '606 '6

"101 = '101 '202 '303 '606 (since 1 '2 '3 '6 equals 2)

Down under algebra

Beginners may carry out all divisions from 2/5 to 2/101 and in so doing learn how to work with unit fraction series. Advanced learners may go a step further and look out for number patterns providing the same conversions:

1 = '2 '1x2 = '2 '2

'2 = '3 '2x3 = '3 '6

'3 = '4 '3x4 = '4 '12

'4 = '5 '4x5 = '5 '20

'5 = '6 '5x6 = '6 '30

'6 = '7 '6x7 = '7 '42

'7 = '8 '7x8 = '8 '56 general form: 'a = 'a+1 'aa+a

1 = '2 '2 -------- "1 = '1 '1

'2 = '3 '6

'3 = '4 '12 ------- "3 = '2 '6

'4 = '5 '20

'5 = '6 '30 ------- "5 = '3 '15 (RMP)

'6 = '7 '42

'7 = '8 '56 ------- "7 = '4 '28

'8 = '9 '72

'9 = '10 '90 ------ "9 = '5 '45 (RMP)

'10 = '11 '110

'11 = '12 '132 ---- "11 = '6 '66 (RMP)

...............................................

The first number pattern generates a pair of remarkable series:

1 = '2 '2

'2 = '6 '3

'3 = '12 '4

'4 = '20 '5

'5 = '30 '6

'6 = '42 ...

1 = '2 '6 '12 '20 '30 '42 ...

1 = '1x2 '2x3 '3x4 '4x5 '5x6 '6x7 ...

1 = '1

1 = '1x2 '2

1 = '1x2 '2x3 '3

1 = '1x2 '2x3 '3x4 '4

1 = '1x2 '2x3 '3x4 '4x5 '5

1 = '1x2 '2x3 '3x4 '4x5 '5x6 '6

1 = '1x2 '2x3 '3x4 '4x5 '5x6 '6x7 '7

........................................

1 = '2 '2

'2 = '3 '6

'6 = '7 '42

'42 = '43 1806

1 = '1

1 = '2 '2

1 = '2 '3 '6

1 = '2 '3 '7 '42

1 = '2 '3 '7 '43 '1806

1 = '2 '3 '7 '43 '1807 '3263443 ...

The equation "3 = '2 '6 can be used for many simple conversions:

"9 equals '6 '18 (RMP)

"15 equals '10 '30 (RMP)

"21 equals '14 '42 (RMP)

..........................

"87 equals '58 '174 (RMP)

"93 equals '62 '186 (RMP)

"99 equals '66 '198 (RMP)

The principle of the first number pattern may be expanded as follows:

1 = '2 '2 = '3 '3 '3 = '4 '4 '4 '4 ...

'2 = '3 '6 = '4 '8 '8 = '5 '10 '10 '10 ...

'3 = '4 '12 = '5 '15 '15 = '6 '18 '18 '18 ...

'4 = '5 '20 = '6 '24 '24 = '7 '28 '28 '28 ...

'5 = '6 '30 = '7 '35 '35 = '8 '40 '40 '40 ...

................................................

Modifying the third column:

1 = '4 '4 '2 ----------- "1 = '2 '2 '1

'2 = '5 '10 '5

'3 = '6 '18 '9

'4 = '7 '28 '14

'5 = '8 '40 '20 ---------- "5 = '4 '20 '10

'6 = '9 '54 '27

'7 = '10 '70 '35

'8 = '11 '88 '44

'9 = '12 '108 '54 ---------- "9 = '6 '54 '27

'10 = '13 '130 '65

'11 = '14 '154 '77

'12 = '15 '180 '90

'13 = '16 '208 '104 -------- "13 = '8 '104 '52 (RMP)

A more demanding general pattern:

"1 equals '1 plus '1x1 of 1 (2/1 = 1/1 + 1/1x1)

"3 equals '3 plus '3x3 of 3 (2/3 = 1/3 + 3/3x3) 3=2+1

'2 plus '3x2 of 1 (2/3 = 1/2 + 1/2x3)

"5 equals '5 plus '5x5 of 5 (2/5 = 1/5 + 5/5x5) 5=3+2

'4 plus '5x4 of 3 (2/5 = 1/4 + 3/4x5) 3=2+1

'3 plus '5x3 of 1 (2/3 = 1/3 + 1/3x5)

"7 equals '7 plus '7x7 of 7 (2/7 = 1/7 + 7/7x7) 7=4+3

'6 plus '7x6 of 5 (2/7 = 1/6 + 5/6x7) 5=3+2

'5 plus '7x5 of 3 (2/7 = 1/5 + 3/5x7) 3=2+1

'4 plus '7x4 of 1 (2/7 = 1/4 + 1/4x7)

"9 equals '9 plus '9x9 of 9

'8 plus '9x8 of 7

'7 plus '9x7 of 5

'6 plus '9x6 of 3

'5 plus '9x5 of 1

........................................................

All these and many more number patterns are contained in Milo Gardner's formulas:

2/p - 1/a = (2a - p) / pa

n/p - 1/a = (na - p) / pa

It was Milo Gardner who stimulated my interest in unit fractions, back in Spring 1997. I thank him again for his many patient e-mails.

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