Franz Gnaedinger

2017-04-21 07:02:27 UTC

Irrationals not being numbers, and 1 not being 0.999..., are everlasting

topics in sci.math, popping up all the time. Very simple yet forgotten

methods of early mathematics can dissolve the problem.

Approximating the square root of 2 by a number column which I reconstructed

in 1979

1 1 2

2 3 4

5 7 10

12 17 24

29 41 58

70 99 140

and so on

99/70 is already a fine value for the square root of 2. (Analogous number

columns approximate the square roots of 3 and 5 and the cube root of 2.)

Now let us look at the square root of 4 by drawing up the analogous number

column

1 1 4

2 5 8

7 13 28

20 41 80

61 121 244

182 365 728

and so on

The ratios, for example 41/20 or 121/61, get ever closer to 2. This means

that 2 as the square root of 4 is equally mysterious as the square root of 2.

Or that the square root of 2 is no less mysterious than 2.

As for 1 = 0.999... let me begin with the Horus eye series of Ancient Egypt,

1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 or simply '2 '4 '8 '16 '32 '64

1 = '1

1 = '2 '2

1 = '2 '4 '4

1 = '2 '4 '8 '8

1 = '2 '4 '8 '16 '16

1 = '2 '4 '8 '16 '32 '32

1 = '2 '4 '8 '16 '32 '64 '64

From this you get

'2 '4 '8 '16 '32 '64 ... = 1

(The Horus eye series had an astronomical function, since a month of 30

days multiplied by '2 '4 '8 '16 '32 '64 yields 29 '2 '32 days or 29 days

12 hours 45 minutes for one lunation of 29 days 12 hours 44 minutes 2.9

seconds, modern average from 1989 AD - mistake of the old value less than

one minute per lunation, or half a day in a lifetime. One eye of the Horus

falcon was the sun, standing for a month of 30 days, his other eye was

the moon, standing for one lunation or synodic month.)

Another infinite series

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 '7x8 '8x9 '9x10 '10x11 '11x12 ...

'1x2 '2x3 '5x6 '6x7 '9x10 '10x11 ... = pi/4

Now for 0.999... being 1

1 = 10/10

1 = 9/10 + 10/100

1 = 9/10 + 9/100 + 10/1000

1 = 9/10 + 9/100 + 9/1000 + 10/10000

and so on

1 = 0.999...

Mathematical education should begin with really simple methods, not on

the level of Euclid. Would prevent some people from wasting their life

on kooky notions.

topics in sci.math, popping up all the time. Very simple yet forgotten

methods of early mathematics can dissolve the problem.

Approximating the square root of 2 by a number column which I reconstructed

in 1979

1 1 2

2 3 4

5 7 10

12 17 24

29 41 58

70 99 140

and so on

99/70 is already a fine value for the square root of 2. (Analogous number

columns approximate the square roots of 3 and 5 and the cube root of 2.)

Now let us look at the square root of 4 by drawing up the analogous number

column

1 1 4

2 5 8

7 13 28

20 41 80

61 121 244

182 365 728

and so on

The ratios, for example 41/20 or 121/61, get ever closer to 2. This means

that 2 as the square root of 4 is equally mysterious as the square root of 2.

Or that the square root of 2 is no less mysterious than 2.

As for 1 = 0.999... let me begin with the Horus eye series of Ancient Egypt,

1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 or simply '2 '4 '8 '16 '32 '64

1 = '1

1 = '2 '2

1 = '2 '4 '4

1 = '2 '4 '8 '8

1 = '2 '4 '8 '16 '16

1 = '2 '4 '8 '16 '32 '32

1 = '2 '4 '8 '16 '32 '64 '64

From this you get

'2 '4 '8 '16 '32 '64 ... = 1

(The Horus eye series had an astronomical function, since a month of 30

days multiplied by '2 '4 '8 '16 '32 '64 yields 29 '2 '32 days or 29 days

12 hours 45 minutes for one lunation of 29 days 12 hours 44 minutes 2.9

seconds, modern average from 1989 AD - mistake of the old value less than

one minute per lunation, or half a day in a lifetime. One eye of the Horus

falcon was the sun, standing for a month of 30 days, his other eye was

the moon, standing for one lunation or synodic month.)

Another infinite series

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 '7x8 '8x9 '9x10 '10x11 '11x12 ...

'1x2 '2x3 '5x6 '6x7 '9x10 '10x11 ... = pi/4

Now for 0.999... being 1

1 = 10/10

1 = 9/10 + 10/100

1 = 9/10 + 9/100 + 10/1000

1 = 9/10 + 9/100 + 9/1000 + 10/10000

and so on

1 = 0.999...

Mathematical education should begin with really simple methods, not on

the level of Euclid. Would prevent some people from wasting their life

on kooky notions.