Reducing Scales - Recurring numbers & Patterns

[tomos]

I work with scaled drawings.

Sometimes I have to produce various versions of a drawing at differnet scales.
I can do it fine, but to be honest I find it hard to get my head around some of the concepts.

Here's an example - the reason I even thought of posting it is the numbers with the recurring numbers after the decimal point.
I reduce a drawing/figure @ 1:00 to 70% of original size.

1:100 * 70% =
7:1000 =
1:142.857142857142∞

The 857142 bit is repeating so I added the infinity symbol ∞ (not sure if that's correct)

Later, I need to reduce the new modified version to 1:50 (which would be 50% of the first figure).
i.e. I want to reduce (1:100 * 70%) to 1:50  i.e. to (1:100 * 50%)

100 * 70% = 70
100 * 50% = 50

70 * X = 50
X = 50/70 = 0.7142857142857142∞
which would be 71.42857142857142∞%

which number is exactly half of the number for the first 70% reduction (1:100 * 70%) = 1:142.857142857142∞

I cannot explain that - I possibly could figure it out, but I'll just enjoy it and move on.
Re the recurring number - if I divide or multiply by two, the result seems to be always recurring at some stage after the decimal point.
BTW is that the correct name - simply "recurring digit" ?


PS thank god I dont work with imperial scales :p

[tomos]

You can get the first recurring digit there by dividing 1000 by 7.
(If possible I show a 10 metre scale - that's partly why I look at what 1000 is:
7cm on drawing is 1000cm =10m)

Here a bit of experimenting with halving the recurring number:

[attachimg=#]

[TaoPhoenix]

Heh I used to like this stuff as a young'un 30 years ago! (     )
Notice you stopped just before the pattern broke! Last I knew of these kinds of patterns as a kid, it matters whether the front whole number is odd or even, which determines if the decimal "can divide by 2 all by itself" or else a flow-over 0.5 amount is given/taken back and forth from it.

The whole Casting-Out-Nines series of tricks might be involved here too, but I'm not sure.

[Renegade]

That's a funky little pattern there. (I love number magic and the like.)

[tomos]

As said, what got me is the way the number cropped up again.

Let's see:

To reduce from 70% to 50% I must multiply the 70% version by half the scaled-value (? terminology again) of one [1] at that scale (70%)

Just seems freaky to me - might try to look closer at it when I have time.
Might be something to do with area - 50% of something from the scale POV is 25% of the area.
If I figured that out for 70% ...

[Renegade]

Here's a fun one:

1/81

[vlastimil]

It is called repeating decimal or recurring decimal, so you were very very close. The infinity sign is not used - instead there should be a horizontal line above the digits that are repeating ( not easily done in html ;-) ).

Rational numbers (M/N, where M and N are integral) do have this property when they are written in the usual decimal notation. Look these examples:
1.2 = 12/10
1.23 = 123/100
1.234 = 1234/1000
These "normal" decimal numbers can be expressed as fractions, but note that there always is a power of 10 in denominator. Our usual decimal notation can be considered just a shortcut of the full M/N notation.

Every fractional number (that cannot be further simplified) that has something else than a product of 2s and 5s (10=2*5) in the denominator will have infinite number of repeating digits when written in decimal notation.
1/3, 1/6, 1/7, 1/9, 1/11, 1/13, ...

Here is a provocative number: 1/2 = 0.5 = 0.500000000000000000000000000000000000... 

[Renegade]

It is called repeating decimal or recurring decimal, so you were very very close. The infinity sign is not used - instead there should be a horizontal line above the digits that are repeating ( not easily done in html ;-) ).

But CSS works!

Code: (css)

.bar {border-top: 1px solid #000000;}

[tomos]

Thanks for the explanations Vlastimil
it's interesting - I've never really thought about these basics before...

[tomos]

Every fractional number (that cannot be further simplified) that has something else than a product of 2s and 5s (10=2*5) in the denominator will have infinite number of repeating digits when written in decimal notation.
1/3, 1/6, 1/7, 1/9, 1/11, 1/13, ...

so, I think the reason I got the same recurring decimal twice, was simply because, in both cases, the number was divided by 7.
(Also the numbers being divided are related.)

the first instance was:
1000/7
and in the second example it would have needed
500/7 to get the correct percentage.

So naturally the second result is half the first 1000 -> 500

Very simple really - once I get a break from it - and get a maths lesson

[Renegade]

Just thought of this as it's related to number patterns:

I wrote some gematria software a while back. The site domain name is NSFW, so here's the link.

The gematria software is the "Anti-Christ Hunter v6.6.6":

[attachimg=#]

It does a few different versions of gematria. I've been thinking about doing another version with more analysis of things like roots, squares, triangle numbers, etc. One of these days I may redo a comprehensive gematria and numerology application.

Here's a little fun fact: Stone Henge encodes the square root of 153, which is the number of full moons in a year. Also the number of fish some of Jesus' disciples pull out of the sea in one verse. etc. etc.

[Renegade]

This video goes through a lot of architecture and history to show recurring numerical patterns:

http://www.secretsinplainsight.com/

Playlist of 23 episodes: http://www.youtube.com/watch?feature=player_embedded&list=PL5680DDCC23C0216A&v=JTA_EkGwUE0

Full video:

http://www.youtube.com/watch?v=L777RhL_Fz4

https://www.youtube.com/watch?v=L777RhL_Fz4

I won't comment on the conclusions or content, but he does an excellent job of showing numerical patterns.

[Jibz]

http://en.wikipedia.org/wiki/Repeating_decimal

Rational numbers are numbers that can be expressed in the form a/b where a and b are integers and b is non-zero. This form is known as a common fraction. On the one hand, the decimal representation of a rational number is ultimately periodic, as explained below. On the other hand every real number which has an eventually periodic decimal expansion is a rational number. In other words the numbers with eventually repeating decimal expansions are exactly the rational numbers (i.e.: those that can be expressed as ratios).