Fun with maths!

[Arugela]

This is something I do when I'm bored. It's a fun way to analyse number patterns and possibly decimals. I used to have a guaranteed way to get a fraction from a decimal but I have since forgot. I'm attempting to remember it. It may have involved subtracting values from dividing by 0.9 or similar and finding the answer. I can't remember. But here is one of the other things I like to do sometimes for fun. Number line evaluations!

Take any number sequence. Write it out and apply a basic function to it(or any function). In this case I will find the difference with no negatives.

0 1 2 3 4 5 6 7 8 9 10

Between those lines on paper write the difference:

  1 1 1 1 1 1 1 1 1

From this you can extrapolate the difference is X+1.(And in as simple or as complex a way as one desired. Explore all the possibilities.)

Now try a different function.

Division:

0/1 1/2 2/3 3/4 4/5 5/6 6/7 7/8 8/9 9/10

Find the difference or any other desired function.

Difference:(with negatives)

-.5 -.166... -.083... -.05 -.03... -

-1/2 -1/6 -1/12 -1/20 1/30

to be continued...

Difference of numberators: (no negatives)

4 6 8 10

X+2

with negatives:

-8 -18 -32 -50

Difference:

10 14 18

x+4?

or

26 40 82

Keep trying combos. Any single logic followed far enough(Even farther than demonstrated here preferably.) usually does something interesting like go back to the original number sequence, another known number sequence, or shows some other aspect of the original sequence in the process.

Example of fractions: (I remembered how!)

( .166... is 1/10 plus 6/90=15/90=1/6) x/9 is always x repeated infinitely(.x...) in a fraction. use x/9 or x/11 or similar to increase decrease and subtract from base fraction to get a simpler fraction and add. (0.083... 8/100+3/900=75/900=1/12)

Continued:

Difference of difference:

-.3... -.249... etc.

-1/3  -1/4 etc.

(1/11+10/11=0.99...=11/11=1<-proof!)  1/110+1/1100=0.009...+24/100=0.249...=11/1100(=0.01)+264/1100=275/1100=11/44=1/4=0.25!<-Proof

Note: 1/111 adds extra zeroes! ;p (0.009009... fun math trick!) Likewise, 1/99=0.0101.. and 1/9= 0.111...! Multiples of 9 and 11 are easily combinable.(likewise any other paired combos!)

You then analyse the results and extract answers about various things in the number line. One fun one is unknown/questionable factors like 0/1 or 1/0.

dividend of the dividend:

0/1/1/2 1/2/2/3 2/3/3/4 3/4/4/5 4/5/5/6

Difference of the dividend of the dividend:

0/.5=0? .5/.6... .6.../.75 .75/.8 .8/.83...

? .75 .8... .9375 .96

? 3/4 8/9 15/16 24/25

x-1/x where the next variable is x+ 5 then 7 then 9 aka the odd number sequence. x+2

Difference of Numerators:

4 9 16 24

  5 7 9

If this goes on endlessly does this mean 0/1/1/2 = 0/1(applied the logic wrong! ><) 2/3? AKA 0/1=.5*.6...=0/1=.3... or multipying on each side by 1/0 you get 0/0 or ?/?= 1/3(not sure on this one)

 

You could go on to use a flipped difference to do the same thing so you have 1/0 instead of 0/1 and see how it comes out: (But I'm not sure of the implications)

1/0/2/1 2/1/3/2 3/2/4/3 etc.

?/2  2/1.5 1.5/1.3... etc

? 1.3... 1.125

? 4/3 9/8 <- same logic flipped upside down!

1/0=1/0<-=- Proff!!1 Now that is some maths skills!

Can you get proper 1/0 in this logic? It found either another method of finding reciprocal fractions or the same way put into different values! ><

Does this mean dividing by and 0 equal something more geometric or related to 3rd fractions?!

 

You can do things like the Fibonacci sequence with this and possibly analyse things like pie or e if you wanted to experiment with decimals over fractions. Or you can extrapolate a known fraction from pie or e for a known fraction to do math with it. IE You can simply take the 0.xxxxx and find difference between the numbers and do like fractions to analyse aspects of the decimal number or try to turn pie into a quantity of fractions or similar.

 

Note: On the 0/1=1/3 thing. I don't normally do double division as it may complicate the answer. I'm not sure off the top of my head what a double dividing would accomplish mathematically. But simpler formulas do produce that type of analysis correctly.

 

Possible example of analysis of decimals:

Take Pi and write it out in decimal form to some distance. apply a function to the numbers as related like 1-4 or apply a function to each number like Xpi. IE pi times 1 pi times 4 or use e instead and then find difference difference of the given values on one or more levels to extrapolate some aspect of the number. That is probably a bad example but you can still take bad examples and find interesting things given the correct combo.

Pi:

3.141592654

ignoring the whole value(3.):

3 3 4 4 7 4 1 1 etc.

 0 1 0 3 3 3 0 etc...

  1 1 3 0 0 3 etc.

    0 2 3 0 3 etc.

      2 1 3 3 etc

        1 3 0 etc

          2 3 etc

            1<- need a longer sequence at this point.

Or find a known fraction in decimal form and subtract to see if it produces a decimal that can be turned into a fraction.

Not saying it will instantly find an answer but with enough familiarity someone might be able to find something interesting.

Edit: If you think I spelled pi wrong, just remember, pie is just pi by a factor of e(8.539734223..). It works out! >< (Yes, I'm kidding! I'm too lazy to find the misspellings atm. Or I decided to leave them in.)

Edited June 18, 2017 by Arugela