reply to post by yampa
What you did seems really interesting. What made you decide to actually plot it out in that way?
It took me quite a while of looking at your how-to chart to figure out what you were doing exactly to produce the numbers.
I would explain it like this:
Step 1: Calculate the Euler sequence to some arbitrary level, like level 13 as you did in your example, or level 5 as I'll do in this example. You
showed good examples of how to do that, and the 5th level is 1, 26, 66, 26, 1.
Step 2: Convert the decimal numbers for that level into binary. For example, level 5 of the Euler sequence looks like this: 1, 26, 66, 26, 1 in
decimal, and like 1, 11010,100010,11010,1 in binary.
Step 3: make all the numbers the same length as the longest number by adding in a bunch of zeros. The longest number is 6 digits long. therefore,
Step 4: Design a simple grid such as on graph paper to be the same width as the number of numbers in the sequence. There are five numbers in the
sequence shown in step three. Therefore the grid is five wide.
Step 5: Design the simple grid so that the height is the total number of digits in each number. Each number is six digits long. Therefore, the grid is
six high. You now have a grid box five x six in size.
Step 6: Go through each number in the sequence. The first number is 100000. The first number will correspond with the first column of the grid. The
number 1 tells us to fill in the box. The number 0 tells us to leave it empty. therefore you fill in the first box only and leave the ones above that
Step 7: Continue with each number in the sequence. So 110100 (the second number) means in the second column, the first, second, and forth grid boxes
will be filled in. The rest will be empty. And in the last column of the grid, which corresponds with the last number in the sequence 100000, only the
first box is filled in (on the bottom).
Hopefully this makes sense to people who don't quite understand how the OP did what he did.
Note on step 2 with converting decimal numbers to binary:
Decimal numbers use 10 digits, and decimal numbers powers of ten multiplied and added to form numbers. Decimal therefore uses the symbols
0,1,2,3,4,5,6,7,8, and 9. After 9, instead of coming up with another symbol, you add another column to the left like 10 to mean (10 x 1) for the first
digit + (0 x 1) for the second digit. 14 means (10 x 1) for the first digit + (1 x 4) for the second digit. 123 means (100 x 1) for the first digit,
(10 x 2) for the second digit, and (1 x 3) for the last digit. So there are ten symbols, and after those ten symbols are used up, they are simply
re-used starting over in the next column with TEN times the number, then another ten times that in the next column, and so on. So you have one, ten,
one hundred, one thousand, etc.
Binary numbers only use the symbols 0 and 1. There are two symbols in the binary system and therefore the power multiple is two for each column.
Instead of One, Ten, One Hundred, One thousand, etc going from left to right, you have One, Two, Four, Eight, etc as you go from left to right. So
instead of using a third symbol, you simply go to the next column for larger numbers as with the decimal system. So, you have 0 then 1. But then you
have 10 to represent (2 x 1) for the first digit + (1 x 0) for the second digit. 1001 in binary means (8x1) + (4x0) + (2x0) + (1x1) because again, the
multiples for each additional digit are two instead of ten.
The reason that numbers are difficult to read is that people have been writing numbers backwards for their whole life. By backwards I mean people
normally write things in ascending order, not descending order. But math is in fact written in descending order. It would make more sense to have 123
to mean 321 why? 1x1 + 10x2 + 100x3 is in ascending order vs. 100x1 + 10x2 + 1x1 is in descending order for the power of tens. When people first start
looking at binary numbers they'll want to read 1011 as (1x1) + (2x0) + (4x1) + (8x1) which would written in ordinary language be the number 13. But in
fact 1011 means (8x1) + (4x0) + (2x1) + (1x1) which written in ordinary language is 11.
The reason that decimal numbers are written in descending factor-of-ten order is because the most significant digit is first instead of last.
Apparently the "wizards" out there who developed the math systems decided that it was more important to have the most significant digit first than to
have the number system in a ascending order. They are probably right, but it still makes learning new number systems slightly more complicated in my
edit on 19-9-2011 by seachange because: (no reason given)
edit on 20-9-2011 by seachange because: (no reason
edit on 20-9-2011 by seachange because: (no reason given)