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An Anomaly of the Logic of Age Ratios This year on Feb. 8, my daughter turned 36.
The previous May 14, I had turned 72. So on her birthday, in whole
number years, I was twice her age, and, of course, conversely she was half my
age. On her birthday part of what I wrote her was: "As I pointed
out previously, for some three months, starting on your birthday until mine
in May, I will be twice your age for the only times in our lives, and that is
cool." I was roughly 36.75 years old when you were born. So to be twice your age, I need to be
roughly 73.5 years old when you are roughly 36.75. You will be 36.75 somewhere around November
of 2018, but since we are talking only whole number years for birthdays, you
will be 36 and I will be 73 then. So
the double is masked by those whole numbers.
The prior whole number match is 36/72 and the subsequent one is
37/74. Doing this in whole numbers
means that my age always has to be an even number for it to be a double of
yours. In the following table, we are whole number doubles during
blue periods and not whole number doubles during red periods.
This shows we are double whole-number-wise from the time
we are roughly 36/72.75 until we are roughly 36.25/73, and from the time we
are 37.25/74 until 38/. So the gap is
from our being 36.25/73 until we are 37.25/74, a gap of one year between by
73^{rd} and 74^{th} birthdays, which is when the fractions of
your birthday are from .25 in 2019 to
.25 again in 2019. And it is during
that time period that I am 73, which doesn’t allow any whole number you are
to be half that. I can see why we cannot be doubles during the gap, but I
am not seeing why we should be able to be doubles on both sides of the gap,
other than clearly we are, but I am not seeing the reason that is. Now if we were both born on January 1, me
in 1945, and you in 1982, I would have been 37 when you were born, and then
in 2019, the whole year we would be double and half, 74/37. And it seems to me that, plus intuition in
general, means there should always be a whole year when parents and children
are double/half in age, but I am not able to see how that means the year
gap. I can see why there is a gap caused
by different birthdays during the year, but I am not seeing how there is a
double/half on both sides of that gap, to fill out the part of the year that
goes with the other gap on the front side of it. So then I figured out the following: Double Age of
Parent/Child Explained Okay, I think I have it now. This is the general rationale; consider the
chart from yesterday (above) with your birthday and mine to be an example: Explanation A:
Rounded off explanation first, just talking about age in years: 1) Whenever the child is born, the parent will already be
a particular age, call that age N. 2) In N more years then, on the date of the child’s N^{th}
birthday, the child will become N years old (same age as parent was when
child was born). The parent will already
have turned 2N years old and so will be twice the child’s age. 3) The parent will stay twice as old as the child until the parent’s next birthday,
which will make the parent then be (2N + 1) years old, while the child is
still N years old, so they are no longer twice/half each other’s age. 4) The next birthday will be the child’s, at which time
the child will become N + 1 years old. 5) At that time, the parent will still be 2N + 1 years
old, and that is not 2(N + 1), so they will still not be double/half each
other’s age. 6) The next birthday will be the parent’s birthday (exactly one year after the birthday in step
3), and the parent will then be 2N + 2, which is 2(N + 1), so once again they will be double/half in age until
the next birthday, which will be the child’s N + 2 birthday, at which point
they will never be twice/half again. 7) Since the date in 6 and the date in 3 are the parent’s
successive birthdays exactly one year apart, they will have been not
double/half for exactly one year, though they were double/half for the part
of the year between the parent’s previous birthday and the child’s N^{th}
birthday. This explains the one year gap between when they were
first double/half until they are double/half again. Explanation B: More
precise explanation, using not just years, but days, considering just
non-leap years with the standard 365 days.
Same explanation will work for leap years, using 366 days instead of
365: 1) 1) Whenever the child is born, the parent will already
be a particular age, call that age (N years + X days) (X being the number of
day since the parent’s most recent birthday). 2) In N more years then, on the date of the child’s N^{th}
birthday, the child will become N years old (same age in years as the parent
was when child was born). The parent
will already have turned 2N years old and so will be twice the child’s age,
plus the same number of days, X, since the parent’s birthday. The parent will be (2N years + X days) old
on the child’s N^{th} birthday. 3) The parent will stay twice as old, in whole number
years, as the child until the parent’s
next birthday, which will make the parent then be (2N + 1) years old
exactly, while the child is still N years old (though more precisely, the
child will now be [N years + (365 – X) days] old (since the child’s previous
birthday was X days after the parent’s, leaving (365 – X) days till the
parent’s subsequent birthday, so they are no longer twice/half each other’s
age in whole number years. Thus, the length
of time here that they are double/half each other’s age is (365 - X)
days. (In other words: there are
always X days from the parent’s birthday to the child’s ensuing birthday
because the child was born X days after the parent’s birthday, and in normal,
non-leap years then, that will mean there are (365 – X) days each non-leap
year from the child’s birthday to the parent’s ensuing birthday. 4) The next birthday will be the child’s (X days after the
parent’s), at which time the child will become N + 1 years old. 5) At that time, in whole number years, the parent will
still be (2N + 1) years old, and that is not 2(N + 1), so they will still not
be double/half each other’s age. 6) The next birthday will be the parent’s birthday (exactly one year after the birthday in step
3), and the parent will then be 2N + 2, which is 2(N + 1), so once again they will be double/half in age in
whole number years, until the next birthday, which will be the child’s N + 2
birthday (X days later), at which point they will never be twice/half
again. 7) Since the date in 6 and the date in 3 are the parent’s
successive birthdays exactly one year apart, they will have been not
double/half for exactly one year, though they were double/half for the part
of the year between the parent’s previous birthday and the child’s N^{th}
birthday. This explains the exact one
year gap between when they were first double/half until they are double/half
again. 8) Since they are double/half each other’s age in whole
number years the first time, for (365 – X) days, and they are double/half
each other’s age the second time until child’s (N + 2) birthday, which is X
days after the parent turned double the child’s age for the second time, the
total time they are double/half each other’s age is [(365 – X) + X] days,
which is 365 days or one year exactly,
Hence, the total time they are double/half each other’s age will be
one year, with a one year gap in between, from the parents’ birthday in step
3 until the parent’s birthday in step 6.
9) When the child is born on the parent’s birthday, X = 0,
and (365 – X) = 365, so there will then be no whole number year gap and the amount of time they
are double/half will still be one year, just a continuous one. 10) Also, when they have the same birthday, the N^{th }birthday
of the child will be at the exact same time as the 2N^{th} birthday
of the parent and they will be exactly double/half when they are double/half
in whole number year, but if X > (365/2) – meaning if the time from the
parent’s birthday to the child’s birthday is more than half a year, the exact
double/half age will fall in the time of the gap year between their whole
number double/half ages. That is
because the initial double/half period is between the child’s birthday and
the parent’s, so that time period has to be more than half a year (and thus
the period between the parent’s birthday and the child’s next birthday, which
is X, has to be less than half a year, or the exact double/half date will
fall in the gap when there is no double/half relationship in whole years,
because it cannot wait until the following whole year later when that date
would fall in the second interval of the double/half ratio.
A)
The reason there are two and only two whole number year ages where the
parent's age is twice the child's (when the parent and child do not have
the same birthday) is that:
B) The age pairings and gap
will be different for different children born in other years, and/or
for the other parent with any given child if the parents are not the
same whole number ages as each other. C)
If we calculate all this in ages in whole number months, instead of
whole number years, the above principles will be the same, but the gap
will be smaller because the whole numbers will change faster.
E.g., I was 440 whole number months old when my older child was born, so
I was 880 months old when she turned 440 months old. But her
birthday is on the 8th of her birth month, February, and mine is on the
14th of mine, May. She turned 440 months old Oct.8, and I turned 880 months old Sept. 14.
So in the same month she turned 440 months old to my 880, I
added another whole number month to my age, making me 881 to her 440,
ending the double temporarily. In the following month, November,
on the 8th, she
would add a whole number month to make the ratio 881/441,
still not double. But then 6 days later, on November 14, I
added another month to
my whole number month age, making the ratio 882/441 and thus double
again until her whole month birthday, December 8. Notice, that
still does not make our birth months line up double/half with our actual
day of the year we are double/half in age, November 5, but it gets it
much closer. So
basically, the unit of measure, if whole numbers are used, not rounded
off, determines the length of the gap and where it occurs. The
same would be true if we compared whole number of days we are
double/half with the whole number hours in the day, or with the whole
number minutes, or with the whole number seconds, or whole number
microseconds, etc. of when our ages are double/half. The smaller
the number used, the smaller the gap and the closer to the actual
half/double exact time. D) When the child is born within the same calendar year as the parent's 'relevant' birthday -- the one that determines the parent's whole age in whole number years at the birth of the child, the parent's birthday will be first in the calendar year when they first become double/half ages, because that happens on the child's birthday. But, if like my daughter and my case, the child is born in the subsequent calendar
year from the parent's relevant birthday, the child's birthday will
always be earlier in the ensuing calendar years. Her birthday is
always in February each year; mine, in May. But since a whole year is a cyclical event, my relevant age-determining May birthdays come before her relevant age-determining ones. If calendar years were not numbered with a starting point each January 1 and an ending point each December 31, but were just twelve months repeated cycles, my relevant birthdays would always be before hers, because I had already had a the relevant birthday when she was born. And the time periods between our birthdays, no matter how large or small, would always be repeated, with, for example, my ('non-relevant') birthday following hers by 95 days (in non-leap years; 96 in leap years), and hers always following mine by 270 days, leap years or not. E)
Similar arguments and conclusions will apply to other temporal ratios
also -- one event's being 3 times older than, 8 times older than, 15 or
100 or 42.35 times older than another. F) Application of this to physics: The red and blue colors above correspond artificially, because assigned somewhat arbitrarily (except for aesthetics and font readability within them) to the parts of the cycle that are relevant. But in physics, in the world of nature, ratios involving cycles or cyclical events coincide with or determine
actual physical properties. And if those ratios change depending
on the size of the measure being used or the size of smallest observable unit being perceived by the senses or any instrument, then that can change our perception and/or understanding of the phenomenon being observed
or studied. For example, physicists may already take all this
into account in their study, understanding, and experimenting with wave
cycles and frequencies, but if not, then doing so might make a
difference. The relative starting times and ages of any waves and
their temporal components may be significant. This is the best I can do with this. A real mathematician would see an even
larger principle and/or have seen this all more readily and one who is also a physicist would probably
know to apply it to something actually important and more useful. I cannot do that, which is why I am not a
real mathematician nor a physicist, but just a person able to calculate some things by brute
force and persistence to see at least some logic in the calculations. |
This work is available here free, so that those who cannot afford it can still have access to it, and so that no one has to pay before they read something that might not be what they really are seeking. But if you find it meaningful and helpful and would like to contribute whatever easily affordable amount you feel it is worth, please do do. I will appreciate it. The button to the right will take you to PayPal where you can make any size donation (of 25 cents or more) you wish, using either your PayPal account or a credit card without a PayPal account. |