In a groundbreaking

article,

T. L. Freeman discusses the
relationship between

*actual age* and

*effective
age*^{1}. His conclusion is that the passing of the years
goes faster as we grow older. This makes sense; for instance when you
are 10 years of

age, a year represents 10% of your

life, and seems like
a very long time. However, when you are 50 years old, one year has
reduced to only 2% of your life, and hence seems only one-fifth as long.

Summarizing this work, Freeman comes to the conclusion that the *actual
age* (AA) needs to be corrected for the apparent length of a year (AY). The
apparent length of a year is inversely proportional to one person's *actual
age*:

AY= `α`/AA

The constant of proportionality `α` is rather loosely
defined by Freeman as *the age at which a year really seems to last a
year*, and it was arbitrarily set at 20 years (`α`=20).

Now Freeman determines the concept of *Effective
age*, which is simply the integral over time of the *Apparent Year*
from age 1 to the *actual age* (AA) of interest:

AA AA
EA = ∫ AY d(AA) = ∫ 20/AA d(AA) = 20 ln(AA)
1 1

Although this formula results in some interesting conclusions, there are
several

flaws with this concept. As mentioned above, the choice of the
proportionality constant is rather

arbitrary. There is no rational
justification for the choice of this age, but it was solely chosen based
on Freeman's own perception of (the passing of) time. Next, the
evaluation of the integral seems incorrect, since its lower limit was
set at 1, and not at 0. Obviously, the choice of zero as lower
integration boundary yields can not be evaluated due to the

logarithmic term in the expression. Because of the obvious problems
with Freeman's concept of time

perception, it is necessary to redefine
the

*Effective Age* on a sounder basis.

In the traditional concept of time perception, one person's
*Actual Age* is proportional to the passing of time (t).

AA = `β`t + `γ`

Note the occurrence of two parameters `β` and
`γ` that are traditionally set to one and zero,
respectively. However, each has a clear (though usually
underappreciated) function in time perception. The
`β`-parameter describes the rate at which one person ages;
some persons remain annoying little crybabies during their life, while
others become boring old farts at 20. The `γ`-parameter
describes the origin of one person's time perception. Did you ever
meet those proud parents boasting about their little one who is only
`x` months old, and already walks, writes obfuscated C, or
recently sold his first dot.com? No, these youngsters aren't bright
for their age; they simply have a high `γ`-factor.

It is clear that with this definition, one person's *Actual Age* may
already be non-synchronous with time. However, analogous to Freeman's
work, the apparent length of a year (AY) is not constant:

AY= `α`/AA = `α`/(`β`t + `γ`)

We can remove one of the parameters by defining two parameters

`δ` and

`ε`.

AY= `α`/(`β`t + `γ`) = (`α`/`β`)/(t + `γ`/`β`) = `δ`/(t + `ε`)

The actual values of

`δ` and

`ε` will become
clear from the

boundary conditions.

In order to obtain the *Effective Age*, the integral of AY is
evaluated. Note that the integral is evaluated over time, and not over
*Actual Age*, since AA is a function of time:

t t
EA = ∫ AY d(t) = ∫ `δ`/(t + `ε`) d(t)
0 0
EA = `δ` ln(t + `ε`) - `δ` ln(`ε`)

The lower boundary condition (t=0) should yield an

*Effective Age* of zero years (EA=0). Therefore

`ε` = 1.

The upper boundary is less apparent. It should be chosen so that at
t=t_{max}, EA = t. At death, the *Effective Age* and real
time are again equal. However, no person knows for sure his or her
personal life expectancy. This is clearly an issue for molecular
biologists to address. However, if we assume for a person a life
expectancy of 80 years (t=80, EA=80), we obtain:

`δ` = 80/ln(81)
80 ln(t + 1)
EA = ----------
ln(81)

This formula can now be used to calculate the

*Effective Age*
(and the Effective percentage Completion of Life) as a function of time.
This is shown in the following table:

time (yrs.) EA (yrs.) Life%
0 0.0 0
1 12.6 16
2 20.0 25
3 25.2 32
4 29.3 37
5 32.6 41
10 43.7 55
15 50.5 63
20 55.4 69
30 62.5 78
40 67.6 85
50 71.6 89
60 74.8 94
70 77.6 97
80 80.0 100

And thus, the bold statement in the title is justified. Life is half
over at age ten, and three quarters over at age thirty. Note the
rapid increase at very young ages: in the initial stages of life, life
itself makes big strides forward. For instance, consider the concepts of
speech, eating and walking; skills that are learned at a young age
and are carried on throughout a person's life.

Another interesting observation that we can make is the *age at
which one year really seems to last one year*. This can be
calculated quite easily from the derivation above. For a life
expectancy of 80 years, it is equal to 80/ ln(81) - 1 = 17.2 years.
Quite close to Freeman's original assumption of 20 years.

## Consequences:

The concept of

*Effective Age* has far stretching implications.
Some of these I have summarized below:

**"Summer vacations lasted almost forever when I was in grammar school"**:

*True*, they did. In fact, when you were six years old, an
*Apparent Year* would be close to three years. That would make a
three week summer vacation feel like almost nine weeks!
**"Now that I am older, I can communicate better with my
parents"**

*Right*. As you can see, you're catching up with them! Closing
the "generation gap", so to speak.
**"Life starts after 65**"

The credo of many people close to their pension age.
*Wrong*: at 65, you only have about 5% of your
*Effective Age* left. Choose your time wisely; start working
late, and retire early.
**"Old people are slow"**

That is such an insensitive comment. Old people aren't slow at all,
they simply have a different time perception.
**"Those annoying birthdays seem to roll around faster every
year**

*True*, they do. Better start celebrating your *Effective
Age*.

T. L. Freeman,

Why it's later than you think,

J. Irr. Res., 1983.