Note that there is a largest a*n! for a from 0 to n smaller than X:

here

X=443.125

and a =3 and n = 5

3*5*4*3*2 = 360

because 4*5*4*3*2 = 480 which is larger than X

any a times n=6 6*5*4*3*2 is too large and a=4 and n = 4 is 4*4*3*2=126 is too small

So we consider the a=3 in front of 5*4*3*2 to be the first digit

3

Now X-360 = 83.125 so we iterate:

that is a=3 times n=4 4*3*2 = 72 and again a =4 would be too large and 3*3*2 is too small

so the second digit is 3

3

83.125 - 72 = 11.125

just one 3*2

1

5.125

two 2's

2

1.125

one 1

1

.125

Now the factorials invert the above is 0 1/2's

0

.125

And also 0 1/2*3

0

.125

And 3 1/4*3*2

3

0

So the number can be written as 3,3,1,2,1x0,0,3 where x is the radix point (decimal point but more general)

which to go back to decimal is:

Every number can be transformed into this "factorial base" number where the possible size of the digits increases by 1 further to the left and the right of the radix point, uniquely, except for the usual caveat that 0,0,3 is the same as 0,0,3,3,4,5,6...

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