A whole shifted inverse is an integer-scaled reciprocal used to replace division by multiplication
and digit shifts while keeping the arithmetic in the integers (Watt 2023). Let 
be an integer base. The following
equations define the base-
whole shift for integers 
and 
, and the whole shifted inverse for 
and a nonzero integer 
.
 |  |  |
(1)
|
 |  |  |
(2)
|
When 
,
the whole shift is multiplication by 
. When 
, it is integer division
by a power of 
.
The whole shifted inverse is a reciprocal shifted into the integer domain. For example,
.
Given positive integers 
and 
with 
, there is a 
such that
 |
(3)
|
Thus multiplication by the whole shifted inverse gives either the quotient or one less than the quotient.
The whole shifted inverse can itself be computed by an integer version of Newton's method. One refinement step has the form
 |
(4)
|
Under suitable initial conditions, the iterates remain integers and converge to 
. For 
, 
, and 
, the target is 
, and the initial value 
gives
 |
(5)
|
An analogous construction for univariate polynomials replaces 
by 
and can be used to compute a polynomial quotient
(Watt 2023).
Marchioro et al. (2026) implemented whole-shifted-inverse division on a GPU for integer sizes from 
through 
bits. Their cost model for classical multiplication counts
five to seven full multiplications for a complete quotient-and-remainder computation.
They identify clipped products as a possible way
to reduce this cost.
