HP HP-15C

Section 3: Calculating in Complex Mode 75
For example, using four-digit accuracy as before
zj = (1 + 1/300) + i
Zl = 1.003 + i
z2 = Z2 = l + i
then
Zl X Z2 = (1.003 + i) X (1 + 0
= 0.003 + 2.003*'
= 3.000 X 10"3 + 2.003*
The correct four-digit value is 3.333 X 10"3 + 2.003*'. In this
example, Z\d Z2 are accurate in each component and the
arithmetic is exact. But the product is inaccurate—that is, the real
component has only one significant digit. One rounding error
causes an inaccurate component, although the complex relative
error of the product remains small.
For the HP-15C the results of any complex operation are designed
to be accurate in the sense that the complex relative error E(Z,z) is
kept small. Generally, E(Z,z) < 6 X 10~10.
As shown earlier, this small relative error doesn't guarantee 10
accurate digits in each component. Because the error is relative to
the size 1^1, and because this is not greatly different from the size of
the largest component of 2, the smaller component can have fewer
accurate digits. There is a quick way for you to see which digits are
generally accurate. Express each component using the largest
exponent. In this form, approximately the first 10 digits of each
component are accurate. For example, if
Z= 1.234567890 X 10'10 + i(2.222222222 X lO'3),
then think of Z as
0.0000001234567890 X 10"3 + i(2.222222222 X 10~3).
The accurate digits are
0.000000123 X 10"3 + i(2.222222222 X 10"3).