Exercise 1: Would Higher Precision Fix the Ill-Conditioned Sensor System? — Possible Solution
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WHY THIS WOULD NOT FIX THE PROBLEM
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Chapter 6 verified the near-singular system's sensitivity using
50-digit exact Decimal arithmetic - a computation with dramatically
more precision than even 128-bit floating point could provide - and
the amplification was still there: a 10^-10 relative input
perturbation still produced a roughly 10,000x larger relative output
change. This proves the sensitivity comes from the mathematical
problem itself (the two sensors giving nearly redundant information,
reflected in the matrix's own near-zero determinant), not from
insufficient precision in the arithmetic used to solve it.

WHAT HIGHER PRECISION WOULD ACTUALLY DO
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Using 128-bit floats instead of standard doubles would reduce the
amount of ROUNDING error introduced while performing the
calculation itself - this is a real, if modest, benefit, since
double precision does introduce its own additional rounding on top
of the problem's inherent sensitivity. But this chapter's Step 5
diagnosis specifically identified the calibration system as
ill-conditioned, meaning the DOMINANT source of the unreliable
output is amplification of whatever imprecision already exists in
the INPUT readings themselves (the two sensors' own raw
measurements), not rounding error introduced during the solve.
Higher precision arithmetic cannot make the input readings
themselves more precise - it can only avoid adding a small amount of
extra rounding error on top of an already-amplified result.

THE ACTUAL SIZE OF THE REMAINING PROBLEM
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Even with arithmetic precise enough to introduce essentially zero
rounding error of its own, the ~40,000x condition number verified in
Chapter 6/8 would still amplify any uncertainty in the sensors' own
raw readings by that same factor. If the physical sensors have even
a small amount of real-world measurement noise (which every physical
sensor does), that noise would still be amplified roughly 40,000-fold
in the computed calibration constants, regardless of how many bits of
floating-point precision are used to perform the arithmetic.

WHY THIS WORKS AS AN ANSWER
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The answer distinguishes precision of the ARITHMETIC (which higher-
precision floats would genuinely improve, if only marginally) from
precision of the INPUT DATA and the problem's own inherent
sensitivity (which no arithmetic precision can fix), grounding the
distinction directly in Chapter 6's own exact-arithmetic experiment
rather than treating "more precision" as a universal cure.