A Sum of “Complex” Numbers

We really want to get on to other things, but this needs to be done. Below is pretty much a complete cataloging of Nelson‘s use of the adjective “complex” in the five recent WitCHes (here, here, here, here and here). To be clear, there is tons more wrong, and bad, in the selected excerpts in the WitCHes: the proper WitCH updating, currently scheduled for late 2029, will be long and painful. But the use of “complex” warrants particular attention. It reflects VCAA’s complex madness, and we doubt that it is a coincidence.

WitCH 115

Complex numbers are introduced as a + bi, with \boldsymbol{i=\sqrt{-1}}. In the Year 11 text excerpts, it is explicitly stated that the set R of real numbers is a subset of the set C of complex numbers. In the Year 12 text excerpts, that R is a subset of C is implicit rather than explicit, but it is clear enough and is never contradicted.

WitCH 116

All uses of “complex” are consistent with R being a subset of C.

WitCH 117

All uses of “complex” are consistent with R being a subset of C. In particular, it is implicit that “complex (nth) roots” may include real roots.

WitCH 118

In the first box:

“complex” is considered to be non-overlapping with “purely imaginary”.

“complex” root is twice used to mean a non-real root.

“complex factor” (sic) is used to mean a non-real factor (i.e root).

“complex conjugates” is used to mean non-real conjugates.

In Worked Example 12, it is implicit that “complex factors” (sic) refers to non-real roots (although the implication is unimportant for the working that follows).

The preamble to the factor theorem and remainder theorem effectively defines “complex polynomials” as polynomials where “the coefficients and variable are complex numbers”. The only self-plausible meaning of “complex numbers” in this definition is that “complex numbers” (correctly) includes the real numbers, and thus a real (coefficient) polynomial is also (correctly) a “complex polynomial”.

The preamble to the fundamental theorem of algebra implies that equation has z4 -1 = 0 has only two “complex roots” (sic), thus using “complex” to mean non-real.

The statement of the fundamental theorem seemingly uses “complex root” to also include also the possibility of a real root (and otherwise the statement of the theorem is false). The proof of the existence of n roots, however, notes that “[s]ome of the factors may be complex”, implying “complex factors” (i.e. complex roots) are non-real.

Worked examples 15 and 16 imply that at least one (but not necessarily all) of the coefficients of a “complex polynomial/quadratic/cubic” must be non-real.

WitCH 119

The introduction to quadratic equations uses “complex roots” to mean non-real roots, and “complex conjugates” to mean non-real conjugates.

Worked example 18 uses “complex root” to mean non-real root.

Worked example 19 uses “complex quadratic equation” to mean the quadratic must have at least one (but not necessarily all) of the coefficients be non-real.

The preamble to the conjugate root theorem uses “complex solutions” to mean non-real solutions.

The statement of the conjugate root theorem seemingly uses “complex root” to mean non-real root, but the statement is correct as worded.

The preamble to polynomial equations twice uses “complex roots” to mean non-real roots.

The preamble to Worked example 23 uses “complex root” to mean non-real root.

Worked examples 23 and 24 use equations/polynomials with “complex coefficients” and “complex cubic/quartic” to mean at least one (but not necessarily all) of the coefficients must be non-real.

So, how did anybody get confused?

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