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<Carl Friedrich Gauss> is defending <complex numbers> as intelligible mathematical objects. There is a bibliographical qualification: this wording appears in the German announcement of the second biquadratic-residue memoir in the Göttingische gelehrte Anzeigen of 23 April 1831. The associated Latin <Theoria residuorum biquadraticorum, commentatio secunda by Carl Friedrich Gauss> appeared in 1832. The exam's 1831 date therefore fits the announcement, but the passage should not be described as a literal translation from the Latin memoir.<Announcement of the second biquadratic-residue memoir by Carl Friedrich Gauss>[23 April 1831, concluding general remarks on terminology and planar representation]{ref}<Theoria residuorum biquadraticorum, commentatio secunda by Carl Friedrich Gauss>[ETH-Bibliothek original-edition record: 1832]{ref}
The proposed directional vocabulary is mathematically specific. Represent $z=a+bi$ by the point $(a,b)$ in the <complex plane>. The addition of <complex numbers> adds components. Multiplication of <complex numbers> is
$$
(a+bi)(c+di)=(ac-bd)+(ad+bc)i.
$$
Multiplication by $-1$ reverses direction through a half-turn, while multiplication by $i$ sends $(a,b)$ to $(-b,a)$, a quarter-turn. Consequently $i^2=-1$ says that two quarter-turns make a half-turn. More generally the <moduli> multiply and the <complex arguments> add. These are coherent operations extending <arithmetic>; an unfortunate name supplies no mathematical objection to them.
The memoir's setting is <number theory>, specifically the question of whether a <integer congruence> $x^4\equiv a\pmod p$ is soluble, and the reciprocal relations between such questions. Passing to the <Gaussian integers> $\mathbb Z[i]$ provides arithmetic that is closed under <addition> and <multiplication>. For example,
$$
\boxed{5=(2+i)(2-i),\qquad N(a+bi)=a^2+b^2,\qquad N(zw)=N(z)N(w).}
$$
The <Gaussian integer norm> follows directly by multiplying by the <complex conjugate>. Here it gives $N(2+i)=5$ and $N(5)=25$, consistent with the factorization. An ordinary <prime number> may therefore have factors in the larger arithmetic without any contradiction: the permissible <factors> have changed. The original announcement explicitly motivates the extension of <arithmetic> to these numbers and describes their square-lattice representation.<Announcement of the second biquadratic-residue memoir by Carl Friedrich Gauss>[Opening explanation of complex integral numbers and the later lattice discussion]{ref}
It is helpful to separate two domains. Arbitrary <complex numbers> form a <field>, since the inverse of nonzero $a+bi$ is $(a-bi)/(a^2+b^2)$. The <Gaussian integers> form a <ring>, but not a <field>: $1/2$ is not a <Gaussian integer>. Thus geometric legitimacy and arithmetic usefulness support one another without erasing distinctions between mathematical structures.
\b[The terminology argument accompanies a constructive account of the numbers.] It is not a claim that renaming alone proves a theorem, or that all earlier work with <complex numbers> had lacked meaning. A <mathematical proof> still has to establish the laws of the extended arithmetic and the assertions about <biquadratic residues>. <Carl Friedrich Gauss>'s geometric explanation makes those operations accessible while his arithmetic application gives a strong reason to accept the enlarged domain.
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