Past exam of the history of mathematics course of the University of Oxford / 2023 / 5

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.^[ref][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 isMultiplication 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 $x^4\equiv a(\mathrm{mod}p)$ 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,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.^[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.

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.