Complex Numbers

Complex numbers as couples

By the early nineteenth century complex numbers had been in use for three hundred years, since Cardano and Bombelli, and Euler had worked wonders with them. Wessel, Argand and Gauss had drawn them as points of the plane. But the unease remained: what, after all, is $\sqrt{-1}$? A number whose square is negative seemed a useful fiction, not an object of mathematics. Hamilton settled the question without drawing anything. In the essay Theory of Conjugate Functions, or Algebraic Couples, presented to the Royal Irish Academy in 1833 and 1835 and published in 1837, he proposed that a complex number is an ordered couple of real numbers, and nothing more.

October 27, 2026 Read

Riemann surfaces

Riemann’s doctoral thesis, of 1851, has a modest title: Foundations for a General Theory of Functions of a Complex Variable. Inside it are two ideas that changed mathematics: the conditions a complex function must satisfy in order to have a derivative, and a geometric way of dealing with functions that have several values. Functions that can be differentiated A complex function $f(z)$, with $z = x + iy$, can be written as $f = u + iv$, where $u$ and $v$ are real functions of $x$ and $y$. For $f$ to have a derivative in the complex sense, that is, for the limit $\frac{f(z + h) - f(z)}{h}$ to be the same whatever the direction in which $h$ goes to zero, the parts $u$ and $v$ must satisfy the Cauchy–Riemann equations:

October 31, 2026 Read

Quaternions: the formula on the bridge

After showing that complex numbers are couples of real numbers and that their multiplication rotates the plane, Hamilton wanted to do the same in space. He was looking for triplets $a + bi + cj$ with a multiplication that respected the law of the moduli: the size of the product should be the product of the sizes. He tried for years, above all in 1843, so much so that his children asked him every morning whether he could multiply triplets yet.

October 27, 2026 Read

ITA 2026 – 1st phase: Mathematics solved

Source: ITA – 1st phase exam and official answer key. Files on the ITA past exams page. The 1st phase of ITA 2026 had 48 multiple-choice questions: 12 in Mathematics, 12 in Physics, 12 in Chemistry and 12 in English. This post covers the mathematics questions, numbered 1 to 12 as in the exam. Exam conventions. $i^2 = -1$; $\bar{z}$ is the conjugate of $z$; $]a, b[$ is the open interval $a < x < b$.

October 28, 2026 Read

ITA 2025 – 1st phase: Mathematics solved

Source: ITA – 1st phase exam and official answer key. Files on the ITA past exams page. The 1st phase of ITA 2025 had 48 multiple-choice questions: 12 in Mathematics, 12 in Physics, 12 in Chemistry and 12 in English. This post covers the mathematics questions, numbered 1 to 12 as in the exam. Exam conventions. The horizontal and vertical axes are $O_x$ and $O_y$, and $O$ is the origin; $\mathbb{N} = \{1, 2, 3, \dots\}$; $i^2 = -1$; $\overline{AB}$ is the segment and $AB$ the line through $A$ and $B$.

October 28, 2026 Read

ITA 2024 – 1st phase: Mathematics solved

Source: ITA – 1st phase exam and official answer key. Files on the ITA past exams page. The 1st phase of ITA 2024 had 60 multiple-choice questions, 12 per subject (Physics, Portuguese, English, Mathematics and Chemistry). This post covers the mathematics questions, numbered 37 to 48 as in the exam. Exam conventions. $\mathbb{N} = \{1, 2, 3, \dots\}$; $A - B$ is the difference of the sets $A$ and $B$; $i^2 = -1$; $M_n(\mathbb{R})$ is the set of $n \times n$ real matrices; $\overline{AB}$ is the segment and $AB$ the line through $A$ and $B$; $m(\overline{AB})$ is the length of $\overline{AB}$; $A\hat{B}C$ is the angle with vertex $B$; $\det A$ is the determinant and $A^T$ the transpose of $A$.

October 28, 2026 Read

ITA 2023 – 1st phase: Mathematics solved

Source: ITA – 1st phase exam and official answer key. Files on the ITA past exams page. The 1st phase of ITA 2023 had 60 multiple-choice questions, 12 per subject (Physics, Portuguese, English, Mathematics and Chemistry). This post covers the mathematics questions, numbered 37 to 48 as in the exam. Exam conventions. $\mathbb{N} = \{1, 2, 3, \dots\}$; $i^2 = -1$; $M_n(\mathbb{R})$ is the set of $n \times n$ real matrices; $\overline{AB}$ is the segment and $AB$ the line through $A$ and $B$; $m(\overline{AB})$ is the length of $\overline{AB}$; $A\hat{O}B$ is the angle with vertex $O$; $\det A$ is the determinant of $A$.

October 28, 2026 Read

ITA 2022 – 2nd phase: Mathematics solved

Source: ITA – 2nd phase Mathematics exam. Files on the ITA past exams page. The 2nd phase Mathematics exam of ITA 2022 has 10 written questions. ITA does not publish official solutions for the 2nd phase; the solutions below are ours, and each answer was checked by a second route (substitution, another formula or a numerical check). The notation conventions are the same as in the 1st phase: $i^2 = -1$, $m(\overline{AB})$ is the length of segment $\overline{AB}$ and $A\hat{O}B$ is the angle with vertex $O$.

October 28, 2026 Read

ITA 2022 – 1st phase: Mathematics solved

Source: ITA – 1st phase exam and official answer key. Files on the ITA past exams page. The 1st phase of ITA 2022 had 70 multiple-choice questions: 15 in Physics, 15 in Portuguese, 10 in English, 15 in Mathematics and 15 in Chemistry. This post covers the mathematics questions, numbered 41 to 55 as in the exam. Exam conventions. $\mathbb{N} = \{1, 2, 3, \dots\}$; $i$ is the imaginary unit ($i^2 = -1$); $M_n(\mathbb{R})$ is the set of $n \times n$ real matrices; $\overline{AB}$ is the segment with endpoints $A$ and $B$, and $m(\overline{AB})$ is its length; $A\hat{O}B$ is the angle with vertex $O$ formed by the rays $\overrightarrow{OA}$ and $\overrightarrow{OB}$.

October 28, 2026 Read

Argand's plane: the imaginary number that found a place

Since the sixteenth century, the Italian algebraists had known that, to solve certain cubic equations, one had to pass through square roots of negative numbers, even when the final answer was a real number. They used $\sqrt{-1}$ uncomfortably. Descartes called such numbers imaginary, and the name stuck. Leibniz described them as “an amphibian between being and non-being”. The problem was semiotic. $\sqrt{-1}$ was a piece of writing that obeyed the rules of algebra but pointed to nothing that could be seen or measured. A real number has a place on the line; $\sqrt{-1}$ had no place at all.

October 19, 2026 Read

FUVEST 2023 – 2nd phase: Mathematics solved

Source: FUVEST – 2nd phase test, 2nd day (questions M01 to M06) and document expected approaches. The 2nd phase of FUVEST 2023 lasted two days. The 1st day was Portuguese and Writing; in 2nd day, the test had 6 questions from each subject, and Mathematics goes from M01 to M06. Each candidate answered only the subjects of their career (in some careers, only the first questions of each subject). FUVEST does not publish a template for the 2nd phase; the resolutions are ours. She disclosed an “expected approaches” document with the path expected of the candidates. Each resolution below was checked by a second path and compared to this document.

September 27, 2026 Read