Geometric meaning of operations on complex numbers

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Problem 1.5.21 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Предел. Непрерывность. Дифференцируемость." [ККЧ+03]More problems in Subsection 1.5.2 "Problems"

Where is the point $z^{2}$ located, if the point $z = x+i y$ belongs to the line $y = 1$ ?

Problem 1.5.22 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Предел. Непрерывность. Дифференцируемость." [ККЧ+03]More problems in Subsection 1.5.2 "Problems"

Where is the point $z$ of the complex plane located, if the point $z^{2}$ belongs to the imaginary axis?

Problem 1.5.23 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Предел. Непрерывность. Дифференцируемость." [ККЧ+03]More problems in Subsection 1.5.2 "Problems"

Let $z \neq \pm 1$ . Prove that the point $(z-1) /(z+1)$ belongs to the imaginary axis if and only if the point $z$ belongs to the circle of radius $R = 1$ centered at the point $z = 0$ .

Problem 1.5.28 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Предел. Непрерывность. Дифференцируемость." [ККЧ+03]More problems in Subsection 1.5.2 "Problems"

Under a clockwise rotation through an angle $\pi / 2$ and a doubling in length, the vector $z_{1} = 2+5 i$ is taken to a vector $z_{2}$ . Find the complex number corresponding to the vector $z_{2}$ .

Problem 1.5.29 from Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. "Предел. Непрерывность. Дифференцируемость." [ККЧ+03]More problems in Subsection 1.5.2 "Problems"

The vector $z = -2+3 i$ is rotated by $180^{\circ }$ and lengthened by a factor of 1,5. Find the complex number corresponding to the resulting vector.

Problem III.5.13 from Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. "Избранные задачи по вещественному анализу" [МГЛ+04]More problems in Section III.5 "Functional Equations"

Let $\bar{B}=\left\{ z \in \mathbb {C}\left| \right| z \mid \leqslant 1\right\}$ . Find all continuous bijective mappings $f$ of the disk $\bar{B}$ onto itself satisfying the condition $f\left(z^{2}\right)=f^{2}(z)$ for all $z \in \bar{B}$ .

Problem 12.19 from Беклемишева Л.А., Беклемишев Д.В., Петрович А.Ю., Чубаров И.А. "Сборник задач по аналитической геометрии и линейной алгебре" [ББП+23]More problems in Subsection 12.1 "General properties of mappings (12.1-12.24)"

(1)(2)

A transformation $f$ of the plane is given, in a rectangular coordinate system, by the formulas: $x^{ * } = x^{2} - y^{2}, y^{ * } = 2xy$ .

(1)

Is the transformation $f:$

(1) a surjection,

(2) one-to-one?

(2)

Find the complete preimage of an arbitrary point $\left(x^{ * }, y^{ * }\right)$ of the plane.

Problem 12.20 from Беклемишева Л.А., Беклемишев Д.В., Петрович А.Ю., Чубаров И.А. "Сборник задач по аналитической геометрии и линейной алгебре" [ББП+23]More problems in Subsection 12.1 "General properties of mappings (12.1-12.24)"

(1)(2)(3)

A transformation $f$ of the plane is given, in a rectangular coordinate system, by the formulas: $x^{ * } = e^{x} \cos y, y^{ * } = e^{x} \sin y$ .

(1)

Is the transformation $f$ one-to-one?

(2)

Identify regions in the plane on which $f$ is one-to-one.

(3)

Let $\bar{f}$ be the restriction of the transformation $f$ to the strip $0 < y < \pi$ . Find the formulas defining the mapping inverse to $\bar{f}$ .

Problem 12.88 from Беклемишева Л.А., Беклемишев Д.В., Петрович А.Ю., Чубаров И.А. "Сборник задач по аналитической геометрии и линейной алгебре" [ББП+23]More problems in Subsection 12.5 "Operations with linear transformations. The structure of orthogonal and affine transformations. The coordinate system is assumed to be rectangular unless otherwise specified. (12.63-12.89)"

Each point $M(x, y)$ of the plane is identified with the complex number $z = x + iy$ . Prove that:

(1)

the transformation $z\mapsto \operatorname {Re} z = x$ is the orthogonal projection onto the $x$ -axis

(2)

the transformation $z\mapsto \bar{z} = x - iy$ is the reflection with respect to the $x$ -axis

(3)

the transformation $z\mapsto z + z_{0}$ , where $z_{0} = x_{0} + iy_{0}$ is a fixed complex number, is the translation by the vector $\mathbf{a}\left(x_{0}, y_{0}\right)$

(4)

the transformation $z\mapsto az$ , where $a$ is a real number not equal to zero, is the homothety with center at the origin and ratio $a$

(5)

the transformation $z\mapsto (\cos \varphi + i\sin \varphi ) z = e^{i\varphi } z(\varphi$ is a fixed real number) is the rotation through angle $\varphi$ about the origin.

Problem 12.89 from Беклемишева Л.А., Беклемишев Д.В., Петрович А.Ю., Чубаров И.А. "Сборник задач по аналитической геометрии и линейной алгебре" [ББП+23]More problems in Subsection 12.5 "Operations with linear transformations. The structure of orthogonal and affine transformations. The coordinate system is assumed to be rectangular unless otherwise specified. (12.63-12.89)"

(1)(2)

Determine the geometric meaning of the transformation $f$ of the complex plane (see problem 12. 88):

(1)

$f(z) = az$ , where $a = r(\cos \varphi + i\sin \varphi ), r > 0$

(2)

$f(z) = az + b$ , where $a, b$ are complex numbers, $a \neq 0$ .

Problem 24.1 from Кострикин А.И. "Сборник задач по алгебре" [Кос15]More problems in Section 24 "Complex numbers and geometry of a plane"

Depict on the plane the points corresponding to the numbers $5,-2,-3 i, \pm 1 \pm i \sqrt{3}$ .

Problem 24.3 from Кострикин А.И. "Сборник задач по алгебре" [Кос15]More problems in Section 24 "Complex numbers and geometry of a plane"

Indicate the geometric meaning of the expression $\left|z_{1}-z_{2}\right|$ , where $z_{1}$ and $z_{2}$ are given complex numbers.

Problem 24.4 from Кострикин А.И. "Сборник задач по алгебре" [Кос15]More problems in Section 24 "Complex numbers and geometry of a plane"

Indicate the geometric meaning of the number $\arg \frac{z_{1}-z_{2}}{z_{2}-z_{3}}$ , where $z_{1}, z_{2}, z_{3}$ are distinct complex numbers.

Problem 24.21 from Кострикин А.И. "Сборник задач по алгебре" [Кос15]More problems in Section 24 "Complex numbers and geometry of a plane"

Determine the geometric meaning of the fractional-linear transformation $w=1 / z$ .

Problem 24.22 from Кострикин А.И. "Сборник задач по алгебре" [Кос15]More problems in Section 24 "Complex numbers and geometry of a plane"

Determine the geometric meaning of the transformation of the complex plane given by the formula $w=z^{n} \quad (n \geqslant 2)$ .

References

[ККЧ+03] Кудрявцев Л.Д., Кутасов А.Д., Чехлов В.И., Шабунин М.И. Предел. Непрерывность. Дифференцируемость. 2-е изд., ФИЗМАТЛИТ, 2003.

[МГЛ+04] Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. Избранные задачи по вещественному анализу. 2-е изд., Невский Диалект; БХВ-Петербург, 2004.

[ББП+23] Беклемишева Л.А., Беклемишев Д.В., Петрович А.Ю., Чубаров И.А. Сборник задач по аналитической геометрии и линейной алгебре. Лань, 2023.

[Кос15] Кострикин А.И. Сборник задач по алгебре. МЦНМО, 2015.