People's Education Press: High School Mathematics, Compulsory Course Second Semester (Version A): From Number Line to Complex Plane: Algebraic Definition and Geometric Mapping of Complex Numbers

Imagine you can only move left and right along a thin string—this is the world of the real number line. If you want to jump upward, the string cannot support you. Introducingcomplex numbersis like adding a whole new dimension to your world. Each complex number in the form $z = a + bi$ is no longer just a point on the number line—it becomes a coordinate $(a, b)$ on the plane, or a vector emanating from the origin. This perfect correspondence between 'number' and 'shape' is one of the greatest leaps in the history of mathematics.

Algebraic Definition and Geometric Mapping of Complex Numbers

In the second semester of the compulsory course, we studied the complex number system. A complex number consists ofreal partandimaginary partcomposed, with its standard algebraic form being $z = a + bi$ ($a, b \in \mathbb{R}$).

To understand complex numbers intuitively, we established thecomplex plane:

real axiscorresponds to the $x$-axis, representing the real part of a complex number.
imaginary axiscorresponds to the $y$-axis, representing the imaginary part of a complex number.
Point and Complex NumberThe complex number $z = a + bi$ corresponds uniquely to the point $Z(a, b)$.
Vector and Complex NumberThe complex number $z = a + bi$ corresponds uniquely to the two-dimensional vector $\vec{OZ}$.

The modulus of a complex number $|z| = \sqrt{a^2 + b^2}$ geometrically represents the distance from point $Z$ to the origin in the complex plane. Meanwhile, $|z_1 - z_2|$ represents the distance between two points.

$$z = a + bi \iff Z(a, b) \iff \vec{OZ}$$

QUESTION 1

In the complex plane, $O$ is the origin. The vector $\vec{OA}$ corresponds to the complex number $2+i$. If point $A$ has a symmetric point $B$ about the real axis, find the complex number corresponding to vector $\vec{OB}$.

$2-i$

$-2+i$

$-2-i$

$1+2i$

QUESTION 2

Following the previous question, if point $B$ has a symmetric point $C$ about the imaginary axis, find the complex number corresponding to point $C$.

$2+i$

$-2-i$

$-2+i$

$2-i$

QUESTION 3

If the real part of a complex number $z$ is positive and its imaginary part is $3$, where does the point corresponding to $z$ lie in the complex plane?

A ray within the first quadrant

A segment on the imaginary axis

The entire first quadrant

A line above the real axis

QUESTION 4

Compute the result of the complex number operation: $(-3-4i) + (2+i) - (1-5i)$.

$-2+2i$

$-2-8i$

$0$

$-4+2i$

QUESTION 5

Given that the imaginary part of complex number $z$ is $\sqrt{3}$ and the modulus of the vector corresponding to $z$ in the complex plane is $2$, find this complex number $z$.

$1 + \sqrt{3}i$ or $-1 + \sqrt{3}i$

$1 + \sqrt{3}i$

$\pm 1 - \sqrt{3}i$

$\sqrt{7} + \sqrt{3}i$

QUESTION 6

The necessary and sufficient condition for the product of complex numbers $(a+bi)$ and $(c+di)$ to be real is:

$ad + bc = 0$

$ac + bd = 0$

$ac = bd$

$ad = bc$

QUESTION 7

Solve the equation $4x^2 + 9 = 0$ in the set of complex numbers $\mathbb{C}$.

$x = \pm \frac{3}{2}i$

$x = \pm \frac{9}{4}i$

$x = \frac{3}{2}i$

No solution

QUESTION 8

If the modulus of complex number $z$ is $5$ and its imaginary part is $-4$, then $z$ is:

$3-4i$ or $-3-4i$

$3-4i$

$\pm 3 + 4i$

$\pm 9 - 4i$

QUESTION 9

Find the distance between the two points corresponding to complex numbers $z_1 = 8+5i$ and $z_2 = 4+2i$ in the complex plane.

$5$

$\sqrt{13}$

$25$

$7$