Imagine being able to move only left and right along a thin string—this is the world of the real number line. If you try to jump upward, the string cannot support you. Introducingcomplex numbersis like adding a brand-new dimension to your world. Each complex number of the form $z = a + bi$ is no longer just a point on the number line, but rather a coordinate $(a, b)$ in the plane, or a vector emanating from the origin. This perfect correspondence between 'number' and 'shape' represents one of the greatest leaps in mathematical history.
Algebraic Definition and Geometric Mapping of Complex Numbers
In the first volume of the elective compulsory textbook, we learned about the complex number system. A complex number consists ofthe real partandthe imaginary partwith its standard algebraic form given as $z = a + bi$ ($a, b \in \mathbb{R}$).
To gain an intuitive understanding of complex numbers, we establishedthe complex plane:
- the real axis: corresponds to the $x$-axis, representing the real part of a complex number.
- the imaginary axis: corresponds to the $y$-axis, representing the imaginary part of a complex number.
- point and complex number: The complex number $z = a + bi$ forms a one-to-one correspondence with the point $Z(a, b)$.
- vector and complex number: The complex number $z = a + bi$ forms a one-to-one correspondence with the plane 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.
$$z = a + bi \iff Z(a, b) \iff \vec{OZ}$$