Find the complex numbers corresponding to the points: 1^(°)…
Find the complex numbers corresponding to the points: $$ 1^{\circ}\,(0,1); \quad 2^{\circ}\,(1,0); \quad 3^{\circ}\,(0,-1); \quad 4^{\circ}\,(-1,0); \quad 5^{\circ}\,(-2,3). $$
Explanation of Method
Every point $(x, y)$ in the Cartesian plane can be identified with a complex number $z = x + iy$, where $x$ is the real part, $y$ is the imaginary part, and $i$ is the imaginary unit with $i^2 = -1$. This identification turns the plane into the complex plane, with the horizontal axis as the real axis and the vertical axis as the imaginary axis.
- Read off the coordinates $(x, y)$ of each point.
- Form the complex number $z = x + iy$.
Step 1° · Point $(0, 1)$
Here $x = 0$ and $y = 1$, so:
$$ z = 0 + i \cdot 1 = i $$
This is a purely imaginary number on the positive imaginary axis.
Step 2° · Point $(1, 0)$
Here $x = 1$ and $y = 0$, so:
$$ z = 1 + i \cdot 0 = 1 $$
This is a purely real number on the positive real axis.
Step 3° · Point $(0, -1)$
Here $x = 0$ and $y = -1$, so:
$$ z = 0 + i \cdot (-1) = -i $$
This is a purely imaginary number on the negative imaginary axis.
Step 4° · Point $(-1, 0)$
Here $x = -1$ and $y = 0$, so:
$$ z = -1 + i \cdot 0 = -1 $$
This is a purely real number on the negative real axis.
Step 5° · Point $(-2, 3)$
Here $x = -2$ and $y = 3$, so:
$$ z = -2 + i \cdot 3 = -2 + 3i $$
This complex number lies in the second quadrant (negative real part, positive imaginary part).
Summary Table
| Point $(x, y)$ | Complex number $z$ | Real part | Imaginary part |
|---|---|---|---|
| $(0, 1)$ | $i$ | $0$ | $1$ |
| $(1, 0)$ | $1$ | $1$ | $0$ |
| $(0, -1)$ | $-i$ | $0$ | $-1$ |
| $(-1, 0)$ | $-1$ | $-1$ | $0$ |
| $(-2, 3)$ | $-2 + 3i$ | $-2$ | $3$ |
Additional Properties
Modulus. For $z = x + iy$, the modulus is $|z| = \sqrt{x^2 + y^2}$:
- $|i| = \sqrt{0^2 + 1^2} = 1$
- $|1| = \sqrt{1^2 + 0^2} = 1$
- $|-i| = \sqrt{0^2 + (-1)^2} = 1$
- $|-1| = \sqrt{(-1)^2 + 0^2} = 1$
- $|-2 + 3i| = \sqrt{(-2)^2 + 3^2} = \sqrt{13}$
Argument. The first four points lie on the unit circle at special angles:
- $\arg(i) = \dfrac{\pi}{2}$ (90°)
- $\arg(1) = 0$ (0°)
- $\arg(-i) = -\dfrac{\pi}{2}$ or $\dfrac{3\pi}{2}$ (−90° or 270°)
- $\arg(-1) = \pi$ (180°)
- $\arg(-2 + 3i) = \pi - \arctan\!\left(\dfrac{3}{2}\right)$ (quadrant II)
The four points $i, 1, -i, -1$ are the fourth roots of unity, forming the vertices of a square on the unit circle.
Result
Applying $z = x + iy$ to each point gives: