Find the points corresponding to the complex numbers and…
Find the points corresponding to the complex numbers and determine their position in the complex plane: $$ 1^{\circ}\; i; \quad 2^{\circ}\; -2i; \quad 3^{\circ}\; -1+i; \quad 4^{\circ}\; 2-3i. $$
Explanation of Method
A complex number $z = x + iy$ corresponds to the point $(x, y)$, where $x = \text{Re}(z)$ is plotted on the horizontal (real) axis and $y = \text{Im}(z)$ is plotted on the vertical (imaginary) axis. The plane is split into four quadrants:
- Quadrant I: $x > 0,\ y > 0$
- Quadrant II: $x < 0,\ y > 0$
- Quadrant III: $x < 0,\ y < 0$
- Quadrant IV: $x > 0,\ y < 0$
- Extract the real part $x$ and imaginary part $y$ of $z$.
- Plot the point $(x, y)$ and identify which quadrant or axis it falls on.
Step 1° · $z = i$
Writing $z$ in standard form: $z = i = 0 + i \cdot 1$, so $x = 0$, $y = 1$.
$$ \boxed{(0, 1)} $$
This point lies on the positive imaginary axis — on the boundary between Quadrants I and II.
Modulus: $|i| = \sqrt{0^2 + 1^2} = 1$. Argument: $\arg(i) = \dfrac{\pi}{2}$ (90°).
Step 2° · $z = -2i$
Writing $z$ in standard form: $z = -2i = 0 + i \cdot (-2)$, so $x = 0$, $y = -2$.
$$ \boxed{(0, -2)} $$
This point lies on the negative imaginary axis — on the boundary between Quadrants III and IV.
Modulus: $|-2i| = \sqrt{0^2 + (-2)^2} = 2$. Argument: $\arg(-2i) = -\dfrac{\pi}{2}$ (−90° or 270°).
Step 3° · $z = -1 + i$
Writing $z$ in standard form: $z = -1 + i = -1 + i \cdot 1$, so $x = -1$, $y = 1$.
$$ \boxed{(-1, 1)} $$
This point lies in Quadrant II (negative real part, positive imaginary part).
Modulus: $|-1 + i| = \sqrt{(-1)^2 + 1^2} = \sqrt{2}$. Argument: $\arg(-1+i) = \pi - \arctan\!\left(\dfrac{1}{1}\right) = \pi - \dfrac{\pi}{4} = \dfrac{3\pi}{4}$ (135°).
Step 4° · $z = 2 - 3i$
Writing $z$ in standard form: $z = 2 - 3i = 2 + i \cdot (-3)$, so $x = 2$, $y = -3$.
$$ \boxed{(2, -3)} $$
This point lies in Quadrant IV (positive real part, negative imaginary part).
Modulus: $|2 - 3i| = \sqrt{2^2 + (-3)^2} = \sqrt{13}$. Argument: $\arg(2-3i) = -\arctan\!\left(\dfrac{3}{2}\right) \approx -56.31°$ (or $\approx 303.69°$).
Summary Table
| Complex number | Point $(x, y)$ | Position | Modulus | Argument |
|---|---|---|---|---|
| $i$ | $(0, 1)$ | Positive imaginary axis | $1$ | $\frac{\pi}{2}$ |
| $-2i$ | $(0, -2)$ | Negative imaginary axis | $2$ | $-\frac{\pi}{2}$ |
| $-1 + i$ | $(-1, 1)$ | Quadrant II | $\sqrt{2}$ | $\frac{3\pi}{4}$ |
| $2 - 3i$ | $(2, -3)$ | Quadrant IV | $\sqrt{13}$ | $-\arctan(\frac{3}{2})$ |
Graphical Representation
The four points plotted on the complex plane (real axis horizontal, imaginary axis vertical).
Result
Applying $z = x + iy \rightarrow (x, y)$ to each number gives:
Position determination: for $z = x + iy$: Quadrant I if $x>0, y>0$; Quadrant II if $x<0, y>0$; Quadrant III if $x<0, y<0$; Quadrant IV if $x>0, y<0$; on the imaginary axis if $x=0, y\neq 0$; on the real axis if $y=0, x\neq 0$.