Write the complex numbers that represent the following points in the plane:
(i) (0,0)
(ii) (3,0)
(iii) (-1,0)
(iv) (0,-1)
(v) (1,-2)
(vi) (4,-1)
(vii) `(-(1)/(3),(1)/(5))`

Answer

To find the complex numbers that represent the given points in the plane, we will use the standard representation of a complex number, which is given by \( z = x + iy \), where \( x \) is the real part and \( y \) is the imaginary part.

Let's solve the problem step by step for each point:

### Step 1: Represent the point (0,0)
- The real part \( x = 0 \)
- The imaginary part \( y = 0 \)
- Thus, the complex number is:
\[ z = 0 + 0i = 0 \]

### Step 2: Represent the point (3,0)
- The real part \( x = 3 \)
- The imaginary part \( y = 0 \)
- Thus, the complex number is:
\[ z = 3 + 0i = 3 \]

### Step 3: Represent the point (-1,0)
- The real part \( x = -1 \)
- The imaginary part \( y = 0 \)
- Thus, the complex number is:
\[ z = -1 + 0i = -1 \]

### Step 4: Represent the point (0,-1)
- The real part \( x = 0 \)
- The imaginary part \( y = -1 \)
- Thus, the complex number is:
\[ z = 0 - 1i = -i \]

### Step 5: Represent the point (1,-2)
- The real part \( x = 1 \)
- The imaginary part \( y = -2 \)
- Thus, the complex number is:
\[ z = 1 - 2i \]

### Step 6: Represent the point (4,-1)
- The real part \( x = 4 \)
- The imaginary part \( y = -1 \)
- Thus, the complex number is:
\[ z = 4 - 1i = 4 - i \]

### Step 7: Represent the point \(\left(-\frac{1}{3}, \frac{1}{5}\right)\)
- The real part \( x = -\frac{1}{3} \)
- The imaginary part \( y = \frac{1}{5} \)
- Thus, the complex number is:
\[ z = -\frac{1}{3} + \frac{1}{5}i \]

### Final Summary of Complex Numbers:
1. (0,0) → \( 0 \)
2. (3,0) → \( 3 \)
3. (-1,0) → \( -1 \)
4. (0,-1) → \( -i \)
5. (1,-2) → \( 1 - 2i \)
6. (4,-1) → \( 4 - i \)
7. \(\left(-\frac{1}{3}, \frac{1}{5}\right)\) → \( -\frac{1}{3} + \frac{1}{5}i \)