To solve the problem of finding the number of integral solutions satisfying the inequality \( |Re(z)| + |Im(z)| < 21 \), we can follow these steps:
### Step 1: Understand the inequality
We start with the inequality:
\[
|Re(z)| + |Im(z)| < 21
\]
Let \( Re(z) = x \) and \( Im(z) = y \). Thus, we rewrite the inequality as:
\[
|x| + |y| < 21
\]
### Step 2: Geometric interpretation
The inequality \( |x| + |y| < 21 \) represents a region in the complex plane. Specifically, it describes a diamond (or rhombus) shape centered at the origin with vertices at the points (21, 0), (0, 21), (-21, 0), and (0, -21).
### Step 3: Count integral solutions in one quadrant
To find the number of integral solutions, we will first count the points in the first quadrant where \( x \geq 0 \) and \( y \geq 0 \). Here, the inequality simplifies to:
\[
x + y < 21
\]
The integer points satisfying this inequality can be counted by fixing \( x \) and determining the possible values for \( y \).
- For \( x = 0 \): \( y < 21 \) gives 21 possible values (0 to 20).
- For \( x = 1 \): \( y < 20 \) gives 20 possible values (0 to 19).
- For \( x = 2 \): \( y < 19 \) gives 19 possible values (0 to 18).
- ...
- For \( x = 20 \): \( y < 1 \) gives 1 possible value (0).
The total number of integral points in the first quadrant can be calculated as:
\[
21 + 20 + 19 + \ldots + 1 = \frac{21 \times 22}{2} = 231
\]
### Step 4: Multiply by the number of quadrants
Since the diamond shape is symmetric across all four quadrants, the total number of integral solutions in all quadrants is:
\[
4 \times 231 = 924
\]
### Step 5: Account for points on the axes
Now we need to consider the points on the axes. The points on the axes include:
- The origin (0, 0)
- Points along the x-axis from (1, 0) to (20, 0) and (-1, 0) to (-20, 0) which gives 20 points on each side.
- Points along the y-axis from (0, 1) to (0, 20) and (0, -1) to (0, -20) which gives 20 points on each side.
Thus, the total points on the axes are:
\[
1 + 20 + 20 + 20 + 20 = 81
\]
### Step 6: Final count
Adding the points from the quadrants and the axes, we get:
\[
924 + 81 = 1005
\]
### Conclusion
The total number of integral solutions satisfying the inequality \( |Re(z)| + |Im(z)| < 21 \) is:
\[
\boxed{1005}
\]
