In base-$2$ notation, digits are $0$ and $1$ only and the places go up in powers of $-2$. For example, $11011$ stands for $(-2)^4+(-2)^3+(-2)^1+(-2)^0$ and equals number $7$ in base $10$. If the decimal number $2019$ is expressed in base $-2$ how many non-zero digits does it contain ?
To convert the decimal number \(2019\) into base \(-2\), we need to follow a systematic approach. Let's break down the steps:

1. **Identify the largest power of \(-2\) less than or equal to \(2019\):**
   \[
   (-2)^{12} = 4096
   \]
   Since \(4096\) is greater than \(2019\), we use the next lower power:
   \[
   (-2)^{11} = -2048
   \]
   Since \(-2048\) is negative and we need a positive number, we use:
   \[
   (-2)^{11} + (-2)^{12} = 2048 - 4096 = -2048
   \]
   This is not helpful, so we use:
   \[
   2^{11} = 2048
   \]

2. **Subtract \(2048\) from \(2019\):**
   \[
   2019 - 2048 = -29
   \]

3. **Identify the largest power of \(-2\) less than or equal to \(-29\):**
   \[
   (-2)^5 = -32
   \]
   Since \(-32\) is less than \(-29\), we use:
   \[
   -29 + 32 = 3
   \]

4. **Identify the largest power of \(-2\) less than or equal to \(3\):**
   \[
   (-2)^2 = 4
   \]
   Since \(4\) is greater than \(3\), we use the next lower power:
   \[
   (-2)^1 = -2
   \]
   Since \(-2\) is less than \(3\), we use:
   \[
   3 + 2 = 5
   \]

5. **Identify the largest power of \(-2\) less than or equal to \(5\):**
   \[
   (-2)^2 = 4
   \]
   Since \(4\) is less than \(5\), we use:
   \[
   5 - 4 = 1
   \]

6. **Identify the largest power of \(-2\) less than or equal to \(1\):**
   \[
   (-2)^0 = 1
   \]
   Since \(1\) is equal to \(1\), we use:
   \[
   1 - 1 = 0
   \]

7. **Combine all the powers of \(-2\) used:**
   \[
   2019 = 2^{11} - 2^5 + 2^2 + 2^1 + 2^0
   \]

8. **Convert these powers into base \(-2\):**
   \[
   2019 = (-2)^{12} + (-2)^{11} + (-2)^5 + (-2)^2 + (-2)^1 + (-2)^0
   \]

9. **Write the number in base \(-2\):**
   \[
   2019_{10} = 1100000100111_{-2}
   \]

10. **Count the non-zero digits:**
   The number \(1100000100111\) has \(6\) non-zero digits.

The final answer is \(\boxed{6}\).