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# Compute the modulus and principal argument of given complex numbers

For each of the following complex numbers, compute the modulus and principal argument.

1. .
2. .
3. .
4. .
5. .
6. .
7. .
8. .
9. .
10. .

1. First, the modulus is given by

The principal argument is the unique such that

In this case we have , and so the principal argument satisfies

2. First, the modulus is given by

The principal argument satisfies

3. The modulus is given by

The principal argument satisfies

4. The modulus is given by

The principal argument satisfies

5. The modulus is given by

The principal argument satisfies

6. The modulus is given by

The principal argument satisfies

7. First, we write in the form ,

Therefore, the modulus is

The principal argument satisfies

8. First, we rewrite in the form ,

Therefore, the modulus is

The principal argument satisfies

9. First, we rewrite in the form ,

Therefore, the modulus is

The principal argument satisfies

10. First, we rewrite in the form ,

The modulus is

The principal argument satisfies