Animated Solution for Mathematics - Complex Numbers: Let $z$ and $w$ be complex numbers such that $\bar{z}+i\bar{w}=0$ and $argzw=\pi$. Then $argz$ equals

We are given the equation $\bar{z}+i\bar{w}=0$. To isolate the variables, we take the conjugate of the entire equation:

Using the fundamental properties of conjugates, where the conjugate of a sum is the sum of the conjugates and the conjugate of a product is the product of the conjugates, we simplify the expression. Recall that $\overline{\bar{z}}=z$, $\overline{i}=-i$, and $\overline{\bar{w}}=w$.

This transforms our equation into:

In the complex plane, multiplying a number by $i$ is a geometric transformation. Since $i=e^{i\frac{\pi }{2}}$, multiplying by $i$ is equivalent to a counter-clockwise rotation by $\frac{\pi }{2}$ radians (or $90^{\circ }$).

This means the vector representing $z$ is simply the vector $w$ rotated by $\frac{\pi }{2}$. Whenever you encounter $i$ in a complex number problem, always consider the geometric rotation it represents.

We are given the condition $\arg (zw)=\pi$. Using the property that the argument of a product is the sum of the arguments, we have:

From our earlier relation $z=iw$, we apply the argument property again:

Since $i$ lies on the positive imaginary axis, its argument is $\frac{\pi }{2}$. Substituting this, we get:

We now have a system of two linear equations: 1) $\arg (z)+\arg (w)=\pi$ 2) $\arg (z)-\arg (w)=\frac{\pi }{2}$

Adding these two equations together yields:

Dividing by 2, we find the final result:

The key to mastering these problems is to look for the geometric meaning behind the algebraic symbols. By combining the properties of conjugates with the rotational nature of complex multiplication, we arrive at the solution of $\arg (z)=\frac{3\pi }{4}$.