To determine which statements are false regarding the principal argument of complex numbers, we will analyze each statement step by step.
### Step 1: Analyze Statement (a)
**Statement (a):** `arg(-1, -i) = pi/4`
To find the argument of the complex number `-1 - i`, we can represent it in the Cartesian plane. The point `(-1, -1)` lies in the third quadrant. The angle corresponding to this point can be calculated as follows:
1. The reference angle in the third quadrant is given by:
\[
\theta = \tan^{-1}\left(\frac{-1}{-1}\right) = \tan^{-1}(1) = \frac{\pi}{4}
\]
2. Since the point is in the third quadrant, we need to add `π` to the reference angle:
\[
\text{arg}(-1 - i) = \pi + \frac{\pi}{4} = \frac{5\pi}{4}
\]
Thus, the statement `arg(-1, -i) = pi/4` is **FALSE**.
### Step 2: Analyze Statement (b)
**Statement (b):** The function `f: R -> (-π, π]` defined by `f(t) = arg(-1 + it)` for all `t ∈ R`, is continuous at all points of `R`.
To analyze the continuity of `f(t)`, we need to consider how `arg(-1 + it)` behaves as `t` varies:
1. For `t > 0`, the point `(-1, t)` is in the second quadrant, where:
\[
f(t) = \pi - \tan^{-1}\left(\frac{t}{1}\right)
\]
2. For `t < 0`, the point `(-1, t)` is in the third quadrant, where:
\[
f(t) = -\pi + \tan^{-1}\left(\frac{t}{1}\right)
\]
3. At `t = 0`, the point is `(-1, 0)`, which corresponds to an angle of `π`.
As `t` approaches `0` from the left, `f(t)` approaches `-π`, and from the right, it approaches `π`. Since `f(0) = π`, the function is not continuous at `t = 0`. Therefore, this statement is also **FALSE**.
### Step 3: Analyze Statement (c)
**Statement (c):** For any two non-zero complex numbers `z1` and `z2`, `arg((z1)/(z2)) - arg(z1) + arg(z2)` is an integer multiple of `2π`.
Using the properties of arguments:
\[
arg\left(\frac{z_1}{z_2}\right) = arg(z_1) - arg(z_2) + 2n\pi
\]
for some integer `n`. Rearranging gives:
\[
arg\left(\frac{z_1}{z_2}\right) - arg(z_1) + arg(z_2) = 2n\pi
\]
Thus, this statement is **TRUE**.
### Step 4: Analyze Statement (d)
**Statement (d):** For any three distinct complex numbers `z1`, `z2`, and `z3`, the locus of the point `z` satisfying the condition `arg\left(\frac{(z-z_1)(z_2-z_3)}{(z-z_3)(z_2-z_1)}\right) = \pi` lies on a straight line.
The condition `arg\left(\frac{(z-z_1)(z_2-z_3)}{(z-z_3)(z_2-z_1)}\right) = \pi` implies that the ratio is negative, meaning that the points `z`, `z1`, `z2`, and `z3` are collinear. Therefore, this statement is **TRUE**.
### Conclusion
The false statements are:
- (a) `arg(-1, -i) = pi/4`
- (b) The function `f(t) = arg(-1 + it)` is continuous at all points of `R`.
