For a non-zero complex number z, let arg(z) denote the principal argument with -π

Q: Wait, why is $\arg(-1-i) = -\frac{3\pi}{4}$ and not $\frac{5\pi}{4}$? They represent the same angle!

That is a brilliant observation! You are absolutely right that $\frac{5\pi}{4}$ and $-\frac{3\pi}{4}$ represent the same ray in the complex plane. However, the definition of the *principal argument* is strict: it must lie in the interval $(-\pi, \pi]$. Since $\frac{5\pi}{4}$ is greater than $\pi$, it falls outside the allowed range. We must always shift our angle by integer multiples of $2\pi$ until it lands within $(-\pi, \pi]$. Thus, $-\frac{3\pi}{4}$ is the only valid principal argument.

Q: Why is the function $f(t) = \arg(-1 + it)$ discontinuous? It looks like a smooth line!

It is a common trap to mistake the geometric locus for the function's behavior. While the locus is indeed a vertical line $x = -1$, the *argument* function measures the angle from the positive real axis. As you move along the line $x = -1$ from $t > 0$ (2nd quadrant) to $t < 0$ (3rd quadrant), you are crossing the negative real axis. At this boundary, the angle jumps instantly from $\pi$ to $-\pi$. That 'jump' is exactly what defines a discontinuity!

Q: How can I tell if $\arg(\dots) = \pi$ means a line or a circle?

Think of the argument as an angle subtended by a chord. If $\arg(\frac{z-z_1}{z-z_3}) = \text{constant}$, the locus of $z$ is an arc of a circle passing through $z_1$ and $z_3$. If the constant is $0$ or $\pi$, the arc becomes a straight line segment connecting $z_1$ and $z_3$. However, in Option D, we have a difference of two arguments equal to $\pi$. This is the condition for the points to be concyclic (forming a cyclic quadrilateral), which implies the locus is a circle, not a line.

Question

For a non-zero complex number $z$, let $\arg(z)$ denote the principal argument with $-\pi < \arg(z) \le \pi$. Then, which of the following statement(s) is (are) FALSE ?

Options:

(A) $\arg(-1-i) = \frac{\pi}{4}$, where $i = \sqrt{-1}$

(B) The function $f : R 	o (-\pi, \pi]$, defined by $f(t) = \arg(-1 + it)$ for all $t \in R$, is continuous at all points of $R$, where $i = \sqrt{-1}$

(C) For any two non-zero complex numbers $z_1$ and $z_2$, $\arg \frac{z_1}{z_2} - \arg(z_1) + \arg(z_2)$ is an integer multiple of $2\pi$

(D) For any three given distinct complex numbers $z_1, z_2$ and $z_3$, the locus of the point $z$ satisfying the condition $\arg \left( \frac{(z-z_1)(z_2-z_3)}{(z-z_3)(z_2-z_1)} ight) = \pi$, lies on a straight line

Sigma Percentile · Accepted Answer

### The Complex Dance: Navigating Arguments and Loci

Welcome, fellow traveler of the complex plane! Today, we are not just solving a problem; we are exploring the subtle, often treacherous, but incredibly beautiful landscape of complex numbers.

This problem is a classic JEE Advanced trap—it tests not just your ability to calculate, but your conceptual depth regarding the principal argument and geometric loci.

### Phase 1: The Quadrant Trap (Option A)

Let us begin with Option A. We are asked to evaluate $\arg(-1-i)$. It is tempting to simply compute $	an^{-1}(\frac{-1}{-1}) = 	an^{-1}(1) = \frac{\pi}{4}$ and call it a day. But stop! Pause for a moment. Where does this point live?

If you plot $z = -1 - i$ on the Argand plane, you see that both the real part ($x = -1$) and the imaginary part ($y = -1$) are negative. This places our point firmly in the **3rd quadrant**.

The principal argument, by definition, must lie in $(-\pi, \pi]$. In the 3rd quadrant, the angle is measured clockwise from the negative real axis, given by the formula $\arg(z) = -(\pi - \alpha)$, where $\alpha = 	an^{-1}|\frac{y}{x}|$.

Substituting our values: 
$$\arg(-1-i) = -(\pi - \frac{\pi}{4}) = -\frac{3\pi}{4}$$

Comparing this to the option's claim of $\frac{\pi}{4}$, we see a massive discrepancy. Option A is undeniably **FALSE**.

### Phase 2: The Discontinuity Dance (Option B)

Now, let us look at $f(t) = \arg(-1 + it)$. This function describes the argument of points on the vertical line $x = -1$. As $t$ varies, we move up and down this line.

Imagine standing at $t = 0$, which is the point $-1 + 0i$. If you approach this point from $t > 0$ (the 2nd quadrant), your angle is approaching $\pi$. But if you approach from $t < 0$ (the 3rd quadrant), your angle is approaching $-\pi$.

Mathematically, we have:
$$\lim_{t 	o 0^+} f(t) = \pi$$
$$\lim_{t 	o 0^-} f(t) = -\pi$$

Because the left-hand limit does not equal the right-hand limit, the function suffers from a jump discontinuity at $t = 0$. The claim that it is continuous everywhere is **FALSE**.

### Phase 3: The Elegance of Properties (Option C)

Option C asks us to verify the identity $\arg(\frac{z_1}{z_2}) - \arg(z_1) + \arg(z_2)$. This is a test of your fundamental understanding of complex logarithms.

We know that the argument of a quotient is the difference of the arguments, adjusted by an integer multiple of $2\pi$ to keep it within the principal range:
$$\arg\left(\frac{z_1}{z_2}ight) = \arg(z_1) - \arg(z_2) + 2k\pi$$

If we rearrange this, we get:
$$\arg\left(\frac{z_1}{z_2}ight) - \arg(z_1) + \arg(z_2) = 2k\pi$$

This is exactly what the option claims! It is a beautiful, elegant property of complex numbers. Option C is **TRUE**.

### Phase 4: The Geometry of Circles (Option D)

Finally, we arrive at the most intimidating part: the cross-ratio. We are given:
$$\arg \left( \frac{(z-z_1)(z_2-z_3)}{(z-z_3)(z_2-z_1)} ight) = \pi$$

Using the properties of arguments, we can split this into:
$$\arg\left(\frac{z-z_1}{z-z_3}ight) - \arg\left(\frac{z_2-z_1}{z_2-z_3}ight) = \pi$$

Geometrically, this represents the difference between the angle subtended by the segment $z_1z_3$ at point $z$ and the angle subtended by the same segment at point $z_2$. When this difference is $\pi$ (or $0$), it is the classic condition for the four points $z, z_1, z_2, z_3$ to be concyclic—meaning they all lie on the circumference of a circle.

Option D claims the locus is a straight line. We have proven it is a circle. Thus, Option D is **FALSE**.

### Conclusion

We have navigated the quadrant traps, identified the jump discontinuities, verified the algebraic properties, and uncovered the geometric truth behind the cross-ratio. Remember, in complex numbers, the math is never just algebra; it is geometry in disguise. Keep visualizing, keep questioning, and you will master this beautiful subject!

Animated Solution for Mathematics - Complex Numbers: For a non-zero complex number $z$ , let $ar g (z)$ denote the principal argument with $- π < ar g (z) \leq π$ . Then, which of the following statement(s) is (are) FALSE ?

Select Answer:

Visualized Solution

Analyzing the Options

Option A: $ar g (- 1 - i)$

Calculating $ar g (- 1 - i)$

Conclusion for Option A

Option B: Function $f (t)$

Right Hand Limit at $t = 0$

Left Hand Limit at $t = 0$

Conclusion for Option B

Option C: Argument Property

Conclusion for Option C

Option D: Concyclic Points

Geometric Interpretation

Conclusion for Option D

Final Summary

The Sigma Insight: Geometrical Applications of Complex Numbers

The Complex Dance

Phase 1

Phase 2

Phase 3

Phase 4

Conclusion

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