$|z_1+z_2|=|z_1|+|z_2|$ is possible if: (A) $z_2=\bar{z}_1$ (B) $z_2=\dfrac{1}{z_1}$ (C) $\arg(z_1)=\arg(z_2)$ (D) $|z_1|=|z_2|$

|z₁+z₂|=|z₁|+|z₂| is possible if: (A) z₂=barz₁ (B) z₂=(1)/(z₁) (C) arg(z₁)=arg(z₂) (D) |z₁|=|z₂|

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|z_1+z_2|=|z_1|+|z_2|

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Concept understanding — Complex Number Arithmetic

Complex Number Arithmetic: A First Look

Imagine you're trying to solve $x^2 + 1 = 0$ . You know that no real number squared gives $-1$ . The square of any real number is either zero or positive. So this equation has no real solution. But what if we invent a number whose square is $-1$ ? That's exactly what mathematicians did — and that invention is the imaginary unit $i$ , defined by:

i^2 = -1

A complex number is any number of the form $a + bi$ , where $a$ and $b$ are real numbers. Here $a$ is called the real part, and $b$ is called the imaginary part. For example, $3 + 4i$ has real part $3$ and imaginary part $4$ .

Note

The name "imaginary" is unfortunate — these numbers are just as real (in the mathematical sense) as the numbers you already know. They're simply a different kind of number.

Why Bother?

Complex numbers let you solve equations that real numbers can't. Every polynomial equation — no matter how complicated — has a solution in the complex numbers. This is the Fundamental Theorem of Algebra, and it's one of the most important results in mathematics.

Arithmetic Operations

The rules are straightforward: treat $i$ like a variable, but remember that $i^2 = -1$ .

Addition and Subtraction

Add (or subtract) real parts with real parts, imaginary parts with imaginary parts.

(a + bi) + (c + di) = (a + c) + (b + d)i

(a + bi) - (c + di) = (a - c) + (b - d)i

Example: $(2 + 3i) + (4 - 5i) = (2 + 4) + (3 - 5)i = 6 - 2i$

Multiplication

Multiply like binomials, then replace $i^2$ with $-1$ .

(a + bi)(c + di) = ac + adi + bci + bdi^2 = (ac - bd) + (ad + bc)i

Example: $(2 + 3i)(4 - 5i) = 2(4) + 2(-5i) + 3i(4) + 3i(-5i)$

$= 8 - 10i + 12i - 15i^2$

$= 8 + 2i - 15(-1)$

$= 8 + 2i + 15 = 23 + 2i$

Watch out

The most common mistake: forgetting that $i^2 = -1$ , not $1$ . Always check your final step.

Division

Division is trickier. The key idea: multiply numerator and denominator by the complex conjugate of the denominator.

The complex conjugate of $a + bi$ is $a - bi$ . When you multiply a complex number by its conjugate, you get a real number:

(a + bi)(a - bi) = a^2 - (bi)^2 = a^2 - b^2 i^2 = a^2 + b^2

So to divide:

\frac{a + bi}{c + di} = \frac{(a + bi)(c - di)}{(c + di)(c - di)} = \frac{(a + bi)(c - di)}{c^2 + d^2}

Example: $\frac{2 + 3i}{4 - 5i} = \frac{(2 + 3i)(4 + 5i)}{(4 - 5i)(4 + 5i)} = \frac{8 + 10i + 12i + 15i^2}{16 + 25} = \frac{8 + 22i - 15}{41} = \frac{-7 + 22i}{41} = -\frac{7}{41} + \frac{22}{41}i$

Tip

To divide quickly: multiply top and bottom by the conjugate of the denominator, then simplify. The denominator always becomes $c^2 + d^2$ , a positive real number.

The Big Picture …

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The equality $|z_1+z_2| = |z_1| + |z_2|$ holds exactly when the complex numbers $z_1$ and $z_2$ have the same argument (direction), i.e., they are positive real multiples of each other. The correct option is (C).

The key here is to understand what the modulus of a sum really means geometrically. Every complex number is a vector in the Argand plane. The modulus $|z|$ is its length. So $|z_1+z_2|$ is the length of the vector sum, and $|z_1|+|z_2|$ is the sum of the individual lengths.

When does the length of the sum equal the sum of the lengths? Only when the two vectors point in exactly the same direction — they are perfectly aligned. If they point in different directions, the triangle formed by $z_1$ , $z_2$ , and $z_1+z_2$ has a side shorter than the sum of the other two (the triangle inequality is strict). The equality case of the triangle inequality is collinearity with no cancellation — both vectors must be non-negative real multiples of each other.

Let’s work through the options systematically.

Option (A): $z_2 = \bar{z}_1$

If $z_1 = re^{i\theta}$ , then $z_2 = re^{-i\theta}$ . Their arguments are $\theta$ and $-\theta$ — opposite directions unless $\theta = 0$ or $\pi$ . In general, they are not aligned, so the equality fails. For example, take $z_1 = 1$ , $z_2 = 1$ (which works), but $z_1 = i$ , $z_2 = -i$ gives $|i + (-i)| = 0 \neq 1+1 = 2$ . So this is not always true.
Option (B): $z_2 = \frac{1}{z_1}$

If $z_1 = re^{i\theta}$ , then $z_2 = \frac{1}{r}e^{-i\theta}$ . The arguments are $\theta$ and $-\theta$ — again opposite unless $\theta = 0$ . So generally not aligned. Example: $z_1 = 2$ , $z_2 = 0.5$ works (both real positive), but $z_1 = i$ , $z_2 = -i$ fails as before. Not a general condition.
Option (C): $\arg(z_1) = \arg(z_2)$ …

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