If two complex numbers are equal, what must be true?

If two complex numbers are equal, what must be true?

• A. Their imaginary parts are equal
• B. Their real parts are equal and their imaginary parts are equal
• C. Their moduli are equal
• D. Their conjugates are equal

Answer: (B)

The key idea is that two complex numbers are equal only when both parts match exactly. If you write them as a + bi and c + di, equality means a = c and b = d. So both the real parts must be equal and the imaginary parts must be equal. Think of it this way: you can’t have the same imaginary part while the real parts differ and claim the numbers are the same; for example, 3 + i and -2 + i share the same imaginary part, but they are not equal because their real parts differ. Similarly, having the same modulus doesn’t guarantee equality, since 1 and i both have modulus 1 but are not equal. And while the conjugates of equal numbers are equal, the most direct way to express equality is that the real and imaginary parts are both equal.

The key idea is that two complex numbers are equal only when both parts match exactly. If you write them as a + bi and c + di, equality means a = c and b = d. So both the real parts must be equal and the imaginary parts must be equal.

Think of it this way: you can’t have the same imaginary part while the real parts differ and claim the numbers are the same; for example, 3 + i and -2 + i share the same imaginary part, but they are not equal because their real parts differ. Similarly, having the same modulus doesn’t guarantee equality, since 1 and i both have modulus 1 but are not equal. And while the conjugates of equal numbers are equal, the most direct way to express equality is that the real and imaginary parts are both equal.