Opposite Complex Numbers

Two complex numbers, \( z \) and \( -z \), are called opposite complex numbers if their real and imaginary parts have opposite signs: $$ z = a + bi $$ $$ -z = -a - bi $$

To find the opposite of a complex number \( z \), simply reverse the signs of both its real and imaginary components.

A Practical Example

Let's take a complex number \( z \):

$$ z = 2 + 3i $$

Its opposite has the same magnitude for both the real and imaginary parts, but with reversed signs:

$$ -z = -2 - 3i $$

While the absolute values remain unchanged, the signs are flipped.

Example 2

Now, consider the complex number:

$$ z = 2 - 3i $$

Its opposite is:

$$ -z = -2 + 3i $$

Note. The opposite of a complex number can be found by multiplying it by -1: $$ z \cdot (-1) = (2 - 3i) \cdot (-1) = 2 \cdot (-1) - 3i \cdot (-1) = -2 + 3i $$

Opposite vs. Conjugate Complex Numbers

Two complex numbers are:

• Opposite complex numbers if both their real and imaginary parts have opposite signs: $$ z = a + bi $$ $$ -z = -a - bi $$
• Conjugate complex numbers if they share the same real part but have opposite imaginary parts: $$ z = a + bi $$ $$ z' = a - bi $$

  Note. In the case of conjugates, the real part (\( a \)) remains unchanged, while the imaginary part (\( b \)) switches sign to \( -b \).

Example

Let's take the complex number \( z = 2 + 3i \):

$$ z = 2 + 3i $$

Its opposite is:

$$ -z = -2 - 3i $$

The complex conjugate is:

$$ z' = 2 - 3i $$

Properties of Opposite Complex Numbers

Here are some key properties of opposite complex numbers:

• The sum of a complex number and its opposite is always zero: $$ z + (-z) = 0 $$

  Proof. Consider two opposite complex numbers, \( z = a + bi \) and \( -z = -a - bi \). Adding them together results in zero since their real and imaginary parts cancel each other out: $$ z + (-z) = (a + bi) + (-a - bi) = a + bi - a - bi = 0 $$

And so on.