Equality of complex numbers

• Definition

  Two complex numbers are said to be equal if and only if their real parts and imaginary parts are equal. That is, if $z_1 = a + bi$ and $z_2 = c + di$, then $z_1 = z_2$ if and only if $a = c$ and $b = d$.

  For example, if $z_1 = 3 + 4i$ and $z_2 = 3 + 4i$, then $z_1 = z_2$ because both complex numbers have a real part of $3$ and an imaginary part of $4$.

  However, if $z_1 = 3 + 4i$ and $z_2 = 4 + 3i$, then $z_1$ is not equal to $z_2$ because their real and imaginary parts are not equal.

Examples

Find the values of $x$ and $y$ when $x + yi = -5+5i$

Solution:

$x + yi = -5 + 5i\\x=-5 \\y = 5\\(x,y) =(-5,5)$

Explanation:

Given that $x + yi = -5 + 5i$.

We can separate the real and imaginary parts of the complex number as:

• Real part: $x = -5$

• Imaginary part: $yi = 5i$

We know that the imaginary unit $i$ is defined as $\sqrt{(-1)}$. Therefore, we can simplify the expression for the imaginary part as:

$y\sqrt{(-1)} = 5\sqrt{(-1)}$

Dividing both sides by $\sqrt{(-1)}$ , we get:

$y = 5$

Hence, the values of x and y are $x = -5$ and $y = 5$.