All About Equality of Complex Numbers

INTRODUCTION:

To begin, let’s review the real and imaginary components of complex numbers. With the Real component of z, Re(z) = a, and the Imaginary part of z, Im(z) = b, a complex number z can be expressed in the form z = a + bi.

Consider the following two complex numbers in standard form: z1 = a + bi and z2 = x + yi.

What would it take to make these two complex numbers equal?

Two complex numbers are equal if their real parts are the same and their imaginary parts are the same.

z1 = z2

If Re(z1) = Re(z2) (a = x) and Im(z1) = Im(z2) (b = y), then

The formal definition of 2 complex numbers being equal is:

If and only if a = x and c = y, two complex numbers a + bi and x + yi are equivalent, where a, b, x, y are real numbers.

This can also be used to determine when two complex numbers are not equal. Two complex numbers are not equal if their Real or Imaginary components disagree.

Note that the complex numbers must be expressed in the conventional form a+bi in order to do any equality checks. This makes it simple to discover the real and imaginary portions of each number, allowing us to do the necessary comparisons.

5 – 2i + 7 and 1 + 4i + 11 – 6i

z1 = 12 – 2i & z2 = 12 – 2i

After that, look at the true components of z1 and z2.

Re(z1) = 12

Re(z2) = 12

Let’s see if the real and imagined components are the same.

Im(z1) = -2

Im(z2) = -2

Because the imaginary components are likewise the same, z1 & z2 are the same.

Complex Number Equality – Examples

Example 1: Find the values of x and y if z1 = 8 + 2yi and z2 = -x + 6i are equal.

Solution: We can compare the real and imaginary components because the numbers are both in standard form. We also know that the real and imaginary parts of the numbers are equal since the two complex numbers are equal.

• 8 = -x

• x = 8 

• Re(z1) = Re(z2)

• 2y = 6 

• y = 3 

Im(z1) = Im(z2)

Example 2: If x + yi = 2y – (3x – 7) find x and y. I

Solution:

Both numbers are in standard form and have also been set to the same value. When the real and imaginary components of two complex numbers are equal, we know they are equal.

Because the real pieces are the same:

x = 2y … (1) Furthermore, because the imaginary portions are equal:

y = -(3x – 7) … (2)

Connecting (1) to (2):

y = -(3(2y) – 7)

y = -(6y – 7)

y = -6y + 7

7y = 7 

y = 1

Put y=1 back into (1) now:

x = 2 (1)

x = 2

Example 3: What must be true of a, b, c, and d if a + bi = c + di?

Solution: a = c, b = d 

Example 4: If x + yi = 3y – 1, find x and y. (2x – 4) i

Solution: This is intriguing: we only have one equation but two variables; there doesn’t appear to be enough information to solve the problem. However, because we can divide this into a real and imaginary half, we can write two equations:

x = 3y & y = 2x – 4

By substituting, we get y = 6y – 4, or y = 45, which equals x = 125.

Conclusion

A Complex Number is the product of a Real as well as an Imaginary Number. A complex number is one that can be written as a + bi, where a & b are real numbers, and I is a solution to the equation 

x2 = 1, which is referred to as an imaginary number because no real number can satisfy this equation. A complex number is defined as a number with the form a + bi, where a & b are both real numbers. The real part of the complex number is a, whereas the imaginary part is b.