equality (of complex numbers)

Equality of Complex Numbers

Definition

In mathematics, equality of complex numbers refers to the comparison of two complex numbers to determine if they are equal or not. A complex number is a number of the form a + bi, where a and b are real numbers and i is the imaginary unit (√-1).

Knowledge Points

To understand equality of complex numbers, one must have knowledge of the following concepts:

1. Complex numbers: Complex numbers are numbers of the form a + bi, where a and b are real numbers and i is the imaginary unit.
2. Real numbers: Real numbers are numbers that can be represented on the number line.
3. Imaginary unit: The imaginary unit, denoted by i, is defined as √-1.

Formula or Equation

The formula for equality of complex numbers is straightforward. Two complex numbers a + bi and c + di are equal if and only if their real parts (a and c) are equal and their imaginary parts (b and d) are equal. Mathematically, this can be expressed as:

a + bi = c + di if and only if a = c and b = d

Application

To apply the formula for equality of complex numbers, we compare the real parts and imaginary parts separately. If both the real and imaginary parts of two complex numbers are equal, then the complex numbers are considered equal.

Symbol

The symbol used to represent equality of complex numbers is the equal sign (=), just like in other mathematical equations.

Methods

There are a few methods to determine the equality of complex numbers:

1. Comparison of real and imaginary parts: Compare the real parts and imaginary parts of the complex numbers separately to check if they are equal.
2. Simplification: Simplify the complex numbers by combining like terms and then compare the simplified forms.

Solved Examples

1. Are the complex numbers (3 + 2i) and (3 + 2i) equal? Solution: Yes, the real parts (3) and imaginary parts (2) of both complex numbers are equal, so they are equal.

2. Determine if the complex numbers (4 + 5i) and (5 + 4i) are equal. Solution: No, the real parts (4 and 5) and imaginary parts (5 and 4) are not equal, so the complex numbers are not equal.

Practice Problems

1. Are the complex numbers (2 + 3i) and (2 + 3i) equal?
2. Determine if the complex numbers (6 + 7i) and (6 + 7i) are equal.
3. Find the value of x if (x + 2i) = (4 + 2i).

FAQ

Q: What is equality of complex numbers? A: Equality of complex numbers refers to comparing two complex numbers to determine if they are equal or not.

Q: How do you determine if two complex numbers are equal? A: To determine if two complex numbers are equal, compare their real parts and imaginary parts separately. If both parts are equal, the complex numbers are considered equal.

Q: What is the formula for equality of complex numbers? A: The formula for equality of complex numbers states that two complex numbers are equal if and only if their real parts are equal and their imaginary parts are equal.