91Ó°ÊÓ

Complex numbers are numbers that have both a real part and an imaginary part. They are usually written in the form \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part.

The imaginary part involves the imaginary unit \( i \), which is defined as the square root of -1. This means \( i^2 = -1 \).

Complex numbers are important because they extend the concept of one-dimensional real numbers to the two-dimensional complex plane by adding an imaginary axis. This helps solve equations that don't have real solutions, like \( x^2 + 1 = 0 \).

The imaginary unit, denoted as \( i \), is a fundamental concept in complex numbers. It is defined by the property that \( i^2 = -1 \). This is not a real number because no real number squared gives -1.

When dealing with imaginary units, you must remember:

• \( i^3 = i \times i^2 = i \times -1 = -i \)
• \( i^4 = i^2 \times i^2 = -1 \times -1 = 1 \)

The imaginary unit allows us to simplify expressions that involve square roots of negative numbers. For example, \( \text{sqrt}(-4) = 2i \).

Rationalizing the denominator involves removing any irrational numbers (like square roots) or imaginary numbers from the denominator of a fraction. This is often done to simplify the expression or to have a standard form.

For complex fractions, you can rationalize the denominator by multiplying both the numerator and the denominator by the conjugate of the denominator.
For instance, consider the problem here:

\( \frac{5 - \text{sqrt}{5} \times i}{\text{sqrt}{5} \times i} \)

To rationalize this, you multiply by \( -i \) (since \( i \times -i = -1 \)). This changes the problem to:

\( \frac{(5 - \text{sqrt}{5} \times i) \times -i}{\text{sqrt}{5} \times i \times -i} = \frac{-5i + \text{sqrt}{5} \times i^2}{-5} \)

Simplifying further, you reach:

\( i - 1 \).

Rationalizing the denominator makes complex arithmetic more manageable and the results easier to understand.