91Ó°ÊÓ

Complex numbers are fundamental in mathematics and engineering, providing solutions to equations that real numbers cannot solve. They are written as a combination of a real number and an imaginary number, and are of the form \( a + bi \), where \( a \) and \( b \) are real numbers and \( i \) is the imaginary unit with the property that \( i^2 = -1 \).

When dealing with complex numbers, it's essential to understand that the real part (\( a \)) and the imaginary part (\( b \)) of the complex number function independently in mathematical operations. This allows for a unique two-dimensional number system that can represent numbers on a plane, often referred to as the complex plane or Argand plane. Complex numbers can be added, subtracted, multiplied, and even divided by applying specific algebraic rules.

The conjugate method is a powerful tool in algebra, especially when working with complex numbers. The conjugate of a complex number \( a + bi \) is \( a - bi \). This creates a unique situation in which the product of a complex number and its conjugate is always a real number. The formula for this product results in \( (a + bi)(a - bi) = a^2 - (bi)^2 = a^2 + b^2 \) because \( i^2 = -1 \).

The conjugate method is particularly useful for rationalizing denominators that contain complex numbers, as seen in the exercise example. By multiplying the numerator and the denominator by the conjugate of the denominator, the result in the denominator becomes a real number, which simplifies the original complex expression.

Expressing complex numbers in \( a + bi \) form is a standard notation that provides a clear representation of both the real and imaginary parts. After rationalizing a complex denominator using the conjugate method, it's common to separate the real and imaginary components for clarity. This is achieved by writing the complex number as the sum of two terms: the real part \( a \) divided by the resultant real number and the imaginary part \( b \) divided by the same real number, followed by \( i \) to signify the imaginary component.

For instance, in the rationalization process shown in the exercise, the final result is expressed as two separate fractions \( \frac{6}{37} - \frac{1}{37}i \), clearly indicating the real portion \( \frac{6}{37} \) and the imaginary portion \( -\frac{1}{37}i \) of the expression.