91Ó°ÊÓ

Imaginary numbers are an extension of the real number system and are used to solve equations that do not have real solutions. They are based on the imaginary unit, represented as \(i\), where \(i\) is the square root of \(-1\). This means that \(i^2 = -1\).

Imaginary numbers help in numerous real-world applications, such as in electrical engineering and signal processing. They allow us to work with complex numbers, which are composed of both a real and an imaginary part.

It's important to understand that operating with imaginary numbers follows certain mathematical rules, just like real numbers. When you add or subtract imaginary numbers, you only focus on the coefficients of \(i\). For example, in the original problem, the expression \(-7i - 9i\) simplifies to \(-16i\) by combining coefficients. Imaginary numbers are essential in expressing complex phenomena easily.

The standard form for a complex number is expressed as \(a + bi\). Here, \(a\) is the real part, and \(b\) is the coefficient of the imaginary part \(i\). This form is particularly useful because it clearly separates the real part and the imaginary part of the number.

Writing a complex expression in standard form makes it easier to perform arithmetic operations, such as addition or multiplication. It also helps in graphing, where the real part corresponds to the horizontal axis, and the imaginary part corresponds to the vertical axis. This makes the standard form integral for visualizing complex numbers.

In this exercise, after solving, the expression \(13 - 16i\) is already in standard form. This clear separation aids in further calculations or interpretations involving complex numbers.

Combining like terms is a fundamental concept in algebra that refers to the process of combining parts of an expression that have the same variable or can be grouped together. In the context of complex numbers, like terms are combined by separately adding or subtracting the real parts and the imaginary parts.

In the given problem, we first identify the real numbers and the imaginary numbers:

• The real numbers: \(12\) and \(1\)
• The imaginary numbers: \(-7i\) and \(-9i\)

By adding the real parts \((12 + 1)\) and the imaginary parts \((-7i - 9i)\), we arrive at the expression \(13 - 16i\).

This simplification is crucial because it allows one to efficiently manage and solve expressions involving multiple terms. Simplifying by combining like terms leads to expressions that are easier to handle and interpret in mathematical equations.