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Imaginary numbers are a mathematical concept that extends our number system beyond the real numbers. They are used to solve equations that have no real solutions. A basic imaginary number is denoted as \(i\), where \(i^2 = -1\). This forms the foundation of imaginary numbers.

Imaginary numbers are useful in a variety of applications, including engineering and physics. For example, they can describe oscillating electrical currents. An imaginary number typically involves \(i\) and can be represented in forms like \(6i\) or \(-2i\).

When paired with a real number, an imaginary number becomes part of a complex number, written in the form \(a + bi\), where \(a\) is the real part, and \(b\) is the imaginary part.

Adding and subtracting complex numbers follows straightforward rules. First, you identify the real parts and the imaginary parts. You then add or subtract each type separately. It's similar to combining like terms in algebraic expressions.

For instance, consider the expression \(6i - (4 - i)\). You first remove parentheses, minding the negative sign, turning it into \(6i - 4 + i\). Next, you combine real numbers and imaginary numbers:

• Imaginary parts: \(6i\) and \(i\) are added to get \(7i\).
• Real parts: \(-4\) is combined with nothing from the second term, remaining \(-4\).

Thus, the sum or difference is expressed neatly as a complex number: \(-4 + 7i\).

Algebraic expressions are combinations of numbers, variables, and operations. They form the building blocks of algebra. In this context, expressions include both real numbers and complex numbers featuring imaginary units.

When dealing with algebraic expressions, it's crucial to properly combine and simplify like terms. This involves managing both the constants and the coefficients linked to the imaginary unit \(i\). With complex numbers, the role of algebraic expressions expands to include these new terms.

By treating each part of an algebraic expression—real and imaginary—separately, you can simplify and solve complex equations. So, embracing this step-by-step approach can help you comfortably navigate through the arithmetic of complex numbers. This approach is pivotal both in solving mathematical problems and in scientific calculations.