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In the world of complex numbers, the standard form is an essential way of expressing results clearly. A complex number in standard form is written as \(a + bi\). Here, \(a\) represents the real part, while \(bi\) represents the imaginary part, where \(b\) is a real number and \(i\) is the imaginary unit. This structure allows for an easier understanding and manipulation of complex numbers.

For instance, when you perform operations like addition or subtraction on complex numbers, you aim to express your final answer in this standard form. This ensures clarity and provides an immediate insight into both the real and the imaginary components of the number. By following the steps of distribution and combining like terms, the complex number \(8i - (14 - 9i)\) simplifies elegantly into the standard form \(-14 + 17i\). This final expression is easy to interpret, with \(-14\) as the real part and \(17i\) as the imaginary component.

The imaginary unit \(i\) is a fundamental concept in dealing with complex numbers. Defined as \(i = \sqrt{-1}\), it enables computations involving negative square roots, which are not possible with real numbers alone.

In expressing complex numbers, \(i\) is paired with a real number to form the imaginary part. For example, in the expression \(8i\), 8 is the coefficient of the imaginary unit \(i\), detailing how many units of \(i\) are present. This scalability of \(i\) makes it a powerful way to expand the reach of mathematical expressions beyond the real number line.

Throughout operations involving complex numbers, maintaining the integrity of the imaginary unit helps to simplify and condense expressions. This can be seen in the original example, where \(8i\) and \(-9i\) were combined, ultimately forming part of the result \(-14 + 17i\). Each step respects the properties of \(i\), ensuring the final expression adheres to the foundations of complex number arithmetic.

In algebra, combining like terms is a technique used to simplify expressions by merging terms that have identical variable parts. This applies not only to real numbers but also to complex numbers, where you deal with both real and imaginary components.

When working with complex numbers, it is important to identify and consolidate terms separately: real terms with real terms, and imaginary terms with imaginary terms. For instance, if we look at the expression given: \(8i - (14 - 9i)\), after distributing the minus sign, the terms were arranged as \(8i - 14 + 9i\). Here:

• The real part is only \(-14\).
• The imaginary parts \(8i\) and \(9i\) were combined to form \(17i\).

By practicing this essential skill of combining like terms, you ensure a crisp and accurate result. This ability not only simplifies complex expressions but also paves the way for advanced mathematical problem-solving, acting as a cornerstone for efficient and effective arithmetic operations in mathematics.