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Complex numbers are numbers that can be expressed in the form \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. Basic operations with complex numbers include addition, subtraction, multiplication, and division. When multiplying complex numbers, such as in the exercise \(12 i(1-9 i)\), you distribute the terms, as we did in our step-by-step solution. Distributing means that you will take each term outside the parenthesis and multiply it by each term inside the parenthesis. In the given problem, we multiply \(12i\) by both \(1\) and \(-9i\), leading to \(12i \cdot 1 - 12i \cdot 9i\). This approach simplifies the process and is crucial when dealing with more complicated expressions.

The standard form of complex numbers is \(a + bi\), where \(a\) is the real number, and \(bi\) is the imaginary part. The real part \(a\) and the imaginary part \(b\) are both real numbers.After performing operations on complex numbers, like addition or multiplication, it's important to rearrange the result into the standard form. For example, in our exercise, after simplifying, you get \(12i + 108\). Reorganizing this as \(108 + 12i\) places \(108\) as the real term and \(12i\) as the imaginary term, correctly fitting the standard form. This practice aids in recognizing the complex number's real and imaginary components easily.

The imaginary unit \(i\) is a fundamental concept in complex numbers. It’s defined as the square root of \(-1\), which is not a real number. This concept is used to represent numbers that are not on the real number line.A crucial property of \(i\) is that \(i^2 = -1\). This property is pivotal when performing operations with complex numbers since any powers of \(i\) can be simplified using this fact. For example, in our exercise, the expression \(-12i \cdot 9i\) becomes \(-108i^2\), which simplifies to \(-108(-1)\) or \(108\). Remembering this property of \(i\) simplifies the arithmetic involved in complex number operations, making the calculations more manageable.