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To tackle problems involving the multiplication of complex numbers, the distributive property is a handy tool. This property allows you to simplify expressions by multiplying each term in the first set of parentheses by every term in the second set. In simpler terms, it helps you "distribute" the multiplication over addition or subtraction. For example, in the expression \((2 + 3i)(4 - 5i)\), we use this property to expand it as follows:

• Multiply \(2\) by \((4 - 5i)\)
• Then multiply \(3i\) by \((4 - 5i)\)

This results in the equation \(2(4 - 5i) + 3i(4 - 5i)\). By applying this property, you can systematically work on each part of the expression, eventually leading to a simplified form that is easier to manage.

The imaginary unit, denoted by \(i\), is a fundamental concept in complex numbers. It is defined by the property \(i^2 = -1\). This simple definition is incredibly powerful because it allows us to work with the square roots of negative numbers.

In our expression \((2+3i)(4-5i)\), the imaginary unit helps us manage operations involving the imaginary part. When multiplying terms like \(3i(4 - 5i)\), you'll reach \(15i^2\) during expansion. Here is where the definition \(i^2 = -1\) becomes crucial. By substituting \(i^2\) with \(-1\), we change \(15i^2\) into \(-15\). This transformation is pivotal because it allows us to turn an expression filled with complex units into something real, simplifying to something like \(12i + 15\), effectively shifting between imaginary and real numbers.

When you multiply complex numbers, you are essentially combining both their real and imaginary components. The multiplication is best done using the distributive property, as shown in our example with \((2 + 3i)(4 - 5i)\). Begin the process by expanding the expression, which splits it into manageable parts:

• Multiply the real numbers together
• Multiply the real with imaginary, and vice versa
• Lastly, multiply the imaginary numbers together

This results in multiple terms like \(8 - 10i + 12i - 15i^2\). After expanding, the next step is to simplify using the imaginary unit \(i\). Once simplified, you combine all the real and imaginary parts to reformat into the standard form of a complex number \(a + bi\).

In our case, the combination leads to the simplified expression \(23 + 2i\). This form tells us everything we need to know about the result: \(23\) is the real part and \(2i\) is the imaginary part.