91Ó°ÊÓ

Complex numbers are an essential part of mathematics, and they have a specific format known as the "standard form." This form is written as \(a + bi\), where:

• \(a\) is the real part of the complex number,
• \(b\) is the imaginary part, and
• \(i\) is the imaginary unit, representing the square root of \(-1\).

In the standard form, both \(a\) and \(b\) are real numbers. The example we are analyzing, \(32 - 72i\), is already in standard form, where \(a = 32\) and \(b = -72\). This format allows us to easily manage complex numbers and perform operations like addition, subtraction, multiplication, and division.

The imaginary unit is a cornerstone of complex numbers, denoted by the letter \(i\). It is defined by the property \(i^2 = -1\). This unique characteristic allows the formation of imaginary numbers, which are fundamentally different from real numbers.
The presence of \(i\) transforms the arithmetic of complex numbers. For instance, in our original exercise, we calculated \(-8i \times 4i = -32i^2\). Thanks to the property \(i^2 = -1\), this expression simplifies to \(32\).
The imaginary unit enables us to work with numbers that seem, at first glance, to "not exist" since they involve the square root of a negative number. However, their utility in solving real-world problems, particularly in fields like engineering and physics, is invaluable.

Multiplying complex numbers involves more than just dealing with real numbers. It requires an understanding of how both the real and imaginary parts interact during multiplication.

• First, distribute each term separately. For example, \(-8i(9 + 4i)\) becomes \(-8i \times 9\) and \(-8i \times 4i\).
• Compute each product. Begin with \(-8i \times 9\), which gives \(-72i\). Then continue with \(-8i \times 4i\), resulting in \(-32i^2\).
• Simplify using the property \(i^2 = -1\), which changes \(-32i^2\) to \(32\).

Finally, combine the results to put the answer back into standard form. This process highlights why understanding the imaginary unit and the algebraic structure of complex numbers is critical in mathematics.