91Ó°ÊÓ

In the realm of complex numbers, the imaginary unit, denoted as \(i\), is a fundamental concept. It is defined by the property \(i^2 = -1\). This unique definition allows for the extension of real numbers into the complex plane, thereby enabling operations involving the square roots of negative numbers.
When working with complex numbers, the imaginary unit is often paired with real numbers to form expressions such as \(a + bi\), where \(a\) is the real part and \(b\) is the coefficient of the imaginary part. In the problem, \(z = -2 + 7i\), \(-2\) is the real part, and \(7i\) is the imaginary part.
Understanding the imaginary unit helps in manipulating and simplifying expressions involving complex numbers. It also plays a key role in electrical engineering and many branches of mathematics.

The difference of squares is a powerful algebraic identity given by the formula \(a^2 - b^2 = (a-b)(a+b)\). This formula is instrumental in simplifying the multiplication of binomials, particularly when dealing with complex numbers.
In the context of our problem, we use this identity to calculate the product of a complex number and its conjugate: \((-2 + 7i)(-2 - 7i)\). This can be expressed as a difference of squares: \((-2)^2 - (7i)^2\).

• The first term \((-2)^2\) gives \(4\).
• The second term \((7i)^2\) is calculated by recognizing that \(i^2 = -1\), resulting in \(49(-1) = -49\).

By combining these, we arrive at the result: \(4 - (-49) = 4 + 49 = 53\). This application underscores the utility of the difference of squares in simplifying expressions in complex number multiplication.

Multiplying complex numbers is similar to multiplying binomials, involving both real and imaginary parts. The key to mastering complex number multiplication is applying basic algebraic principles while considering the special role of the imaginary unit \(i\).
When multiplying the complex number \(z = -2 + 7i\) with its conjugate \(\bar{z} = -2 - 7i\), you apply the distributive property. However, by recognizing the expression as a difference of squares, the multiplication becomes quite straightforward.
Each part of the expression is calculated as follows:

• Real parts: the product of the real components \(-2\times-2 = 4\).
• Imaginary parts: the product includes \(i^2\): \((7i)^2 = -49\) since \(i^2 = -1\).

The combination of these parts results in a real number \(53\), showcasing how effective complex number multiplication can simplify through the use of conjugates and difference of squares.