91Ó°ÊÓ

A complex number is like a binomial consisting of a real part and an imaginary part. The standard form of a complex number is written as \(a + bi\), where \(a\) represents the real part and \(b\) represents the imaginary part, with \(i\) being the imaginary unit. This unit, \(i\), is defined as the square root of -1. For instance, in the given exercise, we encounter the expression \( -5 i^{5} \). To bring this expression into standard form, we need to focus on simplifying the power of \(i\).

As illustrated in the step-by-step solution, the power of \(i\) repeats according to the established cycle every four powers. Once reduced, \(i^5\) simplifies to \(i\). This is a crucial point to understand as it frequents in complex number operations. With \(i\) in its simplest form, the expression -5i can now be seen in the standard form, where the real part is 0 (\(a = 0\)) and the imaginary part is -5 (\(b = -5\)). We would express it simply as \( -5i \) or \( 0 - 5i \) to reflect the absent real part.

The concept of complex conjugates is critical in simplifying and solving complex number expressions. A complex conjugate is created by changing the sign of the imaginary part of the original complex number. For any complex number \(a + bi\), its conjugate is written as \(a - bi\).

The proverbial magic lies in the property that when a complex number is multiplied by its conjugate, the result is always a non-negative real number. This is very useful when we want to eliminate the imaginary part from the denominator of a complex fraction. In our example, the complex number \( -5i \) becomes \( 5i \) once we take its conjugate by altering the sign of the imaginary component. This process expands the realm of possible operations with complex numbers, such as division and finding absolute values, making complex conjugates an essential part of complex number arithmetic.

Familiarity with the powers of the imaginary unit \(i\) significantly simplifies working with complex numbers in various mathematical contexts. We have the following cyclic pattern for the powers of \(i\):

• \(i^1 = i\)
• \(i^2 = -1\)
• \(i^3 = -i\)
• \(i^4 = 1\)

After \(i^4\), the pattern repeats starting back at \(i\). Understanding this pattern helps in quickly determining the value of \(i\) raised to any power. In practice, you divide the exponent by 4 and use the remainder to find the corresponding value of \(i\) in the cycle.

For example, for \(i^5\), we know that \(i^4\) is 1 and so \(i^5 = i^4\times i^1 = 1\times i = i\). This recurring cycle allows us to convert any power of \(i\) into one of the four fundamental values within the cycle, thereby assisting in complex number calculations and making seemingly complicated expressions much more manageable.