Problem 67 - Algebra and Trigonometry Real Mathematics, Real People

Question

Simplify the complex number and write it in standard form. $$-6 i^{3}+i^{2}$$

Step-by-Step Solution

Answer

So, the simplified form of the given complex number in standard form is \(6i-1\).

Step 1: Simplify \(i^{3}\)

In the imaginary unit \(i\), if \(i\) is raised to the power of 3, then it becomes \(-i\). So for \(-6i^{3}\), replace \(i^3\) with \(-i\) to get \(6i\).

Step 2: Simplify \(i^2\)

In the imaginary unit \(i\), if \(i\) is raised to the power of 2, then it becomes \(-1\). So for \(i^2\), replace \(i^2\) with \(-1\). This results in \(-1\).

Step 3: Combine

Combine \(6i\) and \(-1\) to get \(6i-1\).

Key Concepts

Imaginary Unit

The imaginary unit, denoted by \(i\), is a mathematical concept that extends the real number system to include the square root of negative one. The fundamental property of \(i\) is that \(i^2 = -1\). This means when \(i\) is squared, it equals ÷= -1. This property is crucial because it allows for the definition and manipulation of complex numbers, which are numbers that have both real and imaginary parts.

• The sequence of powers of \(i\) cycles every four: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), and \(i^4 = 1\).
• This cyclical property helps simplify expressions involving powers of \(i\). For instance, \(i^3\) simplifies to \(-i\) and \(i^2\) simplifies to \(-1\).

Understanding these properties is essential for working with complex numbers and simplifying expressions that involve the imaginary unit.

Exponentiation

Exponentiation involving complex numbers, especially those with the imaginary unit \(i\), adheres to different rules compared to real numbers due to the cyclical nature of \(i\). It's important to comprehend how the powers of \(i\) influence numbers:

• Raising \(i\) to an integer power leads to a predictable pattern. For example, \(i^3 = -i\) because \(i^2 = -1\) and multiplying by another \(i\) gives \(-i\).
• Exponentiation is essential for simplifying complex expressions. For instance, in the expression \(-6i^3 + i^2\), understanding the powers of \(i\) allows for converting it into an expression of standard form.

By recognizing the power cycle of \(i\), calculations become much simpler, and complex expressions can be easily transformed and understood.

Standard Form in Complex Numbers

The standard form of a complex number is crucial for clarity and is conventionally written as \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. Here are the steps to achieving this form:

• To write a complex number in standard form, identify and simplify both the real and imaginary components.
• For example, in the expression \(-6i^3 + i^2\), you simplify the powers of \(i\) first so you have distinct real and imaginary parts.
• Once simplified, the expression turns into \(6i - 1\), or rearranged, \(-1 + 6i\), clearly showing the real and imaginary components in standard form.

This format is beneficial because it provides a clear and standard way to present complex numbers, making them easier to interpret and utilize in further mathematical computations.