Problem 42 - Algebra 2 and Trigonometry

Question

In \(35-43,\) write each number in simplest form. $$ i+i^{2}+i^{3}+i^{4} $$

Step-by-Step Solution

Answer

0, because each term cancels out.

Step 1: Understand the Imaginary Unit i

The imaginary unit is denoted by the symbol \(i\), and it is defined by the property \(i^2 = -1\). This means that any power of \(i\) can be expressed in terms of a power that is easier to manage.

Step 2: Calculate Each Term

Each term of the expression can be expressed based on the known properties of \(i\).- \(i^1 = i\)- \(i^2 = -1\) (by definition of \(i\))- \(i^3 = i^2 \cdot i = (-1) \cdot i = -i\)- \(i^4 = i^2 \cdot i^2 = (-1) \cdot (-1) = 1\)

Step 3: Substitute and Simplify

Substitute the simplified terms back into the original expression:\(i + i^2 + i^3 + i^4 = i + (-1) + (-i) + 1\).Combine like terms:- "\(i\)" and "\(-i\)" cancel each other out.- "\(-1\)" and "\(+1\)" cancel each other out.Thus, the simplified sum is 0.

Key Concepts

Powers of i

The key to understanding complex expressions involving the imaginary unit lies in mastering the powers of \( i \). The imaginary unit \( i \) is defined as the square root of \(-1\), which establishes the foundational rule \( i^2 = -1 \). This simple fact opens up a fascinating pattern as we progress through successive powers of \( i \):

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

Notice how these powers repeat every four steps, forming a cycle: \( i, -1, -i, \) and \( 1 \). This cyclical behavior is a powerful property as it helps you simplify complex expressions swiftly by reducing any higher power of \( i \) to one of these four basic forms. Recognizing this cycle saves both time and effort while solving multiple problems involving powers of \( i \).

Complex Numbers

Complex numbers extend the idea of regular numbers by including the imaginary unit \( i \). A complex number is usually written in the form \( a + bi \), where \( a \) and \( b \) are real numbers. Here's a friendly way to think about complex numbers:

• If only the \( a \) part exists (i.e., \( b = 0 \)), you are simply dealing with a real number.
• If only the \( b \) part exists (i.e., \( a = 0 \)), it's a purely imaginary number.
• When both \( a \) and \( b \) are non-zero, you have a true complex number.

Complex numbers are plotted on a plane with the real part along the horizontal axis and the imaginary part along the vertical axis, making it easy to visualize their relationships and operations like addition and subtraction. Understanding complex numbers is fundamental not only in mathematics but also in many practical fields like engineering and physics, aiding in solving real-world problems.

Simplifying Expressions

Simplifying expressions, especially those involving complex numbers and the imaginary unit, is all about reducing them to their simplest form. Take each part of an expression, understand its properties—like those derived from the powers of \( i \)—and perform basic arithmetic operations. In the expression \( i + i^2 + i^3 + i^4 \), this simplification process involves identifying each power of \( i \) using known values:

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

Substitute these values back in and carefully combine like terms. Here, \( i \) and \( -i \) cancel out, as well as \( -1 \) and \( 1 \), resulting in the expression simplifying neatly to \( 0 \). Techniques like this ensure clarity and precision, leading to correct and concise solutions. Such methods are valuable across mathematics and its application domains, offering pathways to solve even the most complex equations.