Problem 33
Question
Evaluate the expression and write the result in the form \(a+b i\) $$i^{100}$$
Step-by-Step Solution
Verified Answer
The expression \(i^{100}\) evaluates to \(1+0i\).
1Step 1: Understanding the imaginary unit
The imaginary unit is represented by the symbol \(i\), which is defined as \(\sqrt{-1}\). The powers of \(i\) cycle in a predictable pattern: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), and \(i^4 = 1\). Then the cycle repeats again from \(i^5 = i\).
2Step 2: Determine the pattern cycle length
Since \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), and \(i^4 = 1\), we see the powers of \(i\) repeat every four terms. This is an important observation for simplifying higher powers of \(i\).
3Step 3: Calculate the remainder of the exponent divided by the cycle length
To find \(i^{100}\), we need to find the remainder of 100 when divided by 4 (the length of the pattern cycle). Performing the division, we find \(100 \div 4 = 25\) with a remainder of 0.
4Step 4: Use the cycle pattern to simplify
Since the remainder is 0, \(i^{100}\) corresponds to \(i^4\), which equals 1. Thus, using the cycle pattern of \(i\), \(i^{100} = 1\).
5Step 5: Express the result in the form \(a + bi\)
The result \(1\) can be expressed in the form \(a + bi\) as \(1 + 0i\) where \(a = 1\) and \(b = 0\).
Key Concepts
Imaginary UnitPowers of iComplex Numbers in Standard Form
Imaginary Unit
The imaginary unit, denoted by the symbol \(i\), is a fundamental concept in the field of complex numbers. It is defined as \(i = \sqrt{-1}\). This is a crucial definition because the square root of a negative number isn't a real number, so \(i\) serves as a bridge to understand complex numbers.
The introduction of \(i\) allows mathematicians to solve equations that were previously impossible to solve within the realm of real numbers alone. It forms the simplest type of complex number, which can be expressed as \(0 + 1i\).
The powers of \(i\) cycle through a pattern because each multiplication by \(i\) can be understood in terms of rotation in the complex plane.
Understanding \(i\) is key to unlocking the larger world of complex numbers and provides a new way to express and solve problems in mathematics.
The introduction of \(i\) allows mathematicians to solve equations that were previously impossible to solve within the realm of real numbers alone. It forms the simplest type of complex number, which can be expressed as \(0 + 1i\).
The powers of \(i\) cycle through a pattern because each multiplication by \(i\) can be understood in terms of rotation in the complex plane.
Understanding \(i\) is key to unlocking the larger world of complex numbers and provides a new way to express and solve problems in mathematics.
Powers of i
When dealing with powers of the imaginary unit \(i\), a key point to remember is that it follows a predictable four-cycle pattern. This makes calculations straightforward.
Here's the cyclical pattern of \(i\):
To find higher powers like \(i^{100}\), you simply determine how far 100 goes into the cycle. Since 100 divided by 4 leaves a remainder of 0, \(i^{100}\) corresponds to \(i^4\) in the pattern, which is 1. This simplification process makes calculating powers of \(i\) quick and efficient.
Here's the cyclical pattern of \(i\):
- \(i^1 = i\)
- \(i^2 = -1\)
- \(i^3 = -i\)
- \(i^4 = 1\)
To find higher powers like \(i^{100}\), you simply determine how far 100 goes into the cycle. Since 100 divided by 4 leaves a remainder of 0, \(i^{100}\) corresponds to \(i^4\) in the pattern, which is 1. This simplification process makes calculating powers of \(i\) quick and efficient.
Complex Numbers in Standard Form
Complex numbers are expressed in their standard form as \(a + bi\), where \(a\) and \(b\) are real numbers and \(i\) is the imaginary unit. This structure combines both real and imaginary parts.
For example, a number like \(1 + 0i\) is a complex number, where \(1\) is the real part and \(0i\) is the imaginary part.
Standard form is useful because it visually separates the real component from the imaginary component, making complex numbers simpler to work with.
In the exercise \(i^{100}\), it reduced down to \(1 + 0i\) due to the properties of the powers of \(i\). Expressing it in this form helps to easily combine and compare complex numbers, contributing to the versatility and power of complex numbers in mathematics.
For example, a number like \(1 + 0i\) is a complex number, where \(1\) is the real part and \(0i\) is the imaginary part.
Standard form is useful because it visually separates the real component from the imaginary component, making complex numbers simpler to work with.
In the exercise \(i^{100}\), it reduced down to \(1 + 0i\) due to the properties of the powers of \(i\). Expressing it in this form helps to easily combine and compare complex numbers, contributing to the versatility and power of complex numbers in mathematics.
Other exercises in this chapter
Problem 33
Use transformations of the graph of \(y=\frac{1}{x}\) to graph the rational function, as in Example 2. $$r(x)=\frac{1}{x-1}$$
View solution Problem 33
Factor the polynomial completely, and find all its zeros. State the multiplicity of each zero. $$P(x)=x^{5}+6 x^{3}+9 x$$
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Find all rational zeros of the polynomial, and write the polynomial in factored form. $$P(x)=4 x^{3}+4 x^{2}-x-1$$
View solution Problem 33
Find the maximum or minimum value of the function. $$f(x)=x^{2}+x+1$$
View solution