Problem 3 Write the given number in the fo... [FREE SOLUTION]

Chapter 17: Problem 3

Write the given number in the form $a+i b$. $$ i^{8} $$

Short Answer

Expert verified

$i^8 = 1\,$ which is $1 + 0i$.

Step by step solution

Understand the powers of i

The imaginary unit $i$ has a cyclical pattern for its powers. Recall that: - $i^1 = i$- $i^2 = -1$- $i^3 = -i$- $i^4 = 1$This pattern repeats for every set of four powers.

Calculate $i^8$

To find $i^8$, notice that 8 is a multiple of 4. Since the pattern repeats every four numbers, we know that $i^4 = 1$. Thus, $i^8 = (i^4)^2 = 1^2 = 1$.

Write $i^8$ in the form $a + ib$

Since $i^8 = 1$, it can be expressed in the form $a + ib$, where $a = 1$ and $b = 0$. Thus, $i^8 = 1 + i \times 0$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Imaginary Unit

The concept of the imaginary unit, denoted as $i$, stems from the need to find roots of equations that have no real solutions, such as $x^2 = -1$. In real numbers, this equation doesn't hold, but the introduction of $i$ allows for a solution. Simply put, the imaginary unit is defined as $i = \sqrt{-1}$.
Using $i$, we can express complex numbers, which have both real and imaginary parts. Imaginary numbers complement the real number system, enabling mathematicians and engineers to solve a wide range of problems.
It鈥檚 crucial to differentiate the imaginary unit from real numbers, keeping in mind that the presence of $i$ indicates an imaginary part of a complex number.

Powers of i

The powers of $i$ exhibit a repeating pattern every four exponentiations. This cyclical behavior stems from the fundamental properties of $i$:

$i^1 = i$
$i^2 = -1$
$i^3 = -i$
$i^4 = 1$

Beyond the fourth power, these values start repeating: $i^5 = i$, $i^6 = -1$, and so on.
This cyclical nature makes it simple to calculate higher powers of $i$. For example, $i^8 = (i^4)^2$, which equals $1^2 = 1$. Recognizing this pattern is crucial for simplifying complex numbers and helps in solving equations involving $i$.

Complex Number Representation

Complex numbers are expressed in the form $a + ib$, where $a$ and $b$ are real numbers. In this format, $a$ represents the real part and $b$ denotes the imaginary part.
To write a pure real number as a complex number, you simply set $b = 0$, making it appear as $a + 0i$. Conversely, a purely imaginary number looks like $0 + bi$.
Understanding this representation helps in performing operations, such as addition and multiplication of complex numbers. When calculating powers as shown with $i^8 = 1$, we express $1$ as $1 + 0i$, adhering to the standard complex number form. This format brings out clarity when dealing with real and imaginary sums in mathematics.

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91影视

Write the given number in the form \(a+i b\). $$ i^{8} $$

Short Answer

Step by step solution

Understand the powers of i

Calculate \(i^8\)

Write \(i^8\) in the form \(a + ib\)

Key Concepts

Imaginary Unit

Powers of i

Complex Number Representation

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