Problem 15

Question

Exer. 1-34: Write the expression in the form $a+b i$, where $a$ and $b$ are real numbers. (a) $i^{43}$ (b) $i^{-20}$

Step-by-Step Solution

Verified

Answer

(a) $i^{43} = -i = 0 - i$. (b) $i^{-20} = 1 = 1 + 0i$.

1Step 1: Understand Powers of i

The imaginary unit $i$ has a special property: $i^2 = -1$. This leads to cyclical powers: $i^3 = i^2 \times i = -i$, $i^4 = i^3 \times i = 1$. Then it repeats: $i^5 = i$. Thus any power of $i$ can be simplified using these identities.

2Step 2: Simplify i^43

To simplify $i^{43}$, notice that $43 \div 4 = 10$ remainder $3$. Therefore, $i^{43} = (i^4)^{10} \times i^3 = 1^{10} \times i^3 = i^3 = -i$.

3Step 3: Result of Part (a)

Given the simplification, $i^{43} = -i$. Hence, in the form $a + bi$, where $a$ and $b$ are real numbers: $a = 0$ and $b = -1$.

4Step 4: Simplify i^-20

To simplify $i^{-20}$, use the reciprocal identities: $i^{-1} = \frac{1}{i} = -i$. Similarly, $i^{-2} = \frac{1}{i^2} = -\frac{1}{-1} = 1$, $i^{-3} = \frac{1}{i^3} = i$, and $i^{-4} = \frac{1}{i^4} = 1$. Notice the negative powers also repeat every four. Since $-20$ divided by $4$ leaves a remainder of $0$, $i^{-20} = (i^{-4})^5 = 1^5 = 1$.

5Step 5: Result of Part (b)

Therefore, $i^{-20} = 1$. In the form $a + bi$, $a = 1$ and $b = 0$.

Key Concepts

Powers of Imaginary UnitAlgebraReal and Imaginary Components

Powers of Imaginary Unit

The imaginary unit, denoted as $i$, plays a crucial role in complex numbers. Primarily, it is defined by the equation $i^2 = -1$. This definition results in a repeating cycle every four powers:

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

After $i^4 = 1$, the cycle restarts, so $i^5 = i$, $i^6 = i^2 = -1$, and so on. Understanding this cyclical nature allows us to simplify any power of $i$ by reducing the exponent modulo 4. For instance, to find $i^{43}$, we find the remainder of 43 divided by 4, which is 3. Hence, $i^{43} = i^3 = -i$. Similarly, negative powers follow the same pattern, with values like $i^{-1} = \frac{1}{i} = -i$, showing that they also cycle every four steps.

Algebra

Algebra is a branch of mathematics dealing with symbols and the rules for manipulating those symbols. In the context of the imaginary unit $i$, algebra helps us succinctly express and solve equations involving complex numbers. Utilizing algebraic techniques such as simplifying expressions, recognizing patterns, and applying identities is vital for dealing with powers of $i$.Simplifying expressions like $i^{43}$ and $i^{-20}$ involves recognizing their positions within the sequence of repeated cycles discussed earlier. Algebra allows us to group terms, apply modular arithmetic, and employ arithmetic identities. By understanding these patterns and rules, complex powers become more approachable.

Real and Imaginary Components

Complex numbers are numbers that have both a real and an imaginary component, usually expressed in the form $a + bi$, where $a$ and $b$ are real numbers, and $i$ represents the imaginary unit. For any expression involving $i$, such as $i^{43}$ and $i^{-20}$, your task is to simplify it to this standard form.For $i^{43}$, which simplifies to $-i$, this can be rewritten as $0 + (-1)i$, where: