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Chapter 7: Problem 90

Find each power of i. $$i^{102}$$

Short Answer

Expert verified
i^{102} = -1

Step by step solution

01

Understanding Powers of i

Recall that powers of the imaginary unit i cycle every four exponentiations: i^1 = i, i^2 = -1, i^3 = -i, i^4 = 1. This pattern repeats every four powers.
02

Find the Equivalent Exponent Modulo 4

To find the value of i^{102}, calculate the remainder when 102 is divided by 4. Perform the calculation: 102 梅 4 = 25 remainder 2. Therefore, 102 鈮� 2 (mod 4).
03

Determine the Corresponding Power of i

Using the remainder from Step 2, identify the corresponding power of i. Since 102 modulo 4 is 2, this corresponds to i^2 in the cycle. We know that i^2 = -1.

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Key Concepts

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

Modular Arithmetic
Modular arithmetic is a system of arithmetic for integers, which considers numbers 'wrapping around' after reaching a certain value, known as the modulus. Imagine a clock where after 12 hours, it returns to 1. That鈥檚 a simple way to think about modular arithmetic, with 12 as the modulus.
Here鈥檚 how it works:
  • Perform the usual arithmetic operation (addition, subtraction, multiplication, etc.).
  • Then take the remainder when the result is divided by the modulus.
For instance, to solve 102 modulo 4:
  • Divide 102 by 4.
  • The quotient is 25 with a remainder of 2.
Thus, 102 鈮� 2 (mod 4). Understanding this equivalence is key when working with powers of complex numbers, like the imaginary unit i.
Complex Numbers
A complex number is a number of the form a + bi, where a and b are real numbers, and i is the imaginary unit, satisfying i^2 = -1. Complex numbers extend the idea of one-dimensional real numbers to two dimensions, allowing for the solution of equations that real numbers alone cannot solve.
Every complex number has a real part (a) and an imaginary part (b).
For example, in the complex number 3 + 4i:
  • 3 is the real part.
  • 4i is the imaginary part.
The concept of complex numbers is fundamental in fields like engineering, physics, and applied mathematics, providing a more complete picture of number systems.
Imaginary Unit i
The imaginary unit i is defined as the square root of -1. This makes it possible to extend the real number system to include solutions to equations like x^2 = -1, which have no real solutions.
To work with powers of i effectively, we use the cyclic nature of its powers:
  • i^1 = i
  • i^2 = -1
  • i^3 = -i
  • i^4 = 1
Since these results repeat every four exponentiations, calculating higher powers of i, such as i^{102}, involves reducing the exponent modulo 4. So, i^{102} is the same as i^2, which equals -1. Recognizing this pattern simplifies dealing with complex numbers in algebra and beyond.

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Most popular questions from this chapter

Work each problem. Suppose someone claims that \(\sqrt[n]{a^{n}+b^{n}}\) must equal \(a+b,\) because when \(a=1\) and \(b=0,\) a true statement results: $$\sqrt[n]{a^{n}+b^{n}}=\sqrt[n]{1^{n}+0^{n}}=\sqrt[n]{1^{n}}=1=1+0=a+b$$ Explain why this is faulty reasoning.

Rationalize each denominator. Assume that all radicals represent real numbers and that no denominators are \(0 .\) $$ \frac{5}{\sqrt{m-n}} $$

Find the distance between each pair of points. \((\sqrt{2}, \sqrt{6})\) and \((-2 \sqrt{2}, 4 \sqrt{6})\)

Write each radical as an exponential and simplify. Leave answers in exponential form. Assume that all variables represent positive numbers. $$ \sqrt[5]{x^{3}} \cdot \sqrt[4]{x} $$

Simplify. Assume that all variables represent positive real numbers. \(\sqrt{144 x^{3} y^{9}}\)
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