Find one value of each of the following in \(x+t y\) form. \(i^{23}\)

One of the foundational concepts in complex numbers is understanding the powers of the imaginary unit, represented as \( i \). The imaginary unit \[ i \] is defined by the property \[ i^2 = -1 \]. Given this, we can derive a repeating pattern for higher powers of \[ i \] by repeatedly multiplying by \[ i \]. The pattern for the first few powers of \[ i \] is:

• \[ i^1 = i \]
• \[ i^2 = -1 \]
• \[ i^3 = -i \]
• \[ i^4 = 1 \]

The pattern repeats every four terms. This cyclic behavior helps greatly in simplifying expressions with higher powers of \[ i \]. For example, to find the value of \[ i^{23} \], you can determine the remainder when dividing 23 by 4. Since \[ 23 \div 4 = 5 \] with a remainder of 3, we know that \[ i^{23} \] is equivalent to \[ i^3 \], which equals \[ -i \].

The imaginary unit \( i \) is a fundamental concept in complex numbers. It is not a 'real' number in the traditional sense but is used to extend our number system. By definition, the value of \[ i \] is such that \[ i^2 = -1 \]. This property allows us to write and solve equations that do not have solutions within the set of real numbers. For example, while the equation \[ x^2 = -1 \] has no real solution, it has the solutions \[ x = i \] and \[ x = -i \] in the set of complex numbers.

The use of \[ i \] extends to various fields such as engineering, physics, and computer science, where complex numbers provide convenient ways to solve real-world problems. Understanding the nature of the imaginary unit is crucial for working with complex number operations, including addition, multiplication, and finding powers of \[ i \].

Modular arithmetic is a system of arithmetic for integers, where numbers 'wrap around' upon reaching a certain value—the modulus. In the context of simplifying powers of \( i \), modular arithmetic helps us determine the remainder when an exponent is divided by 4.

Since the powers of \[ i \] repeat every four terms, any power of \[ i \] can be simplified by finding the remainder of the exponent when divided by 4. For example, with \[ i^{23} \], you divide 23 by 4 which gives a quotient of 5 and a remainder of 3. Thus, \[ i^{23} \] simplifies to \[ i^{3} = -i \].

Using modular arithmetic this way reduces the complexity of expressions involving large powers of \[ i \] and helps standardize computations in many mathematical and engineering applications.