Question

Find one value of each of the following in $x+ty$ form. $i^{23}$

Short answer

0 + (-1)i

Step-by-step solution

Understand Complex Numbers and Powers of i

The imaginary unit is represented by the symbol i, where $i^2=-1$. Powers of i repeat every four values. Specifically: $$i^1=i,i^2=-1,i^3=-i,i^4=1$$ and then the cycle repeats. This periodicity will help simplify $i^{23}$.

Find the Remainder when Dividing the Exponent by 4

Since the powers of i repeat every 4 terms, we need to find the equivalent power of i within the range 1 through 4. To do this, compute $23÷4$ and find the remainder. $$23÷4=5\cdot 4+3$$ So, the remainder is 3.

Use the Remainder to Determine the Equivalent Power of i

The remainder tells us that $i^{23}$ is equivalent to $i^3$. Based on the cycle mentioned in Step 1, $i^3=-i$.

Express the Answer in the Form x + t\cdot y

Since we have determined that $i^{23}=-i$, it can be expressed in the form $0+(-1)\cdot i$. Thus, $$x=0\text{ and }t=-1,\text{ so }0+(-1)i\text{ is our final answer}$$

Key concepts

powers of i

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 $$23÷4=5$$ with a remainder of 3, we know that $$i^{23}$$ is equivalent to $$i^3$$, which equals $$-i$$.

imaginary unit

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

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.