Question

Evaluate each power of \(i\). $$i^{21}$$

Short answer

\(i^{21} = i\)

Step-by-step solution

Understanding the Powers of i

The imaginary unit i is defined by its property that \(i^2 = -1\). When raising i to any power, recall the cyclic nature of these powers: \(i^1 = i\), \(i^2 = -1\), \(i^3 = -i\), and \(i^4 = 1\). Then, this cycle repeats every four powers.

Find the Equivalent Exponent Modulo 4

To simplify \(i^{21}\), first determine the remainder when 21 is divided by 4. This is because the powers of i repeat every 4th power. Calculate the remainder of 21 divided by 4: \(21 \mod 4 = 1\).

Equivalent Power of i

Since \(21 \mod 4 = 1\), \(i^{21}\) is equivalent to \(i^1\) because they both have the same remainder when divided by 4.

Conclude with the Simplified Result

Therefore, \(i^{21} = i^1 = i\).

Key concepts

Imaginary Unit

The concept of the imaginary unit, denoted as 'i', is a fundamental building block in the field of complex numbers. It represents the square root of -1, an operation not possible with real numbers. Without the use of 'i', the solution to equations such as 'x² + 1 = 0' would not be possible as no real number squared results in a negative. By introducing 'i' where 'i² = -1', we unlock a new dimension of numbers referred to as imaginary.

Understanding 'i' is not just about knowing its definition but also about how to manipulate it mathematically. Specifically, when 'i' is raised to successive powers, it follows a predictable pattern: 'i¹ = i', 'i² = -1', 'i³ = -i', and 'i⁴ = 1'. This cycle repeats every four terms, which becomes immensely useful when dealing with higher powers of 'i'.

Exponent Modulo

Calculating Powers Using Modulo

When working with powers of the imaginary unit 'i', a technique known as 'exponent modulo' becomes incredibly helpful. Given the cyclic nature of 'i', we can simplify large exponents by finding their equivalent modulo 4. This basically means that for any power of 'i', such as 'i^n', we only need to determine the remainder 'n mod 4'.

To compute 'n mod 4', divide 'n' by 4 and the remainder is what matters. For example, '5 mod 4' would be 1, as 5 divided by 4 leaves a remainder of 1. This remainder then dictates the equivalent power of 'i' to be used since 'i^1', 'i^2', 'i^3', and 'i^4' represent the full cycle.

Complex Numbers Mathematics

Complex numbers are an extension of the real numbers and have a form that includes both a real and an imaginary part, expressed as 'a + bi'. Here, 'a' represents the real part, and 'bi' is the imaginary part where 'b' is a real number and 'i' is the imaginary unit. Complex numbers are crucial in various fields, for example in signal processing or solving certain algebraic equations.

Adding and Multiplying Complex Numbers

Knowledge of how to operate with 'i' also informs how to add, subtract, multiply, and even divide complex numbers. When adding or subtracting, one simply combines or subtracts the real parts and the imaginary parts separately. Multiplication, however, requires applying the distributive property while bearing in mind that 'i² = -1'.

These principles allow for dealing with powers of 'i' in the context of complex number arithmetic, enriching not just algebra, but areas spanning from engineering to physics.