Question

Given that \(i=\sqrt{-1}\), what is the value of \(i^{4}+i^{12}\) ? (A) \(\sqrt{-1}\) (B) \(-1\) (C) 1 (D) 2

Short answer

The value of \(i^4 + i^{12}\) is 2 (Option D).

Step-by-step solution

Understanding powers of i

The imaginary unit, denoted as \(i\), is defined by the property \(i^2 = -1\). From this, we derive the cyclical pattern of powers of \(i\): - \(i^1 = i\) - \(i^2 = -1\) - \(i^3 = -i\) - \(i^4 = 1\) This pattern repeats every four powers, so understanding this cycle helps in evaluating higher powers of \(i\).

Evaluate i^4

Since \(i^4 = 1\), and the cycle of \(i\)'s powers repeats every four, any power that is a multiple of 4 will equal 1.

Evaluate i^12

Notice \(i^{12} = (i^4)^3\) because \(12 = 4 \times 3\). Since \(i^4 = 1\), we have: \((i^4)^3 = 1^3 = 1\). Hence, \(i^{12} = 1\).

Add the results

Now, add the values obtained: \(i^4 + i^{12} = 1 + 1 = 2\).

Choose the correct option

The sum of \(i^4 + i^{12}\) is 2. Therefore, the correct answer is \((D) 2\).

Key concepts

Understanding Imaginary Numbers

Imaginary numbers often intrigue and puzzle due to their unique properties. Imaginary numbers are built around the imaginary unit, denoted as \(i\). This unit is defined by the square root of -1, \(i = \sqrt{-1}\). In reality, we can’t have the square root of a negative number using conventional real numbers. Imaginary numbers come to the rescue here, enabling the solutions of equations that would otherwise be impossible to solve in the realm of real numbers.

Imaginary numbers are essential in various fields, from engineering to complex calculations in physics. They offer a numerical solution where traditional numbers fall short. Understanding \(i\) and its implications opens new doors in mathematics, including dealing with complex numbers, which we will explore next. So, whenever you encounter \(i\), think of it as a gateway to a broader mathematical universe.

Exploring Complex Numbers

To grasp complex numbers, imagine merging real and imaginary numbers. A complex number takes the form \(a + bi\), where \(a\) is the real part, and \(bi\) is the imaginary part. For instance, with the complex number 3 + 4i, 3 represents the real component, and 4i represents the imaginary component.

This combination allows complex numbers to solve equations where real numbers alone wouldn't suffice, significantly expanding the scope of mathematical problem-solving. By accommodating negative square roots and offering a new dimension of understanding, complex numbers simplify calculations across science and engineering, where such numbers are often employed.

Complex numbers also help in visualizing mathematical concepts, as they can be represented graphically on the complex plane, where the x-axis denotes the real part and the y-axis represents the imaginary part, forming a comprehensive landscape of solutions.

Cyclical Pattern in Powers of i

The powers of \(i\) follow a neat cyclical pattern every four numbers, making it easier to manage and predict higher powers without direct calculation:

• \(i^1 = i\)
• \(i^2 = -1\)
• \(i^3 = -i\)
• \(i^4 = 1\)

Once you understand this four-step cycle, it becomes easy to reduce any power of \(i\) to a simpler form. For instance, if you wish to find \(i^{12}\), recognize that 12 is a multiple of 4. Therefore, \(i^{12} = (i^4)^3 = 1^3 = 1\).

This cyclical behavior is not just a mathematical trick; it is the key to simplifying problems in complex arithmetic. It emerges from the unique definition of \(i\) and how its powers interact with each other. Grasping this routine cycle can save time and confusion, allowing for more efficient problem-solving in the realm of complex numbers.