Problem 94 - College Algebra

Question

Will help you prepare for the material covered in the first section of the next chapter. Evaluate \(i^{2}+1\) for all consecutive integers from 1 to 6 inclusive. Then find the sum of the six evaluations.

Step-by-Step Solution

Answer

The sum of the evaluations is 0.

Step 1: Evaluate \(i^{2}+1\) for each integer from 1 to 6

We know \(i^{2}=-1\). So, \(i^{2}+1=-1+1=0\). As the exponent (2) remains the same for each integer from 1 to 6, the evaluations will also be the same (0) for each integer.

Step 2: Find the sum of the six evaluations

Since each evaluation gives us 0, the sum of all six evaluations will be \(0+0+0+0+0+0=0\).

Key Concepts

Imaginary Unit

The concept of the imaginary unit is a fundamental building block in the field of complex numbers. The imaginary unit is denoted by the symbol \(i\). It is defined by the property \(i^2 = -1\). This property is what distinguishes the imaginary unit from real numbers, as no real number squared gives a negative result. Essentially, the imaginary unit allows us to extend the number system to include solutions to equations like \(x^2 + 1 = 0\). Without \(i\), such equations would have no real solutions. This extension helps in various fields such as engineering, physics, and applied mathematics, where complex numbers are frequently used to solve real-world problems involving oscillations and waves.

Integer Evaluation

Evaluating expressions for integer values is a common task in mathematics. It involves substituting integer values into an expression to find specific results. In the given exercise, we are asked to evaluate the expression \(i^2 + 1\) for integers from 1 to 6. Because \(i^2\) has a fixed value of -1, substituting it into the expression simplifies the calculation considerably.

• For \(i^2 + 1\), we know \(i^2 = -1\).
• This makes \(i^2 + 1 = -1 + 1 = 0\).

This evaluation results in 0 for every integer from 1 to 6, demonstrating how mathematical properties can simplify complex expressions when evaluated at specific values.

Exponentiation

Exponentiation is the process of raising a number to a power. It is a fundamental operation in mathematics that describes repeated multiplication. For example, \(a^n\) means multiplying \(a\) by itself \(n\) times. When it comes to the imaginary unit, \(i\), exponentiation follows specific rules due to the unique property \(i^2 = -1\).

• When \(i\) is raised to subsequent powers, it cycles through a set of values:
• \(i^1 = i\)
• \(i^2 = -1\)
• \(i^3 = -i\)
• \(i^4 = 1\)
• And then the pattern repeats.

This cyclic nature of \(i\)'s powers simplifies calculations in many complex number problems. It helps in predicting the outcome of exponentiating the imaginary unit over multiple operations.

Sum of Evaluations

The sum of evaluations is the process of adding up results from multiple evaluations of an expression. In mathematical exercises, this often helps to explore patterns or confirm consistency across multiple values. In this particular exercise, since the evaluation of \(i^2 + 1\) for each integer from 1 to 6 was the same—0 for each—adding these evaluations is straightforward.

• We performed the evaluation for each integer, getting the same result each time: 0.
• Adding these results together: \(0 + 0 + 0 + 0 + 0 + 0 = 0\).

This approach can reveal whether an expression possibly has a constant output over a range and ensures that the calculations align with theoretical expectations.