91Ó°ÊÓ

Complex numbers are an extension of the real numbers. They include a real part and an imaginary part. A complex number is usually written in the form \(a + bi\) where \(a\) is the real part and \(b\) is the imaginary part. The imaginary part includes the imaginary unit \(i\), which is defined as the square root of -1. Complex numbers are useful in many areas of mathematics, especially in solving equations that do not have real solutions. For example, the equation \(x^2 + 1 = 0\) has no real solution, but it has complex solutions \(x = i\) and \(x = -i\). Complex numbers obey all the usual rules for arithmetic, but you must remember to include the imaginary unit appropriately.

The imaginary unit, denoted by \(i\), is a fundamental concept in understanding complex numbers. It is defined to be the square root of -1: \(i^2 = -1\). This means that while real numbers are on a continuous line, imaginary numbers are perpendicular to real numbers. When we raise \(i\) to higher powers, these follow a cyclic pattern:

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

This cycle repeats every four powers, which makes calculations involving powers of \(i\) simpler by reducing them modulo 4. For instance, \(i^{30-2k}\) simplifies based on \(30-2k \mod 4\).

Modulo arithmetic is a system of arithmetic for integers, where numbers 'wrap around' after reaching a certain value. The value is called the modulus. For example, in modulo 4 arithmetic, after reaching 3, the numbers start again at 0. This is helpful in simplifying powers of the imaginary unit \(i\). In our problem, we need to compute \(i^{30 - 2k}\), and instead of calculating each power directly, we use the property that powers of \(i\) repeat every four steps:

• \(30 - 2k \mod 4\)

This enables us to convert the exponent into one of the four standard forms, which simplifies our calculations.

The binomial theorem is a fundamental theorem in algebra that describes the expansion of powers of binomials. It states that




\( (x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k \)
where \( \binom{n}{ k} \) is a binomial coefficient. Binomial coefficients have symmetrical properties and are used extensively in combinatorics and probability. In our exercise, the sum involves binomial coefficients \( \binom{15}{k}\) and powers of \(i\). The symmetry is used to simplify calculations, as \( \binom{15}{k} = \binom{15}{15-k}\). This symmetry often helps in reducing the amount of actual arithmetic needed by observing cancellations or patterns in problems featuring sums of binomial coefficients.