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Complex numbers can be a bit magical because they have both real and imaginary parts. In mathematics, a complex number is often represented as a combination of a real number and an imaginary number. When you see something like \(1 + i\), which is the focus of our problem, you should know:

• The number "1" is the real part.
• The "i" represents the imaginary unit, which is defined as the square root of \(-1\).

An essential thing to remember is the property of \(i\), specifically that \(i^2 = -1\). This property is particularly useful when dealing with powers of \(i\) as it helps simplify expressions. Complex numbers are crucial in various fields such as engineering, physics, and even computer science, as they allow us to solve equations that don't have real solutions.

The idea of expansion in mathematics often relates to expressing an equation or expression as a sum of its parts. When we apply the Binomial Theorem for expansion, especially to a complex number like \((1 + i)^4\), we break it down into its components using powers and coefficients derived from its expansion.For instance, using the Binomial Theorem which is: \[(a+b)^n = \sum_{r=0}^{n} \binom{n}{r} a^{n-r} b^{r}\]you can expand \((1+i)^4\) into separate terms. Each term takes the form of \(\binom{4}{r} \cdot 1^{4-r} \cdot i^r\), where \(r\) ranges from 0 to 4. This means:

• The first term is when \(r = 0\): \(\binom{4}{0} \cdot 1^4 \cdot i^0 = 1\)
• The subsequent terms increase the power of \(i\) and decrease the power of 1, using binomial coefficients.

By adding these terms together while remembering that \(i^2 = -1\), you can fully expand and simplify the original expression.

Binomial coefficients are the numerical factors that arise in the expansion of binomials using Pascal's Triangle or while applying the Binomial Theorem. They are typically written as \(\binom{n}{r}\), which is read as "n choose r." This notation signifies the number of ways you can choose \(r\) elements from a set of \(n\) elements, without worrying about order.In the context of expanding \((1+i)^4\), these coefficients help determine the weight or quantity of each term in the expansion formula. Specifically, for our example:

• The coefficients \(\binom{4}{0}, \binom{4}{1}, \binom{4}{2}, \binom{4}{3},\) and \(\binom{4}{4}\) are calculated based on the combinations and are essential for forming the expanded terms.
• These coefficients are determined using the formula: \(\binom{n}{r} = \frac{n!}{r!(n-r)!}\), where \(!\) denotes factorial, a mathematical operation that multiplies a number by all the positive integers below it.

Once computed, these coefficients multiply each involving term and essentially guide the entire process of binomial expansion ensuring that each part of the equation is correctly formed and balanced.