Problem 89 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the Binomial Theorem to expand the complex number. Simplify your result. (Remember that \(i=\sqrt{-1 .})\) \((1+i)^{4}\)

Step-by-Step Solution

Answer

-4 - 8i

Step 1: Apply Binomial Theorem

First, to apply the Binomial Theorem to expand \((1+i)^{4}\) we need to remember that the Binomial Theorem states that \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\) . Then by simply substituting \(a=1\), \(b=i\), and \(n=4\) into the formula, the result will be \((1+i)^4 = \binom{4}{0} \times 1^{4} \times i^{0} + \binom{4}{1} \times 1^{3} \times i^{1} + \binom{4}{2} \times 1^{2} \times i^{2} + \binom{4}{3} \times 1^{1} \times i^{3} + \binom{4}{4} \times 1^{0} \times i^{4}\).

Step 2: Calculate Binomial Coefficients and Powers of i

Next, calculate the binomial coefficients and the powers of i. The binomial coefficient can be calculated using the formula \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\). Since \(i^2=-1\), \(i^3=-i\), and \(i^4=1\), replace these powers of i in the previous equation and the result ends up as \(1 - 4i - 6 - 4i + 1\).

Step 3: Simplify the expression

Finally, simplify the expression by summing the real and imaginary parts separately. Thus, \(1 - 4i - 6 - 4i + 1 = -4 - 8i\).

Key Concepts

Complex Numbers

Complex numbers are an exciting and fascinating concept in mathematics. These numbers consist of two parts: a real part and an imaginary part.
Imaginary numbers are formed by the square root of negative numbers, represented as multiples of the imaginary unit, denoted by \(i\). The basic definition of \(i\) is that it equals \(\sqrt{-1}\). This means that \(i^2 = -1\). Complex numbers are typically written in the form \(a + bi\), where \(a\) is the real part, and \(bi\) is the imaginary part.
They play a crucial role in various branches of mathematics and engineering, particularly in dealing with signals and waves. Understanding complex numbers allows you to do operations like addition, subtraction, and multiplication easily, similar to working with polynomials. When working with these numbers, always remember to handle the imaginary unit, \(i\), carefully, especially in terms of its powers and what they represent.

Binomial Expansion

The Binomial Theorem provides a powerful way to expand expressions in the form of \((a+b)^n\). Using this theorem, you can break down the polynomial into smaller, manageable parts.
This is essential in expanding expressions like \((1+i)^4\). According to the theorem,

• \((a+b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\)
• The notation \(\binom{n}{k}\) represents a binomial coefficient, which determines how many ways you can choose \(k\) elements from \(n\).
• In our expression, substitute \(a = 1\), \(b = i\), and \(n = 4\).

By expanding using the formula provided by the Binomial Theorem, the expression transforms into a sum involving terms with different powers of \(i\). This allows us to calculate each part separately before simplifying the final result. Mastery of the Binomial Theorem is indispensable in handling polynomial expressions and is widely applicable in higher-level math topics.

Powers of i

The powers of \(i\) follow a specific repeating pattern that is key to simplifying expressions involving complex numbers. Knowing this pattern helps immensely in calculations.

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

The cycle repeats every four powers, which means that for any positive integer \(n\), \(i^n\) can be found by considering \(n \mod 4\) to determine which number in this series it corresponds to.
In our exercise, this pattern simplifies working with complex numbers using the Binomial Theorem. For example, when \(i^2\) appears, you immediately know it equals \(-1\). This fundamental understanding allows for the simplification of terms with higher powers of \(i\), making it easier to add up complex expressions effectively. Remembering these cycles reduces mistakes and streamlines calculations involving complex numbers.