Problem 12 - Advanced Engineering Mathematics

Question

In Problems 1-26, write the given number in the form \(a+i b\). $$ (1-i)^{3} $$

Step-by-Step Solution

Answer

The expression \((1-i)^3\) simplifies to \(-2 + 2i\).

Step 1: Expand the Expression

We need to expand \((1 - i)^3\) using the binomial theorem. The binomial theorem for \((a - b)^n\) is given by:\[(a - b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} (-b)^k\]For \((1 - i)^3\), let \(a = 1\) and \(b = i\), we get:\[(1 - i)^3 = \binom{3}{0} 1^3 - \binom{3}{1} 1^2 i + \binom{3}{2} 1 i^2 - \binom{3}{3} i^3\]

Step 2: Calculate Each Term

Calculate each term using the expression obtained in Step 1:- The first term is \( \binom{3}{0} 1^3 = 1 \).- The second term is \( \binom{3}{1} 1^2 i = 3i \).- The third term is \( \binom{3}{2} 1 i^2 = 3(-1) = -3 \) because \(i^2 = -1\).- The fourth term is \( \binom{3}{3} i^3 = -i \) because \(i^3 = i^2 imes i = -i \).

Step 3: Combine the Terms

Combine all the terms obtained from Step 2:\[ 1 - 3 + 3i - i \] First, combine the real parts: \( 1 - 3 = -2 \).Then, combine the imaginary parts: \( 3i - i = 2i \).So, the expression simplifies to: \( -2 + 2i \).

Step 4: Write in the Form \(a + bi\)

We have derived the simplified expression \(-2 + 2i\). This is already in the form \(a + bi\), where \(a = -2\) and \(b = 2\).

Key Concepts

Binomial Theorem

The Binomial Theorem is a powerful mathematical tool used to expand expressions that are raised to a power. It is especially useful in algebra when dealing with expressions like \((a - b)^n\).

The general form of the Binomial Theorem is given by:

• \[(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k\]

Here, \(\binom{n}{k}\) is the binomial coefficient, which calculates the number of ways to choose \(k\) items from \(n\) items. The expression \(a^{n-k}\) represents the power of \(a\) decreasing with each term, while \(b^k\) represents the power of \(b\) increasing.

In the example \((1 - i)^3\), we use this theorem to expand the expression. Letting \(a = 1\) and \(b = i\), we systematically apply the coefficients and powers to find the expanded form. This technique not only helps with calculations but also gives a clear structured approach to expansions.

Imaginary Unit

The imaginary unit, denoted as \(i\), is fundamental in dealing with complex numbers. It is defined by the equation \(i^2 = -1\).

In mathematics, the existence of \(i\) allows us to solve equations that do not have real number solutions, such as \(x^2 + 1 = 0\). The introduction of \(i\) gives us complex numbers of the form \(a + bi\), where \(a\) and \(b\) are real numbers.

Here are some properties of \(i\) that are useful:

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

These properties repeat in cycles of four, making calculations with powers of \(i\) predictable and manageable. Knowing these can simplify complex equations and help to convert expressions into the standard form \(a + bi\).

Algebraic Expansion

Algebraic expansion is a process used to simplify expressions by breaking them down into multiple terms. This is essential for solving and simplifying equations.

With algebraic expansion, we distribute each term over others, using properties like the distributive property, which states that \( a(b + c) = ab + ac \). When expanding expressions with multiple terms raised to a power, it's important to maintain careful attention to each step.

In the problem \((1-i)^3\), the expression is expanded using the Binomial Theorem, where each term is calculated individually before combining them all together. This step-by-step expansion helps manage complex expressions and is a standard technique in algebra.

• Focus first on calculating individual terms.
• Apply known algebraic identities to simplify.
• Combine like terms to find the solution.

This structured approach makes it easier to follow through with calculations and to understand how the expression simplifies.

Exponents in Algebra

Exponents play a fundamental role in algebra, allowing for the expression of numbers and variables raised to a power. Exponents indicate the number of times a number is multiplied by itself.

The general rules of exponents are simple but crucial:

• \(a^m \cdot a^n = a^{m+n}\)
• \(a^0 = 1\) (where \(aot=0\))
• \(a^{-n} = \frac{1}{a^n}\)

In our problem, exponents are applied when expanding \((1-i)^3\), where the power of \(3\) indicates multiplication three times.

This step involves careful calculation of exponents, including both the real and imaginary components, to ensure accurate simplification of the complex number. Understanding and correctly applying exponent rules help pave the way for solving more complex algebraic equations, enabling students to tackle a wider range of problems.