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Magazine Chapter 17: Problem 12
Write the given number in the form \(a+i b\). $$ (1-i)^{3} $$
Short Answer
Expert verified
The expression \((1-i)^3\) simplifies to \(-2-2i\).
Step by step solution
01
Expand the Expression
To find the cube of a binomial, we use the binomial theorem formula: \[ (x - y)^3 = x^3 - 3x^2y + 3xy^2 - y^3 \]In this context, let \( x = 1 \) and \( y = i \), so the expression becomes:\[ (1 - i)^3 = 1^3 - 3(1)^2(i) + 3(1)(i)^2 - i^3 \]
02
Compute the Terms
Let's compute each term individually:1. \( 1^3 = 1 \)2. \( -3(1)^2(i) = -3i \)3. \( 3(1)(i)^2 = 3(i^2) \) where \( i^2 = -1 \), so this becomes \( 3(-1) = -3 \)4. \( -i^3 = -(-i) \) because \( i^3 = -i \). Thus, this becomes \( i \).
03
Simplify the Expression
Combine all the terms computed in Step 2:\[ 1 - 3i - 3 + i \]Combine like terms:- The real parts: \( 1 - 3 = -2 \)- The imaginary parts: \( -3i + i = -2i \)Thus, the expression simplifies to \( -2 - 2i \).
04
Express in Standard Form
Ensure the solution is in the form \( a + ib \), where \( a \) and \( b \) are real numbers. From the simplification, the expression is:\[ -2 - 2i \]This is already in the form \( a + ib \), where \( a = -2 \) and \( b = -2 \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Binomial Theorem
To understand how to expand expressions like (1-i)^3, we use the Binomial Theorem. This theorem is a powerful tool in algebra that helps to expand expressions raised to a power, especially binomials, which are expressions with two terms. It states that for any positive integer n, (x + y)^n can be expanded using the sum of terms \( \binom{n}{k} x^{n-k} y^k \) for k ranging from 0 to n.
- The coefficients \( \binom{n}{k} \) are calculated using combinations, often referred to as binomial coefficients. These give us the number of ways to select k items from a total of n.
- The pattern follows Pascal's Triangle, where coefficients are arranged in a triangle with each number being the sum of the two directly above it.
Imaginary Unit
The imaginary unit, denoted by i, is a fundamental element in complex number theory. It is defined such that \( i^2 = -1 \). This property allows us to extend the real number system to include numbers that have a real and imaginary part. When working with i, it鈥檚 crucial to remember its cyclical nature:
- \( i^0 = 1 \) 鈥� The zeroth power of i is 1, just like any number.
- \( i^1 = i \) 鈥� The first power is the imaginary unit itself.
- \( i^2 = -1 \) 鈥� By definition of the imaginary unit.
- \( i^3 = -i \) 鈥� Derived from multiplying \( i^2 \) by i.
- \( i^4 = 1 \) 鈥� Cycles back to one, repeating every four powers.
Polynomial Expansion
When you expand a polynomial, like a binomial raised to a power, you break it down into simpler terms. Each term of the polynomial is a simpler expression that separately contributes to the whole.
For the expression \((1-i)^3\), using the binomial expansion method involves realizing:
For the expression \((1-i)^3\), using the binomial expansion method involves realizing:
- A polynomial is made up of several terms. For example, \((x-y)^3\) expands to four terms.
- Each term is made up of a combination of the original terms raised to varying powers, reflecting their contribution to the expansion based on the binomial theorem coefficients.
- The expansions incorporate changes due to the imaginary unit, reflecting how imaginary powers interact with real terms and each other to adjust their signs and magnitudes.
