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Chapter 6: Problem 2

Expand \((1+i)^{2}=?\)

Short Answer

Expert verified
The expansion of \((1+i)^{2}\) is \(2i\).

Step by step solution

01

Binomial Theorem

Use the Binomial Theorem, which states that \( (a+b)^{2} = a^{2} + 2ab + b^{2} \). Hence, for \( (1+i)^{2} \), \( a = 1 \) and \( b = i \).
02

Substituting into the formula

Substitute \( a \) and \( b \) into the formula to get the expanded form, thus it will become \( 1^{2} + 2*1*i + i^{2} \).
03

Simplification

Simplify each term separately. \( 1^{2} = 1 \), \( 2*1*i = 2i \), and most importantly, remember that \( i^{2} = -1 \). Thus, the expression becomes \( 1 + 2i -1 \).
04

Final Answer

Combine like terms to get the final answer. The \( 1 \) and \( -1 \) cancel out, so the answer is \( 2i \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Binomial Theorem
The binomial theorem is a fundamental principle in algebra. It provides a formula to expand expressions that are raised to a power. In simple terms, it helps you break down expressions like \((a+b)^n\) into a manageable sum of terms. For our specific example \((1+i)^2\),
  • The binomial theorem tells us that \((a+b)^2 = a^2 + 2ab + b^2\).
  • By comparing, you can see that both "a" and "b" are parts of our starting expression, \((1+i)\).
  • Here, \(a = 1\) and \(b = i\).
It鈥檚 crucial to understand that this simple formula changes depending on the power n, but the foundational idea鈥攅xpanding into a sum of terms鈥攔emains consistent.
Understanding the binomial theorem is a great starting point in mastering algebra. It holds significance in many mathematical areas and solutions.
Imaginary Unit
The imaginary unit, denoted as \(i\), is a concept from complex numbers used to represent the square root of a negative number. Specifically, \(i\) represents \(\sqrt{-1}\). A major feature of \(i\) is the interesting property that \(i^2 = -1\). This property is crucial in our exercise:
  • When expanding \((1+i)^2\), \(i^2\) becomes \(-1\), which changes our result significantly.
  • This transformation from \(i^2\) to \(-1\) is an essential step that simplifies complex number calculations.
Although it seems abstract, understanding the imaginary unit helps in many fields, such as engineering and physics, where complex numbers are used to solve problems.
Algebraic Expansion
Algebraic expansion transforms expressions into longer ones by removing brackets and combining like terms. In our case of \((1+i)^2\), algebraic expansion plays a key role. Here鈥檚 a breakdown of the process that occurs with our example:
  • The original expression \((1+i)^2\) is expanded using the binomial theorem. Substituting gives \(1^2 + 2 \cdot 1 \cdot i + i^2\).
  • Each term is calculated separately: \(1^2 = 1\), \(2 \cdot 1 \cdot i = 2i\), and \(i^2 = -1\).
  • Combining these results, you simplify to \(1 + 2i - 1 = 2i\).
By following algebraic expansion, we convert complex expressions into simpler forms making mathematical problems easier to solve.

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Most popular questions from this chapter

In Example 6.8, the SFC for a Boeing 787 is \(0.587 \mathrm{lbm} / \mathrm{h} / \mathrm{lbf}\) leading to a fuel consumption rate of \(12,680 \mathrm{lbm} / \mathrm{h}\). If a standard aviation jet fuel (JP-4) costs \( 3.50 / \mathrm{US}\) gallon, has a density of \(6.84 \mathrm{lbm} / \mathrm{US}\) gallon, how much does fuel cost for each hour of level flight?

Simplify: \(\frac{4-i}{7+i}\)

What is \((a+b i)(a-b i)=?\)

A new development in airfoil design is the so called "scimitar winglet" as seen below on a United Airlines airplane. Actually this particular picture shows a double scimitar winglet, one facing upward and one facing downward. Boeing states these winglets save several percent in drag (and thus fuel) by reducing the size of vortices shed from the wingtip in normal flight. On a Boeing 737 these winglets stand as much as \(3.5 \mathrm{~m}\) over the wingtip surfaces. What is the net improvement in lift-over-drag ratio given the unmodified plane had a lift-to-drag ratio of \(17.0\) while the drag has been reduced by \(4.00 \%\) ? Your solution must also compensate for the combined \(220.0 \mathrm{lbm}\) of two winglets. Assume the unmodified weight of the plane was 176,000 lbf.

Simplify: \(\frac{(3+2 i)^{2}}{(-1+3 i)}\)
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