/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 64 Find each quotient. $$ \frac... [FREE SOLUTION] | 91影视

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Chapter 10: Problem 64

Find each quotient. $$ \frac{2}{1+i} $$

Short Answer

Expert verified
\(1-i\)

Step by step solution

01

Multiply by the Conjugate

To simplify the given quotient \(\frac{2}{1+i}\), multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(1+i\) is \(1-i\). So, multiply by \( \frac{1-i}{1-i} \): \(\frac{2}{1+i} \times \frac{1-i}{1-i} = \frac{2(1-i)}{(1+i)(1-i)}\).
02

Simplify the Denominator

Simplify the denominator by using the difference of squares formula \( (a+b)(a-b) = a^2 - b^2 \). For \( (1+i)(1-i) \), the formula gives \( 1^2 - i^2 = 1 - (-1) = 2 \): \(\frac{2(1-i)}{2} \).
03

Simplify the Fraction

Simplify the fraction by dividing the numerator by the denominator: \(\frac{2(1-i)}{2} = 1-i \).

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Key Concepts

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

Complex Numbers
Understanding complex numbers is key to solving problems like the given quotient. Complex numbers have a real part and an imaginary part. For example, in the number \(1 + i\), \1\ is the real part and \i\ is the imaginary part. You can think of the imaginary unit \(i\) as representing the square root of \{-1}\. This means \(i^2 = -1\).
In general, a complex number is in the form \(a + bi\), where \a\ and \b\ are real numbers. When working with complex numbers, you often need to perform operations such as addition, subtraction, multiplication, and division. For division, like in our exercise, you use the concept of rationalization with the conjugate.
Rationalization
Rationalization involves making a complex fraction simpler, usually by eliminating the imaginary part from the denominator. This process uses the conjugate of the denominator. The conjugate of a complex number \(a + bi\) is \(a - bi\).
Here鈥檚 how rationalization works in the given problem \(\frac{2}{1+i}\):
  • First, identify the conjugate of the denominator. For \(1 + i\), the conjugate is \(1 - i\).
  • Next, multiply both the numerator and the denominator by this conjugate to form a fraction equal to one. This ensures the expression鈥檚 value doesn鈥檛 change but allows simplification.
So, we have \(\frac{2}{1+i} \times \frac{1-i}{1-i} = \frac{2(1-i)}{(1+i)(1-i)}\). Don't forget to apply the difference of squares formula to simplify the denominator!
Conjugate in Algebra
The notion of a conjugate is crucial in algebra, especially when dealing with complex numbers. The conjugate of a complex number \(a + bi\) is \(a - bi\). This simple switch of the sign before the imaginary part helps remove the imaginary unit from the denominator during rationalization.
Using the conjugate makes denominators real numbers. In the exercise, multiplying by \(1 - i\) yields:
  • Simplify the denominator using \( (a + bi)(a - bi) = a^2 - (bi)^2 \). For \( (1+i)(1-i) \), you get \(1^2 - i^2 = 1 - (-1) = 2\). So, \(\frac{2(1-i)}{2} = 1-i\).
Rationalized fractions are easier to interpret and use in further computations. Ready for more exercises?

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