Problem 20
Question
Write the expression in the form \(a+b i,\) where \(a\) and \(b\) are real numbers. $$\frac{5}{2-7 i}$$
Step-by-Step Solution
Verified Answer
The expression is \( \frac{10}{53} + \frac{35}{53}i \).
1Step 1: Identify the task
We need to write the expression \( \frac{5}{2-7i} \) in the form \( a + bi \), where \( a \) and \( b \) are real numbers. This involves simplifying the complex fraction.
2Step 2: Determine the conjugate
The denominator is \( 2-7i \). The conjugate is \( 2+7i \). We'll multiply both the numerator and the denominator by this conjugate to eliminate the imaginary part in the denominator.
3Step 3: Multiply numerator and denominator by the conjugate
Multiply \( 5 \) and \( 2-7i \) by the conjugate \( 2+7i \): \[\frac{5(2+7i)}{(2-7i)(2+7i)}\] This results in a fraction involving the product of two complex numbers.
4Step 4: Apply the difference of squares
The denominator \( (2-7i)(2+7i) \) is the difference of squares: \[4 - (7i)^2 = 4 + 49 = 53\] Thus, the denominator simplifies to \( 53 \).
5Step 5: Distribute in the numerator
Distribute the 5 in the numerator: \[5(2+7i) = 10 + 35i\]
6Step 6: Construct the simplified form
Combine the results from Steps 4 and 5 to write the complex number over the real denominator: \[\frac{10 + 35i}{53}\]
7Step 7: Split into real and imaginary components
Rewrite the expression as separate fractions for the real and the imaginary parts: \[\frac{10}{53} + \frac{35}{53}i\] This is in the form \( a + bi \), with \( a = \frac{10}{53} \) and \( b = \frac{35}{53} \).
Key Concepts
Conjugate of Complex NumbersSimplifying Complex FractionsImaginary Unit
Conjugate of Complex Numbers
The concept of conjugates is crucial when dealing with complex numbers. Given a complex number in the form of \( a + bi \), the conjugate is \( a - bi \). Essentially, it involves changing the sign of the imaginary component. This can be particularly helpful in simplifying expressions with complex numbers, especially when they appear in denominators.
Using the conjugate of a complex number allows us to utilize the property that the product of a complex number and its conjugate yields a real number. For a complex number \( z = a + bi \), the product of \( z \) and its conjugate \( \overline{z} = a - bi \) simplifies as follows:
\[ z \times \overline{z} = (a + bi)(a - bi) = a^2 - (bi)^2 = a^2 + b^2 \]
This result is always real and is equal to the sum of the squares of the real and imaginary parts. In the example given, where we have \( \frac{5}{2-7i} \), multiplying numerator and denominator by the conjugate \( 2+7i \) ensures the imaginary unit \( i \) is eliminated from the denominator.
Using the conjugate of a complex number allows us to utilize the property that the product of a complex number and its conjugate yields a real number. For a complex number \( z = a + bi \), the product of \( z \) and its conjugate \( \overline{z} = a - bi \) simplifies as follows:
\[ z \times \overline{z} = (a + bi)(a - bi) = a^2 - (bi)^2 = a^2 + b^2 \]
This result is always real and is equal to the sum of the squares of the real and imaginary parts. In the example given, where we have \( \frac{5}{2-7i} \), multiplying numerator and denominator by the conjugate \( 2+7i \) ensures the imaginary unit \( i \) is eliminated from the denominator.
Simplifying Complex Fractions
Simplifying complex fractions requires a structured approach to remove the imaginary part in the denominator. The original problem, \( \frac{5}{2-7i} \), is simplified through multiplying both the numerator and denominator by the conjugate of the denominator.
By converting the denominator into a real number using the difference of squares, we made the division more straightforward. Here’s the quick process:
By converting the denominator into a real number using the difference of squares, we made the division more straightforward. Here’s the quick process:
- Identify the complex number in the denominator: \( 2 - 7i \).
- Find the conjugate: \( 2 + 7i \).
- Multiply both parts of the fraction by the conjugate:
\[ \frac{5(2+7i)}{(2-7i)(2+7i)} \] - Apply the difference of squares to the denominator:
\[ (2-7i)(2+7i) = 4 - (7i)^2 = 4 + 49 = 53 \] - Simplify the numerator: \( 5(2 + 7i) = 10 + 35i \).
Imaginary Unit
The imaginary unit, denoted by \( i \), is a mathematical concept used to extend the real number system. It is defined through the equation \( i^2 = -1 \). This allows for the existence of complex numbers, which include both a real part and an imaginary part.
Complex numbers are written in the form \( a + bi \), where \( a\) and \( b \) are real numbers. The imaginary unit \( i \) is what distinguishes a complex number from a purely real number.
In the process of simplifying complex fractions, knowing the properties of \( i \) is key. When you square \( i \), you get \( -1 \), which plays a crucial role in converting complex components into real numbers by utilizing squares and conjugates. In our example, when calculating \((2-7i)(2+7i)\), you'll notice the imaginary component squared \( (7i)^2 = -49 \), turning into a real number which is positive when further simplified:
\[ 4 - 49(-1) = 4 + 49 = 53 \].
This step is vital for treating complex fractions intuitively, bringing the entire expression into the desired form with ease.
Complex numbers are written in the form \( a + bi \), where \( a\) and \( b \) are real numbers. The imaginary unit \( i \) is what distinguishes a complex number from a purely real number.
In the process of simplifying complex fractions, knowing the properties of \( i \) is key. When you square \( i \), you get \( -1 \), which plays a crucial role in converting complex components into real numbers by utilizing squares and conjugates. In our example, when calculating \((2-7i)(2+7i)\), you'll notice the imaginary component squared \( (7i)^2 = -49 \), turning into a real number which is positive when further simplified:
\[ 4 - 49(-1) = 4 + 49 = 53 \].
This step is vital for treating complex fractions intuitively, bringing the entire expression into the desired form with ease.
Other exercises in this chapter
Problem 20
Express as a polynomial. $$(2 a+b)^{3}$$
View solution Problem 20
Factor the polynomial. $$x^{3}-25 x$$
View solution Problem 21
The two given numbers are coordinates of points \(A\) and \(B\), respectively, on a coordinate line. Express the indicated statement as an inequality involving
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Simplify. $$\left(8 x^{4} y^{-3}\right)\left(\frac{1}{2} x^{-5} y^{2}\right)$$
View solution