Problem 42 - Precalculus : Building Concepts and Connections

Question

Find \(x+y, x-y, x y,\) and \(x / y\). $$x=2-7 i ; y=11+2 i$$

Step-by-Step Solution

Answer

\(x+y=13-5i, x-y=-9 - 9i, xy=36 - 73i, x/y=0.064 - 0.648i \)

Step 1: Sum of Complex Numbers

To calculate the sum \(x + y\) of two complex numbers, add their real parts and their imaginary parts separately. So \(x + y = (2 + 11) + (-7 + 2)i = 13 - 5i\)

Step 2: Subtract Complex Numbers

For the subtraction \(x - y\) of two complex numbers, subtract the real part of the second complex number from the real part of the first complex number. Then subtract the imaginary part of the second complex number from the imaginary part of the first complex number. This gives \(x - y = (2 - 11) + (-7 - 2)i = -9 - 9i\)

Step 3: Multiply Complex Numbers

To multiply \(x * y\) of two complex numbers, we use the distributive property to expand the product, and then use the fact that \(i^2 = -1\). So, \(x * y = (2 - 7i) * (11 + 2i) = 22 + 4i - 77i - 14i^2= 22 - 73i + 14 = 36 - 73i\)

Step 4: Divide Complex Numbers

To divide complex numbers \(x / y\) we multiply the numerator and the denominator by the conjugate of the denominator and simplify. The conjugate of \(11 + 2i\) is \(11 - 2i\). So, \((2 - 7i) / (11 + 2i) = ((2 - 7i)(11 - 2i))/((11 + 2i)(11 - 2i))= (22 - 4i - 77i +14i^2)/ (121 - 4i^2) = (22 - 81i - 14)/ (121 + 4) = 8 - 81i /125 = 0.064 - 0.648i

Key Concepts

Addition of Complex Numbers

Adding complex numbers is a straightforward process. Complex numbers are expressed in the form \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. To add two complex numbers, like \(x = 2 - 7i\) and \(y = 11 + 2i\), you simply add the real parts together and the imaginary parts together.

In this case:

• Add the real parts: \(2 + 11 = 13\).
• Add the imaginary parts: \(-7 + 2 = -5\).

The resulting sum is \(13 - 5i\).
This method keeps the structure of complex numbers while simplifying their combination.

Subtraction of Complex Numbers

Subtraction of complex numbers follows a parallel process to addition. To determine the difference between two complex numbers, such as \(x = 2 - 7i\) and \(y = 11 + 2i\), subtract the second number's real and imaginary parts from those of the first number separately.

Here's how it's done:

• Subtract the real parts: \(2 - 11 = -9\).
• Subtract the imaginary parts: \(-7 - 2 = -9\).

The result is \(-9 - 9i\).
Subtraction helps in determining the difference between the magnitudes and directions of different complex values.

Multiplication of Complex Numbers

Multiplying complex numbers may seem a bit more complex, but it's easily manageable using the distributive property. When multiplying \(x = 2 - 7i\) by \(y = 11 + 2i\), you distribute each part of one number across the parts of the other. Remember that \(i^2 = -1\), as this identity will simplify your result.

Breaking it down:

• Multiply: \((2)(11) = 22\).
• Multiply: \((2)(2i) = 4i\).
• Multiply: \((-7i)(11) = -77i\).
• Multiply: \((-7i)(2i) = -14i^2 = 14\) (since \(i^2 = -1\)).

Add these results together: \(22 - 73i + 14 = 36 - 73i\).
Multiplication fuses the magnitudes and affects directions in the complex plane.

Division of Complex Numbers

Dividing complex numbers involves the interesting concept of conjugates. To divide \(x = 2 - 7i\) by \(y = 11 + 2i\), multiply the numerator and denominator by the conjugate of the denominator. The conjugate of a complex number \(a + bi\) is \(a - bi\). This helps eliminate the imaginary part in the denominator.

Here's what you do:

• Find the conjugate: \(11 - 2i\).
• Multiply numerators: \((2 - 7i)(11 - 2i) = 22 - 4i - 77i + 14i^2 = 8 - 81i\).
• Denominator multiplication: \((11 + 2i)(11 - 2i) = 121 + 4 = 125\).

Divide: \((8 - 81i) / 125\), which simplifies to \(0.064 - 0.648i\).
This technique keeps complex division manageable by transforming it into a real form in the denominator.