Problem 43 - Precalculus Student Solutions Manual 5th

Question

Find each sum or difference. Write the answer in standard form. $$(3+2 i)+(9-3 i)$$

Step-by-Step Solution

Answer

12 - i

Step 1 - Identify the real and imaginary parts

Look at each complex number and separate the real and imaginary parts. For the first complex number, (3 + 2i), the real part is 3 and the imaginary part is 2i. For the second complex number, (9 - 3i), the real part is 9 and the imaginary part is -3i.

Step 2 - Add the real parts

Add the real parts of the two complex numbers together. In this case, add 3 and 9: 3 + 9 = 12.

Step 3 - Add the imaginary parts

Next, add the imaginary parts together. In this case, add 2i and -3i: 2i + (-3i) = -1i or simply -i.

Step 4 - Combine the sums

Combine the results from the previous steps to write the answer in standard form. The sum of the real parts is 12 and the sum of the imaginary parts is -i. So, the result is:12 - i.

Key Concepts

Complex Numbers

In mathematics, a complex number is a number that comprises a real part and an imaginary part. It is written in the form \(a + bi\), where \(a\) is the real part, and \(bi\) is the imaginary part. The imaginary unit \(i\) is defined as the square root of -1, fulfilling the equation \(i^2 = -1\). Complex numbers are used in many fields including engineering, physics, and applied mathematics. They allow us to solve equations that have no real solutions and provide a deeper understanding of polynomial functions.

Real Part

The real part of a complex number is the component that doesn't involve the imaginary unit \(i\). In the example \(3 + 2i\), the real part is 3. When adding or subtracting complex numbers, we first handle the real parts separately. The given exercise illustrates this:

• Identify the real part of each complex number.
• For the complex numbers \(3 + 2i\) and \(9 - 3i\), the real parts are 3 and 9 respectively.
• Add these real parts: \(3 + 9 = 12\).

Imaginary Part

The imaginary part of a complex number includes the imaginary unit \(i\). It is essential to treat the imaginary part as a factor of \(i\). For instance, in \(3 + 2i\), the imaginary part is \(2i\). Let's consider our exercise again:

• Identify the imaginary parts, which are \(2i\) and \(-3i\).
• When adding these, remember to add the coefficients of \(i\): \(2 + (-3) = -1\).
• So, the sum of the imaginary parts is \(-i\).

Standard Form

Complex numbers are officially written in the standard form: \(a + bi\), where \(a\) and \(b\) are real numbers. The exercise's final solution should be presented this way. Combine the real and imaginary sums:

• The sum of the real parts is 12.
• The sum of the imaginary parts is \(-i\).

Thus, the result in standard form is \(12 - i\). This ensures clarity and uniformity, allowing easy interpretation and further operations with complex numbers.