Problem 40 - Intermediate Algebra

Question

Perform the operations. Write all answers in the form \(a+b i.\) $$ (5+3 i)-(6-9 i) $$

Step-by-Step Solution

Answer

The result is \(-1 + 12i\).

Step 1: Identify Parts of Each Complex Number

The expression given is \( (5+3i) - (6-9i) \).Here, \(5+3i\) is the first complex number, consisting of a real part 5 and an imaginary part 3i. The second complex number is \(6-9i\), with a real part 6 and an imaginary part -9i.

Step 2: Subtract Real Parts

Subtract the real parts of the complex numbers.The real part of the first number is 5, and the real part of the second number is 6.Calculate:\[5 - 6 = -1\]

Step 3: Subtract Imaginary Parts

Subtract the imaginary parts of the complex numbers.The imaginary part of the first number is 3i, and for the second number it is -9i.Calculate:\[3i - (-9i) = 3i + 9i = 12i\]

Step 4: Combine Real and Imaginary Results

Combine the computed real and imaginary parts into the form \(a + bi\).Thus, the result of the subtraction is:\(-1 + 12i\).

Key Concepts

Real and Imaginary Parts

Understanding complex numbers requires identifying two main components: the real part and the imaginary part. A complex number is typically expressed in the form \(a + bi\), where \(a\) represents the real part, and \(b\) with the \(i\) represents the imaginary part. For example, in the complex number \(5 + 3i\):

• The real part is \(5\).
• The imaginary part is \(3i\).

The imaginary unit \(i\) is a mathematical symbol that represents the square root of \(-1\).
This concept is crucial as it allows us to treat complex numbers like coordinates in a plane, where the real part corresponds to the x-axis, and the imaginary part to the y-axis.

Subtraction of Complex Numbers

Subtracting complex numbers involves dealing separately with their real and imaginary parts. For any two complex numbers, say \((a + bi)\) and \((c + di)\), the subtraction process is straightforward:

• Subtract the real parts \(a-c\).
• Subtract the imaginary parts \(b-d\).

Let's see an example using the numbers \((5 + 3i)\) and \((6 - 9i)\):
- Real parts: Subtract \(5 - 6 = -1\).
- Imaginary parts: Subtract \(3i - (-9i) = 3i + 9i = 12i\).
By calculating these separately, the result is easily combined to form a new complex number: \(-1 + 12i\). This process simplifies complex problem-solving by allowing us to focus on one part at a time.

Standard Form a+bi

The standard form of a complex number is \(a + bi\) where \(a\) represents the real part and \(b\) the coefficient of the imaginary part \(i\). This form is used because it offers a simple way to express complex numbers.
To ensure clarity and consistency in mathematical operations, always write the result in this standard form. For instance, after performing the subtraction \((5 + 3i) - (6 - 9i)\), the result is \(-1 + 12i\).

• \(a = -1\) is the real part of the result.
• \(b = 12\) is the coefficient of \(i\), making \(12i\) the imaginary part.

Arranging complex numbers in this form not only makes them easier to interpret but also facilitates further operations such as addition, multiplication, or division.