Problem 3 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-34: Write the expression in the form \(a+b i\), where \(a\) and \(b\) are real numbers. $$ (7-6 i)-(-11-3 i) $$

Step-by-Step Solution

Answer

The expression is \(18 - 3i\).

Step 1: Identify the Expression

The given expression is \((7 - 6i) - (-11 - 3i)\). This is a subtraction problem involving two complex numbers.

Step 2: Distribute the Negative Sign

Distribute the negative sign across the terms in the second complex number. This changes the signs of both terms: \(-( -11 - 3i) = 11 + 3i\).

Step 3: Combine Like Terms

Now, add the resulting expression from the distribution to the first complex number: \((7 - 6i) + (11 + 3i)\).

Step 4: Add Real Parts

Combine the real parts of the complex numbers: 7 + 11 = 18.

Step 5: Add Imaginary Parts

Combine the imaginary parts of the complex numbers: -6i + 3i = -3i.

Step 6: Write the Final Expression

Combine the results from Step 4 and Step 5 to write the expression in the form \(a + bi\). Thus, the solution is \(18 - 3i\).

Key Concepts

Real Numbers

Real numbers are a fundamental part of mathematics. They include both rational numbers, like fractions and integers, and irrational numbers, such as \(ackslash\sqrt{2}\). A few key points about real numbers are:

• They're located on the number line, stretching from negative infinity to positive infinity.
• They can be positive, negative, or zero.
• Operations such as addition, subtraction, multiplication, and division can be performed on real numbers.

When discussing complex numbers, the real part is the component that does not involve the imaginary unit \(i\). For example, in the expression \(7 - 6i\), the number 7 is the real part. This component behaves like any standard real number and is critical in solving complex equations.

Imaginary Numbers

Imaginary numbers are as fascinating as they sound. They arise from the need to consider numbers that square to negative values. The key component is the imaginary unit \(i\), defined as \(i^2 = -1\). Here are some essential details:

• Imaginary numbers help in solving equations where the result can't be a real number, such as \(x^2 = -1\).
• Combining the imaginary unit \(i\) with real numbers forms complex numbers.

In the expression \(7 - 6i\), \(-6i\) is the imaginary part. Imaginary numbers are not found on the real number line but rather help form a plane where both real and imaginary components are combined. This combination creates complex numbers, allowing a rich field of mathematics beyond the real numbers alone.

Addition and Subtraction of Complex Numbers

The addition and subtraction of complex numbers might initially seem confusing, but it's quite straightforward. These operations involve combining like terms from each complex number. Here’s how you can handle them:

• Add or Subtract the Real Parts: Combine the real components from each number. For example, \( (7 - 6i) + (11 + 3i)\) results in adding \(7\) and \(11\), giving \(18\).
• Add or Subtract the Imaginary Parts: Combine the imaginary components. In our example, \(-6i\) is added to \(3i\), resulting in \(-3i\).

Thus, the final expression is formed by combining these sums: \(18 - 3i\). This approach towards complex number arithmetic helps in maintaining the structural integrity of the number’s real and imaginary parts—simplifying and solving complex problems effectively.