Problem 5 - Algebra for College Students

Question

Label each statement true or false. The sum of two complex numbers is always a complex number.

Step-by-Step Solution

Answer

True. The sum of two complex numbers is always a complex number.

Step 1: Understanding Complex Numbers

Complex numbers are numbers of the form \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit with the property that \( i^2 = -1 \). An example of a complex number is \( 3 + 4i \).

Step 2: Identifying the Properties of Addition

The addition of two complex numbers \( (a + bi) + (c + di) \) involves adding their real parts and their imaginary parts separately: \( (a + c) + (b + d)i \). This result is also of the form \( x + yi \), where \( x \) and \( y \) are real numbers, guaranteeing it's a complex number.

Step 3: Verifying the Addition with an Example

Consider two complex numbers, for instance, \( z_1 = 2 + 3i \) and \( z_2 = 4 + 5i \). Their sum is \((2 + 3i) + (4 + 5i) = (2 + 4) + (3 + 5)i = 6 + 8i\), which is also a complex number, affirming the statement.

Step 4: Conclusion on the Statement

Since the form of the result after addition (\( x + yi \)) is consistent with the definition of a complex number, the statement "The sum of two complex numbers is always a complex number" is confirmed to be true.

Key Concepts

Addition of Complex Numbers

When adding complex numbers, each part has its own place. A complex number comprises a real part and an imaginary part. Thus, if you have two complex numbers, say \( z_1 = a + bi \) and \( z_2 = c + di \), adding them means combining like parts: the real parts and the imaginary parts separately. This operation looks like this:

• Add the real parts: \( a + c \)
• Add the imaginary parts: \( b + d \)

The sum is another complex number: \( (a + c) + (b + d)i \). The addition maintains the form \( x + yi \), confirming that the sum of two complex numbers, no matter what those numbers are, always results in another complex number.

Properties of Complex Numbers

Complex numbers have specific properties that are quite similar to regular numbers. They can undergo operations like addition, subtraction, multiplication, and division. Here are some key properties you'll find helpful:

• Closure: The sum of any two complex numbers is a complex number, and so is the product.
• Commutative Property: You can switch the order of complex numbers when adding or multiplying them. For example, \( a + bi + c + di = c + di + a + bi \).
• Associative Property: Regardless of how you group complex numbers, the sum or product remains the same. For instance, \( (a + bi) + (c + di) + (e + fi) = a + (c + di) + (e + fi) \).

These fundamental properties ensure that complex numbers behave predictably across various arithmetic operations. This predictability is why they're so widely used in math and engineering.

Arithmetic of Complex Numbers

Arithmetic with complex numbers extends beyond addition. It's a rich field that involves different operations, much like ordinary arithmetic with real numbers. Let's explore the other basic operations:

• Subtraction: Similar to addition, subtract complex numbers by subtracting corresponding real and imaginary parts: \( (a + bi) - (c + di) = (a-c) + (b-d)i \).
• Multiplication: Use distributive properties, and remember that \( i^2 = -1 \). For example, multiply \( (a + bi) \) by \( (c + di) \): \[ (a + bi)(c + di) = ac + adi + bci + bdi^2 \] which simplifies to \( (ac-bd) + (ad+bc)i \).
• Division: Divide by multiplying the numerator and the denominator by the conjugate of the denominator. This clears the imaginary part from the denominator for easy division.

Understanding these operations provides a comprehensive grasp of how complex numbers function, allowing for more advanced problem-solving.