Problem 3 - College Algebra

Question

Add or subtract as indicated and write the result in standard form. $$ (3+2 i)-(5-7 i) $$

Step-by-Step Solution

Answer

-2 + 9i

Step 1: Identify the Real and Imaginary parts

In the given complex numbers, for (3+2i) '3' is the real part and '2i' is the imaginary part, while for (5-7i), '5' is the real part and '-7i' is the imaginary part.

Step 2: Subtract the Real parts and the Imaginary parts separately

Subtract the real parts (3-5 = -2) and subtract the imaginary parts (2i - (-7i) = 2i + 7i = 9i).

Step 3: Write the Solution

Combine steps 1 and 2 to write the result in standard form. The real part '-2' and the imaginary part '9i' constitute the resultant complex number, which is -2 + 9i.

Key Concepts

Standard Form of Complex Numbers

Understanding the standard form of complex numbers is foundational for working with these numbers in mathematics. The standard form is expressed as \( a + bi \), where \( a \) is the real component and \( b \) is the imaginary coefficient, with \( i \) representing the square root of -1. This neat arrangement allows for clear identification of the different parts of the complex number, which is crucial when performing operations such as addition and subtraction.

For instance, in our exercise \( (3+2i)-(5-7i) \), both \( 3+2i \) and \( 5-7i \) are in standard form. It's this consistent structure that enables us to manage the elements of the complex numbers separately, ensuring accurate arithmetic operations.

Real and Imaginary Parts

Breaking down a complex number into its real and imaginary parts makes it easier to perform arithmetic operations. Every complex number consists of a real part (\( a \)) and an imaginary part (\( bi \)), where \( b \) is a real number that accompanies the imaginary unit \( i \). It's the combination of these parts that forms the complete complex number.

In the context of our example \( (3+2i)-(5-7i) \), '3' and '5' are the real parts of the complex numbers, whereas '2i' and '-7i' are the imaginary parts. Recognizing these components allows us to apply arithmetic operations on the corresponding parts independently from each other, leading to a more organized and comprehensible calculation process.

Complex Number Arithmetic

Just as with real numbers, arithmetic with complex numbers follows specific rules. When adding or subtracting complex numbers, combine the real parts with real parts and the imaginary parts with the imaginary parts. This is facilitated by the standard form, which aligns like terms for easy identification and calculation.

In our exercise where we need to subtract \( (3+2i) \) from \( (5-7i) \), the process is straightforward: subtract the real parts, \( 3-5 \), to get '-2' for the real component of the result, and subtract the imaginary parts, \( 2i - (-7i) \), to get '9i' for the imaginary component. By adhering to these orderly steps, the fundamental operations of complex number arithmetic don't have to be complicated, making it accessible for students and learners to work with complex numbers.