Problem 85 - Calculus for Biology and Medicine

Question

Simplify each expression and write it in the standard form \(a+b i\). \((3-2 i)-(-5+2 i)\)

Step-by-Step Solution

Answer

The simplified expression is \(8 - 4i\).

Step 1: Identify the Expression Components

The expression given is \((3 - 2i) - (-5 + 2i)\). It consists of two complex numbers: the first one is \((3 - 2i)\), and the second one is \((-5 + 2i)\). Note the subtraction operation between them.

Step 2: Distribute the Negative Sign

Apply the negative sign to the second complex number. This changes the sign of each part: \((-5 + 2i)\) becomes \(+5 - 2i\). The expression now is \((3 - 2i) + (5 - 2i)\).

Step 3: Combine the Real Parts

Add the real numbers from each complex number: \(3 + 5 = 8\).

Step 4: Combine the Imaginary Parts

Add the imaginary numbers from each complex number: \(-2i - 2i = -4i\).

Step 5: Form the Standard Complex Expression

Combine the results of Step 3 and Step 4 to write the expression in the standard form \(a + bi\). The expression simplifies to \(8 - 4i\).

Key Concepts

Real Part

The real part of a complex number is the component that does not have the imaginary unit, which is denoted by \(i\). In the expression \((3 - 2i)\), the real part is 3, since it is not associated with the \(i\) term. Similarly, for the expression \((-5 + 2i)\), the real part is \(-5\). These components represent actual numbers as opposed to the imaginary parts, which involve the square root of negative one. Recognizing the real part is essential for performing operations with complex numbers. When you add or subtract complex numbers, you combine their real parts separately from their imaginary parts. This keeps operations straightforward:- Identify the real parts of the numbers involved.- Add or subtract these real numbers as you would with any integers or fractions.In our step-by-step solution, the operation performed on the real parts was \(3 + 5 = 8\), demonstrating how combining real numbers follows the basic rules of arithmetic.

Imaginary Part

The imaginary part of a complex number involves the imaginary unit \(i\). The number \(i\) is defined such that \(i^2 = -1\), making it possible to represent numbers that are not real. In our given expression \((3 - 2i)\), the imaginary part is \(-2i\). For the second expression \((-5 + 2i)\), the imaginary part is \(+2i\). When dealing with imaginary components in complex numbers:- Always consider the signs in front of the numbers as they indicate the direction on the imaginary plane.- Perform operations separate from the real parts, treating \(i\) as a constant factor with its special property. For example, adding \(-2i\) and \(+2i\) results in \(-4i\), as demonstrated in the original exercise. Recognizing and separately calculating imaginary parts ensures accuracy when simplifying complex expressions. Understanding how to handle \(i\) can be tricky, but keep practicing by identifying the imaginary portions in complex numbers and combining or subtracting them as needed.

Standard Form

Complex numbers are typically expressed in their standard form, \(a + bi\), where \(a\) represents the real part and \(b\) the imaginary coefficient. This form is essential because it clearly separates the real and imaginary components of a complex number.The exercise solution aimed to simplify the expression and convert it into this standard form. Initially, we had \((3 - 2i) - (-5 + 2i)\). After algebraic operations, the expression was rewritten as the simplified form \(8 - 4i\).Writing the expression in standard form involves several steps:- Resolve the subtraction by distributing signs across each complex component.- Combine like terms by joining real parts and imaginary parts separately.- Present the combined result appropriately in the \(a + bi\) form. Being able to express a complex number in the standard form is crucial for performing further mathematical operations like addition, subtraction, or even multiplication. It provides a clearer picture of each part's magnitude and impact. Practice simplifying complex expressions into \(a + bi\) to get comfortable identifying and working with each component.