Problem 88 - Calculus for Biology and Medicine

Question

Simplify each expression and write it in the standard form \(a+b i\). \((6-4 i)+(2+5 i)\)

Step-by-Step Solution

Answer

The simplified expression is \(8+i\).

Step 1: Identify Real and Imaginary Parts

Examine the expression \((6-4i) + (2+5i)\). Identify the real parts and the imaginary parts. The real parts are \(6\) and \(2\), and the imaginary parts are \(-4i\) and \(5i\).

Step 2: Add the Real Parts

Add the real parts of the expression from Step 1: \(6 + 2 = 8\).

Step 3: Add the Imaginary Parts

Add the imaginary parts of the expression from Step 1: \(-4i + 5i = 1i\) or simply \(i\).

Step 4: Combine Results in Standard Form

Combine the results from Steps 2 and 3 to write the expression in the standard form of a complex number. The expression becomes \(8 + i\), where \(a = 8\) and \(b = 1\).

Key Concepts

Real and Imaginary Parts

In the realm of complex numbers, each number is composed of two distinct parts: the real part and the imaginary part. These parts collaborate to form the complete picture of what a complex number truly is. Consider a complex number like \( 6 - 4i \) and \( 2 + 5i \) from the original exercise. Here, the real components are simply the numbers without the 'i', namely \( 6 \) and \( 2 \). The imaginary components carry the 'i' and are \( -4i \) and \( 5i \).

It's worth noting that 'i' is an imaginary unit, which is defined as the square root of -1. Therefore, being able to identify and separate these parts is a fundamental step in working with complex numbers. Understanding this foundation helps you perform operations like addition, subtraction, and even multiplication of complex numbers with greater ease.

Simplification

Simplification in the context of complex numbers refers to combining like terms to create a more concise and manageable expression. When handling an expression like \((6-4i)+(2+5i)\), it is important to address both the real and imaginary components separately.

• **Real Parts:** Combine the real parts \(6 + 2\), which equals \(8\).
• **Imaginary Parts:** Combine the imaginary parts \(-4i + 5i\), which simplifies to \(1i\) or simply \(i\).

When you bring together the results from these separate calculations, you achieve a streamlined form of the original expression. This practice simplifies the work with complex numbers, making future manipulations and interpretations clearer.

Standard Form a+bi

The standard form of a complex number is expressed as \( a + bi \), where 'a' is the real part, and 'b' is the coefficient of the imaginary part 'i'. This representation helps in expressing complex numbers in a consistent and recognizable manner.

From our simplified expression, we've identified \( a = 8 \) and \( b = 1 \). Therefore, the expression \( 8 + i \) perfectly fits into the standard form. This form is not only aesthetically pleasing but also aids in computational operations, understanding, and communication of complex numbers.

• Aids in operations: Consistent format simplifies addition, subtraction, and even more complex operations.
• Improves understanding: Recognizable format enhances comprehension of underlying mathematics.
• Facilitates communication: Standardized form allows for clear communication in a mathematical context.

Embracing the standard form enables mathematicians and students alike to handle complex numbers with confidence and clarity.