Problem 34 - College Algebra and Calculus: An Applied Approach

Question

Perform the indicated operation and write the result in standard form. $$(6-5 i)(1+i)$$

Step-by-Step Solution

Answer

The result is \(11 + i\)

Step 1: Distribute

The first step is to distribute (multiply) each term in the first complex number by each term in the second complex number. This gives: \((6)(1) + (6)(i) + (-5i)(1) + (-5i)(i)\), which simplifies to \(6 + 6i - 5i - 5i^2\)

Step 2: Simplify

The next step is to simplify the expression by combining like terms and simplifying \(i^2\) to \(-1\). This give: \(6 + i - 5(-1)\), which simplifies to \(6 + i + 5\).

Step 3: Combine Like Terms

We then combine the real number terms together and the imaginary number terms together to get this in standard form, i.e., \(a+bi\). So, the answer is \(11 + i\).

Key Concepts

Standard Form

In the realm of complex numbers, the standard form is an essential concept. It allows us to express complex numbers neatly and consistently. The standard form of a complex number is written as \(a + bi\), where:

• \( a \) represents the real part
• \( b \) represents the imaginary part
• \( i \) is the imaginary unit, satisfying \(i^2 = -1\)

Writing a complex number in standard form makes it easier to perform operations like addition, subtraction, multiplication, and division. This form is not only standard for representation but also for solving and simplifying complex number-related mathematical problems. A complex number in standard form acts as a foundation for understanding complex arithmetic.

Imaginary Unit

At the heart of complex numbers lies the imaginary unit, denoted as \(i\). This special number allows us to perform operations on numbers that, at first glance, seem unsolvable. The imaginary unit is defined by the property:

• \(i^2 = -1\)

This definition might seem unconventional, but it proves to be tremendously useful.
When we multiply complex numbers, such as in the problem \((6-5i)(1+i)\), the appearance of \(i^2\) prompts a substitution:

• Replace \(i^2\) with \(-1\)

This substitution is a crucial step in simplifying expressions involving complex numbers. Without \(i\), we wouldn't be able to solve quadratic equations with no real solutions or express square roots of negative numbers.

Operations with Complex Numbers

Performing operations with complex numbers involves rules similar to those used with real numbers, but they do require particular attention to the imaginary component. Let's use the multiplication of two complex numbers, as seen in the exercise, to illustrate this:

• Distribute each term of the first complex number to each term of the second.
• Observe any occurrences of \(i^2\) and replace it with \(-1\).
• Combine like terms: Add the real parts together and the imaginary parts together.

For our exercise, multiplying \((6-5i)\) by \((1+i)\) and simplifying using \(i^2 = -1\) results in the standard form \(11 + i\).
This clearly shows how combining real and imaginary parts is important—both in simplifying given expressions and maintaining the standard format. Every operation you perform with complex numbers should ultimately strive for this standard form, ensuring clarity and consistency in your calculations.