Problem 11 - A Graphical Approach to Precalculus with Limits

Question

Perform operations and write the result in standard form. $$3 i+5 i$$

Step-by-Step Solution

Answer

The result is \(8i\).

Step 1: Identify Similar Terms

In the given expression, we need to add the imaginary numbers together. Identify that both terms, \(3i\) and \(5i\), contain the imaginary unit \(i\). Arithmetic operations can be performed between similar terms.

Step 2: Add the Coefficients

Add the coefficients of the imaginary unit. This means we will add the numbers in front of \(i\): \(3 + 5 = 8\).

Step 3: Write the Result

After adding the coefficients, attach the imaginary unit back to the result. This gives us \(8i\) as the sum in standard form for imaginary numbers.

Key Concepts

Imaginary Unit

In mathematics, the imaginary unit is a fascinating concept used to extend our number system. It is denoted as \(i\), where \(i\) represents \(\sqrt{-1}\). This means that when you square \(i\), you get \[i^{2} = -1\]This unique property gives us the ability to work with numbers that include the square roots of negative numbers.Understanding the imaginary unit is crucial because it forms the basis of complex numbers, which consist of a real part and an imaginary part. The concept of \(i\) allows us to solve equations that do not have real solutions, like \(x^2 + 1 = 0\). Without \(i\), such equations would remain unsolvable within the realm of real numbers.

Standard Form

Complex numbers are often expressed in standard form, which makes it easy for us to perform calculations and interpret these numbers. The standard form of a complex number is:\[a + bi\]Here, \(a\) is the real part and \(b\) is the coefficient of the imaginary unit \(i\). For instance, the number \(8i\) from our exercise is already in standard form where the real part \(a\) is 0, and the imaginary component is \(8i\).Using standard form helps simplify operations such as addition, subtraction, and even more complex operations like multiplication and division. Identifying the real part and imaginary part separately can make it easier to organize and solve problems involving complex numbers.

Arithmetic Operations

Arithmetic with imaginary and complex numbers follows principles that are very similar to those used with real numbers. Here is how you can handle them:

• Addition and Subtraction: When adding or subtracting complex numbers, add or subtract the real parts and imaginary parts separately. You can only combine like terms, i.e., real with real and imaginary with imaginary.
• Multiplication: Use the distributive property, remembering that \(i^2 = -1\), to simplify the expression.
• Division: This involves multiplying the numerator and the denominator by the complex conjugate of the denominator to eliminate the imaginary unit from the denominator.

From our example, adding \(3i + 5i\), we see addition in action. Combine the coefficients of \(i\): calculate \(3 + 5\) to get \(8\), and thus the result is \(8i\) in standard form. Understanding these operations will help you handle complex number problems more effectively!