Problem 45 - A Graphical Approach to Precalculus with Limits

Question

Add or subtract as indicated. Write each sum or difference in standard form. $$(3+2 i)+(4-3 i)$$

Step-by-Step Solution

Answer

The sum is \(7 - i\) in standard form.

Step 1: Identify Components

Identify the real and imaginary components from each complex number in the expression. The first complex number is \(3+2i\) with a real part of \(3\) and an imaginary part of \(2i\). The second complex number is \(4-3i\) with a real part of \(4\) and an imaginary part of \(-3i\).

Step 2: Add Real Parts

Add the real components of the complex numbers. We add \(3\) from the first number and \(4\) from the second number: \(3 + 4 = 7\).

Step 3: Add Imaginary Parts

Add the imaginary components from each complex number. We add \(2i\) from the first number and \(-3i\) from the second number: \(2i + (-3i) = -i\).

Step 4: Write in Standard Form

Combine the results of Step 2 and Step 3 to form the sum in standard form. The standard form of a complex number is \(a + bi\). Using our results: Real part = \(7\), Imaginary part = \(-i\). Therefore, the sum is \(7 - i\).

Key Concepts

Addition of Complex Numbers

Complex numbers might seem intimidating at first, but they can be fun to work with once you get the hang of it. In algebra, a complex number is typically expressed in the form \(a + bi\), where \(a\) and \(b\) are real numbers and \(i\) is the imaginary unit. The process of adding complex numbers involves combining both their real and imaginary components separately.

When adding complex numbers like \((3+2i) + (4-3i)\), follow these simple steps:

• Identify the real and imaginary parts of each complex number. In this example, the real parts are 3 and 4, and the imaginary parts are 2i and -3i.
• Add the real parts together: \(3 + 4 = 7\).
• Add the imaginary parts together: \(2i + (-3i) = -i\).

You'll end up with another complex number, which in this case is \(7 - i\). Keeping these steps in mind makes the addition of complex numbers straightforward and manageable.

Imaginary Numbers

Imaginary numbers are a fascinating and essential part of complex numbers. The imaginary unit is denoted by \(i\), where \(i = \sqrt{-1}\). This allows us to work with numbers that can't be found on the usual real number line.It's called 'imaginary,' but it can be quite practical and important in fields like engineering and physics. Imaginary numbers help to expand our mathematical understanding and provide solutions to certain equations that real numbers alone cannot solve.

When you see a number with \(i\), such as 2i or -3i, you are dealing with an imaginary number. These numbers work much like real numbers in algebraic expressions. You add, subtract, multiply, and divide them by applying similar rules as you would with regular numbers, keeping the property that \(i^2 = -1\) in mind.Engaging with imaginary numbers opens up a new realm of possibilities, enabling you to solve equations that involve square roots of negative numbers.

Standard Form Representation

Standard form is the typical way to express complex numbers, represented as \(a + bi\), where \(a\) and \(b\) are real numbers. This format clearly displays both the real and imaginary parts, making it easier to work with complex numbers mathematically.When you add or subtract complex numbers, your result should always be in standard form. For example, after performing the addition \((3+2i) + (4-3i)\), we ended up with the complex number \(7 - i\). Here, 7 is the real part (\(a\)), and -1 is the coefficient of \(b\), indicating that it's \(-i\).Expressing complex numbers in standard form allows you to quickly identify and perform operations on their real and imaginary parts. It also makes it easier to visualize these numbers on the complex plane, where the real numbers form the horizontal axis and the imaginary numbers form the vertical axis.

Always remember that maintaining the standard form \(a + bi\) at each step in your calculations will enhance your understanding and clarity when working with complex numbers.