Problem 2 - Algebra and Trigonometry with Analytic Geometry

Question

Write the expression in the form \(a+b i\), where \(a\) and \(b\) are real numbers. $$ (-5+7 i)+(4+9 i) $$

Step-by-Step Solution

Answer

The expression in standard form is \(-1 + 16i\).

Step 1: Identify Real and Imaginary Parts

Identify the real parts and the imaginary parts of each complex number. The expression \((-5 + 7i)\) has a real part of -5 and an imaginary part of 7i. The expression \((4 + 9i)\) has a real part of 4 and an imaginary part of 9i.

Step 2: Combine the Real Parts

Add the real parts of the two complex numbers. Real part of \((-5)\) added to the real part of \((4)\) is \((-5 + 4 = -1)\).

Step 3: Combine the Imaginary Parts

Add the imaginary parts of the two complex numbers. Imaginary part \((7i)\) added to the imaginary part of \((9i)\) is \((7i + 9i = 16i)\).

Step 4: Write in Standard Form

Combine the results from Steps 2 and 3 to write the complex number in standard form \(a + bi\). Therefore, the sum is \(-1 + 16i\).

Key Concepts

Imaginary Numbers

Imaginary numbers might sound complex at first, but they are quite simple once you get the hang of them. An imaginary number is defined using the imaginary unit, denoted as "\(i\)". The key property of this unit is that \(i^2 = -1\). Imaginary numbers are expressed in the form "\(bi\)" where \(b\) is a real number, and \(i\) is the imaginary unit. This means that whenever you see a term like "7i" or "9i", these are imaginary numbers.
Imaginary numbers are especially useful in mathematics for working with concepts that can't be easily expressed using just real numbers. For example, when solving quadratic equations that do not have real solutions, imaginary numbers come to the rescue. While they might seem abstract, they are a crucial part of the number system that extends beyond the limitations of real numbers.

Real Numbers

Real numbers are what you likely deal with on a daily basis. They include all the integers, fractions, and decimals you can think of. In essence, real numbers encompass both rational numbers (like \(1/2\) or \(3.14\)) and irrational numbers (such as \(\sqrt{2}\) or \(\pi\)).
A real number is any number that can be found on the number line, lacking a complex component such as "i", which represents the imaginary part. For instance, in the complex number \(-5 + 7i\), the real part is \(-5\). Real numbers are direct, tangible, and provide the foundation upon which complex numbers are built. Real numbers are represented by the "Re" part in a complex number, distinguishing them from their imaginary counterparts.

Complex Addition

Adding complex numbers is a straightforward process once you understand that you simply combine like terms. Each complex number consists of a real part and an imaginary part. When performing complex addition, you follow these steps:

• Step 1: Identify and separate the real and imaginary parts of the complex numbers. For example, in \((-5 + 7i)\) and \((4 + 9i)\), separate \(-5\) and \(4\) as the real parts, and \(7i\) and \(9i\) as the imaginary parts.
• Step 2: Add the real parts together. For these numbers, that's \(-5 + 4 = -1\).
• Step 3: Add the imaginary parts together. Here, \(7i + 9i = 16i\).
• Step 4: Combine the results to express the sum in the standard form of a complex number \(a + bi\), which in this case is \(-1 + 16i\).

By treating real and imaginary parts separately, complex addition becomes as simple as adding and subtracting the numbers you're already familiar with.