Problem 28 - Algebra A Combined Function

Question

Add or subtract as indicated. Write your answers in the form \(a+b i .\) $$ (6-2 i)+7 i $$

Step-by-Step Solution

Answer

\(6 + 5i\)

Step 1: Identify the Real and Imaginary Parts

In the given expression \((6-2i)+7i\), identify the real and imaginary parts. The real part is 6, and the imaginary parts are \(-2i\) and \(+7i\).

Step 2: Separate the Expression

Break down the expression into real parts and imaginary parts. Here, the expression is:\(6\) (real part) and \(-2i + 7i\) (imaginary part).

Step 3: Combine the Imaginary Parts

Add the imaginary parts together: \(-2i + 7i = 5i\).

Step 4: Write in the Form \(a+bi\)

Now, combine the real part with the resulting imaginary part:Real part: \(6\) Imaginary part: \(+5i\)Thus, the final expression is \(6 + 5i\).

Key Concepts

imaginary numbers

Imaginary numbers might sound like something out of a storybook, but they serve a fundamental purpose in mathematics. They extend our number system to cover more than just real numbers. An imaginary number is essentially a multiple of the imaginary unit, denoted as \(i\). The imaginary unit \(i\) is defined as the square root of \(-1\), so \(i^2 = -1\). This means any imaginary number can be expressed as a real number multiplied by \(i\). For example, \(3i\) or \(-2i\) are imaginary numbers because they are real numbers (3 and -2) multiplied by \(i\).
Imaginary numbers are part of complex numbers, which have both a real part and an imaginary part. This allows us to solve equations that do not have real solutions, like \(x^2 + 1 = 0\). Understanding imaginary numbers is crucial for working with complex numbers effectively.

real numbers

Real numbers are what most people consider 'regular' numbers. They include the integers (such as \(-1, 0, 1\)), fractions, and decimal numbers (like 2.5 or \(-3.14\)). Real numbers can be any value along the continuous number line, which extends infinitely in both the positive and negative directions. This is why real numbers are called so; they can be found 'in reality' as tangible quantities.
Real numbers are crucial because they form the backbone of everyday arithmetic and mathematics. They represent quantities we measure and calculate, like distance, time, and speed. In the context of complex numbers, the real part of a complex number is its component that does not involve the imaginary unit \(i\).
When dealing with complex numbers, identifying the real component helps simplify the expression. For example, in a problem like \(6 - 2i + 7i\), the real number is 6. Knowing this ensures that calculations are accurate and meaningful.

addition and subtraction of complex numbers

Adding and subtracting complex numbers is quite straightforward once you understand the components involved. A complex number typically has the form \(a + bi\), where \(a\) is the real part and \(b\) is the coefficient of the imaginary part. When adding or subtracting complex numbers, you simply handle the real parts separately from the imaginary parts.
For addition:

• Add the real parts together.
• Add the imaginary parts together.
  For instance, when you add \((6 - 2i) + 7i\), you first combine the imaginary parts \(-2i + 7i\).
• This simplifies to \(5i\), as both terms involve \(i\), then write the result in \(a + bi\) form: \(6 + 5i\).

Subtraction follows the same rules, except you'll subtract corresponding parts instead. Complex numbers let you operate within a broader scope since equations often have solutions that may not be evident only with real numbers. Mastering addition and subtraction is an essential foundation for more advanced operations with complex numbers.