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Chapter 2: Problem 30

Performing Operations with Complex Numbers. Perform the operation and write the result in standard form. $$25+(-10+11 i)+15 i$$

Short Answer

Expert verified
The simplified form of the given expression in standard form is \(15 + 26i\).

Step by step solution

01

Distribute

Before starting with the addition, distribute the negative sign through the second set of parentheses. Thus, making the expression: \(25+ -10 +11i +15i\) .
02

Combine like terms

Group and simplify the real parts and imaginary parts. Adds the real numbers, -10 and 25 and adds imaginary numbers 11i and 15i. The expression then simplifies to: \(15 + 26i\) .
03

Write in Standard Form

The result from step 2 is already in standard form, which is \(a+ bi\) . So, the final result is \(15 + 26i\) .

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Standard Form
Complex numbers have a unique representation called the standard form, which helps make calculations simpler. The standard form of a complex number is written as \( a + bi \), where:
  • \( a \) is the real part
  • \( bi \) is the imaginary part, with \( b \) being the coefficient of the imaginary unit \( i \)
The imaginary unit \( i \) is defined as the square root of -1, which we will explore further in the next section. For instance, the expression from the exercise, \( 15 + 26i \), is already in standard form. Here, 15 is the real part and 26i is the imaginary part. This arrangement makes it easier to identify and perform operations on the real and imaginary components separately.
Imaginary Numbers
Complex numbers include an imaginary part, which involves the unit \( i \). Imaginary numbers stem from the challenge of finding square roots of negative numbers. By definition, \( i \) is equal to \( \sqrt{-1} \).Imaginary numbers, such as \( 11i \) and \( 15i \) from our exercise, represent multiples of \( i \). These numbers extend the real number system and are crucial for solving equations that do not have real solutions. Combining or separating the imaginary part is a key step in operations with complex numbers, such as simplifying \( (11i + 15i) \) to \( 26i \). This operation shows that imaginary numbers follow the same addition rules as real numbers.
Operations with Complex Numbers
Performing operations with complex numbers is straightforward when they're in standard form. In our exercise, we needed to combine both real and imaginary parts.To do this:
  • Distribute any negative signs through terms, as seen when we adjusted \(-10 + 11i\).
  • Group like terms: real parts like 25 and -10; imaginary parts like \(11i\) and \(15i\).
  • Add the real parts together and the imaginary parts together, resulting in \( 15 + 26i \).
These steps not only help in addition and subtraction but also lay the foundation for multiplying and dividing complex numbers. The general rule with these operations is to treat \( i \) just like a variable, except remember that \( i^2 = -1 \), which can be handy especially in multiplication tasks.

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Most popular questions from this chapter

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