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Chapter 10: Problem 23

Perform the operation(s) and write the result in standard form. $$(3+4 i)-(-2-5 i)$$

Short Answer

Expert verified
The operation results in the complex number \(5 + 9i\).

Step by step solution

01

Change the sign

Firstly, change the sign of the second complex number so the operation becomes an addition. So, it turns from \((3+4 i)-(-2-5 i)\) into \((3+4 i)+ (2+5 i)\).
02

Add the real parts

Add the real parts (parts without i) of the complex numbers. So, it turns from \((3+4 i)+ (2+5 i)\) into \(5 + 4i + 5i\).
03

Add the imaginary parts

Add the imaginary parts (parts with i) of the complex numbers. Therefore, \(5 + 4i + 5i\) simplifies to \(5 + 9i\).

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Key Concepts

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

Addition of Complex Numbers
When working with complex numbers, one important operation is their addition. In our example, we initially had the problem \((3 + 4i) - (-2 - 5i)\). To perform the addition, the subtraction needs to be converted into addition by changing the signs of the second complex number. This gives us: \((3 + 4i) + (2 + 5i)\).

Adding complex numbers is straightforward:
  • Add the real parts together. Here, the real parts are 3 and 2. So, \(3 + 2 = 5\).
  • Add the imaginary parts together. The imaginary parts are \(4i\) and \(5i\). Adding these gives \(4i + 5i = 9i\).
Thus, combining these, our result is \(5 + 9i\). The process involves pairing the like parts鈥攔eal with real, imaginary with imaginary鈥攖o form a new complex number.
Imaginary Numbers
Imaginary numbers are a fascinating component of complex numbers. They involve the imaginary unit \(i\), where \(i\) is defined as the square root of -1, satisfying \(i^2 = -1\).

Imaginary numbers form the basis of the imaginary part of complex numbers. In exercises like these, math problems often involve adding or subtracting these imaginary parts after simplifying the overall equation. Imaginary numbers are crucial in mathematical fields such as electrical engineering, signal processing, and quantum mechanics.
  • Pure imaginary numbers have the form \(bi\), where \(b\) is a real number.
  • They do not exist on the traditional number line, but they play important roles in calculations that require complex relationships.
Understanding imaginary numbers expands your ability to solve equations beyond real numbers, offering comprehensive solutions to certain types of polynomial equations.
Standard Form of Complex Numbers
The standard form of a complex number is a way of writing it such that it's easy to recognize its real and imaginary components. This standard form is expressed as \(a + bi\), where \(a\) is the real part, and \(bi\) is the imaginary part.

In our exercise, after performing the operation and simplifying, we arrived at the standard form \(5 + 9i\). Writing complex numbers in this form is essential for:
  • Clearly identifying each component for operations like addition, subtraction, multiplication, and division.
  • Determining the magnitude and phase of the complex number if needed.
It's a vital notation within mathematical practices to ensure clarity and consistency. By keeping complex numbers in this form, you can simplify and systematically solve complex equations more effectively.

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