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Chapter 2: Problem 26

Performing Operations with Complex Numbers. Perform the operation and write the result in standard form. $$(3+2 i)-(6+13 i)$$

Short Answer

Expert verified
The result of the operation is \(-3 - 11i\).

Step by step solution

01

Subtract Real Parts

Subtract the real part of the second complex number (6) from the real part of the first complex number (3). This gives us \(3 - 6 = -3\).
02

Subtract Imaginary Parts

Subtract the imaginary part of the second complex number (13i) from the imaginary part of the first complex number (2i). This gives us \(2i - 13i = -11i\).
03

Write in Standard Form

The standard form for complex numbers is \(a + bi\), where 'a' stands for the real part and 'b' stands for the imaginary part. Plugging the results from steps 1 and 2 into this formula gives \(-3 - 11i\).

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Key Concepts

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

Subtracting Complex Numbers
Understanding how to subtract complex numbers is essential in the world of algebra and beyond. To subtract one complex number from another, you simply subtract the real parts and the imaginary parts separately. Consider the expression ewlineewlineewlineewlineewlineewlineewlineewlineewlineewlineewlineewlineewlineewlineewlineewline The real numbers 3 and 6 are subtracted as usual, yielding -3. Similarly, the imaginary parts 2i and 13i are also subtracted to give -11i. The result is then combined to form a single complex number in standard form.
Standard Form of Complex Numbers
The standard form of a complex number is expressed as ewlineewlineewlineewlineewlineewlineewlineewlineewlineewlineewlineewlineewlineewlineewlineewlineewline The 'a' and 'b' are real numbers, where 'a' represents the real part, and 'b' represents the imaginary part, multipled by 'i', the imaginary unit. In the context of our exercise, once we have subtracted the real and imaginary parts separately, we combine them into the standard form as ewlineewlineewlineewlineewline -3 - 11i, adhering to the standard format.
Imaginary Numbers
The concept of imaginary numbers is fundamental when we deal with complex numbers. An imaginary number is a value that produces a negative number when squared. It's defined by the imaginary unit 'i', which is equal to the square root of -1. In the subtraction process, we deal specifically with the imaginary parts of the complex numbers. We subtract the imaginary components by treating 'i' as a constant and subtracting the coefficients, as demonstrated by the subtraction of 2i from 13i in our provided example.
Real Parts of Complex Numbers
Every complex number consists of a real part and an imaginary part. The real part is a real number, not multiplied by 'i'. When subtracting complex numbers, as shown in the exercise, we separately subtract the real parts of the complex numbers. In essence, the operation ewlineewlineewlineewlineewlineewlineewlineewlineewlineewlineewline -3 is the result of subtracting the second number's real part, 6, from the first number's real part, 3. This distinction between real parts and imaginary parts is crucial in performing arithmetic on complex numbers.

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