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Chapter 8: Problem 421

Add or subtract. \((1-4 i)-(3-6 i)\)

Short Answer

Expert verified
-2 + 2i

Step by step solution

01

- Identify Like Terms

Observe the given expression \((1 - 4i) - (3 - 6i)\). There are two sets of like terms: real numbers and imaginary numbers.
02

- Distribute the Negative Sign

Distribute the negative sign to the terms inside the parentheses: \((1 - 4i) - 3 + 6i\).
03

- Group Like Terms

Group the real and imaginary parts of the expression: \1 - 3\ (real numbers) and \(-4i + 6i\) (imaginary numbers).
04

- Combine Like Terms

Combine the like terms: \[1 - 3 + (-4i + 6i) = -2 + 2i\].

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Key Concepts

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

adding complex numbers
Adding complex numbers involves adding their real and imaginary parts separately. Suppose you have two complex numbers, say, \(a + bi\) and \(c + di\). The addition can be represented as:
\[ (a + bi) + (c + di) = (a + c) + (b + d)i \]
For example, if you have \(1 + 2i\) and \(3 + 4i\), their sum will be:
\[ (1 + 2i) + (3 + 4i) = (1 + 3) + (2 + 4)i = 4 + 6i \]
This is how you add complex numbers, by individually adding their real parts and imaginary parts, respectively. It is a straightforward process that makes working with complex numbers simpler.
subtracting complex numbers
Subtracting complex numbers follows a process similar to adding them but with subtraction. Suppose you want to subtract \(c + di\) from \(a + bi\). The subtraction is represented as:
\[ (a + bi) - (c + di) = (a - c) + (b - d)i \]
For instance, consider the subtraction \(1 - 4i\) and \(3 - 6i\). Distribute the negative sign across the terms of the second complex number and combine like terms:
\[ (1 - 4i) - (3 - 6i) = 1 - 4i - 3 + 6i \]
Group the real and imaginary parts:
\[ (1 - 3) + (-4i + 6i) \]
Simplify to get the final result:
\[ -2 + 2i \]
Thus, subtracting \(3 - 6i\) from \(1 - 4i\) results in \( -2 + 2i \).
like terms in algebra
Like terms in algebra are terms that have the same variables raised to the same power. When dealing with complex numbers, like terms include the real parts and the imaginary parts separately. This helps in simplifying expressions.

For example, in the expression \(1 - 4i - 3 + 6i\), you can identify the like terms:
  • Real terms: 1 and -3
  • Imaginary terms: -4i and 6i

Combine the real terms and the imaginary terms separately to get a simplified expression:
\[ 1 - 3 \] (for real parts) and \[ -4i + 6i \] (for imaginary parts).
This results in \[ -2 + 2i \].

Recognizing and organizing like terms is essential to perform operations smoothly and accurately in algebra, especially when dealing with complex numbers.

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