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Chapter 7: Problem 31

Add or subtract and simplify. Write each answer in the form \(a+b i\). $$ (7-4 i)-(5-3 i) $$

Short Answer

Expert verified
2 - i

Step by step solution

01

Distribute Negative Sign

The given expression is (7 - 4i) - (5 - 3i). To simplify, first distribute the negative sign through the second complex number, turning the expression into: (7 - 4i) - 5 + 3i.
02

Group Real and Imaginary Parts Separately

Separate the real and imaginary parts of the expression: (7 - 5) + (-4i + 3i).
03

Perform Addition/Subtraction

Simplify each group: 7 - 5 = 2 for the real part, and -4i + 3i = -i for the imaginary part. This gives us: 2 - i.

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Key Concepts

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

Subtracting Complex Numbers
In complex number operations, subtraction is straightforward once you understand the interaction between real and imaginary parts. Complex numbers are in the form of \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. To subtract complex numbers, you need to handle the real and imaginary components separately:
For example, if you want to subtract \((7 - 4i)\) and \((5 - 3i)\):
  • First, distribute the negative sign through the second complex number. This step changes the subtraction into:
    $$(7 - 4i) - (5 - 3i) = (7 - 4i) - 5 + 3i.$$
  • Now, separately add or subtract the real parts and the imaginary parts:
    • Real parts: 7 - 5 = 2.
    • Imaginary parts: -4i + 3i = -i.
Hence, the subtraction of the complex numbers \((7 - 4i) - (5 - 3i)\) results in the simplified form \( 2 - i \).
Simplifying Complex Expressions
Simplifying expressions involving complex numbers requires organizing and combining like terms. Complex numbers have both real parts and imaginary parts:
The expression we started with is: $$ (7 - 4i) - (5 - 3i). $$
To simplify this:
  • First, break them into their respective components:
    • Real parts: 7 and -5
    • Imaginary parts: -4i and +3i
  • Then, proceed by combining like terms:
    • Combine the real parts: $$ 7 - 5 = 2. $$

    • Combine the imaginary parts: $$ -4i + 3i = -i. $$
  • So the simplified form is: $$2 - i.$$
Always perform operations step by step to avoid mistakes and ensure clarity in your results.
Distributing Negative Signs in Algebra
Distributing negative signs correctly is essential in algebra, especially with complex numbers. When you subtract a complex number, you effectively distribute a negative sign across its real and imaginary components. Here鈥檚 how to handle this:
  • Consider the operation: $$(7 - 4i) - (5 - 3i).$$
  • The negative sign affects both terms inside the parentheses, so:
    • Distribute the negative: $$(7 - 4i) - 5 + 3i.$$
  • This changes the signs of both the real and imaginary parts of the second complex number.
    Here are the steps for better comprehension:
    • Write out the given problem.
    • Apply the negative sign to each term inside the second complex number.
    • Combine like terms (real with real, imaginary with imaginary).
  • After distributing, continue to simplifying the expression:
    • $$ 7 - 5 = 2$$
    • $$ -4i + 3i = -i.$$
    Thus, the final simplified form is $$ 2 - i.$$
Mastering this technique will enhance your algebra skills, making complex operations less daunting.

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