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Chapter 10: Problem 43

Add or subtract as indicated. Give answers in standard form. $$ (4+i)-(-3-2 i) $$

Short Answer

Expert verified
7 + 3i.

Step by step solution

01

Identify the complex numbers

Determine the two complex numbers involved in the equation. Here, they are (4 + i) and (-3 - 2i).
02

Rewrite the subtraction

Rewrite the subtraction operation as the addition of the opposite: (4 + i) - (-3 - 2i) becomes (4 + i) + (3 + 2i).
03

Combine the real parts

Add the real parts of the two complex numbers: 4 + 3 = 7.
04

Combine the imaginary parts

Add the imaginary parts of the two complex numbers: i + 2i = 3i.
05

Write the result in standard form

Combine the results from Steps 3 and 4 to write the final answer in standard form: 7 + 3i.

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Key Concepts

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

addition of complex numbers
To understand how to add complex numbers, you first need to know what a complex number is. A complex number is written in the form: \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. When adding complex numbers, you handle the real parts and imaginary parts separately. For instance, if you have two complex numbers, \(4+i\) and \(-3-2i\), you add them component-wise:
  • Real parts: \(4 + (-3) = 1\)
  • Imaginary parts: \(i + (-2i) = -i\)
Combining them, we get: \(1 - i\). Don't forget, simplifying in steps can make it easier鈥攈andle the real and imaginary parts one by one.
standard form
Complex numbers should always be written in their standard form, which is \(a + bi\). This format ensures consistency and clarity when manipulating the numbers. For example, after finding the real and imaginary sums, you need to combine them properly:

Consider the problem: \((4+i) - (-3-2i)\). By converting the subtraction to addition, it becomes: \((4+i) + (3+2i)\). Now we add the real parts and imaginary parts separately, reaching \(7+3i\). This represents the standard form: \(a = 7\) and \(bi = 3i\).

Keep the order \(a + bi\) intact, ensuring math problems are solved and presented correctly.
imaginary units
Imaginary units are an essential part of complex numbers. It uses the symbol \(i\), which is defined as \(i^2 = -1\). Imaginary units allow us to work with square roots of negative numbers. Consider the operation involving imaginary parts: in the example \((4+i) - (-3-2i)\), adding the imaginary parts means combining \(i + 2i\):

  • Understand that \(i\) is an imaginary unit
  • The result of \(i\) multiplied by any real number stays an imaginary number
  • Add the coefficients of \(i\) just like you would with regular numbers
In our case, the imaginary parts \(i + 2i\) equals \(3i\). Imaginary units help manage operations involving square roots of negatives and keep expressions in a usable, coherent form.

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