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Chapter 3: Problem 27

Add or subtract. Write the answer in standard form. \(7-(3+4 i)+6 i\)

Short Answer

Expert verified
The result of the operation \(7-(3+4 i)+6 i\) in standard form is \(4+2 i\).

Step by step solution

01

Identify the real and imaginary parts

In the provided expression, \(7-(3+4 i)+6 i\), first, identify the real and the imaginary numbers. Here, 7 and 3 are real numbers while \(4 i\) and \(6 i\) are imaginary numbers.
02

Regroup and perform the operations

Now, group together the real and imaginary parts separately and perform the appropriate addition or subtraction. That will be \(7-3\) for the real parts and \(-4 i + 6 i\) for the imaginary parts. After performing these operations, the result will be \(4+2 i\).
03

Write the answer in standard form

The standard form of a complex number is \(a+bi\). So, after Step 2, the result is in standard form, which is \(4 + 2i\).

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Key Concepts

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

Real Numbers
Real numbers are the building blocks of many mathematical concepts. They include all the numbers that you can find on the number line. This includes integers like -2, 0, 5, fractions like 1/2, and irrational numbers like \(\sqrt{2}\). These numbers have no imaginary component and are denoted simply as \(a\) in a complex number.
  • Real numbers are infinite in both positive and negative direction on the number line.
  • They can be added, subtracted, multiplied, and divided, abiding by the real number axioms.
  • In the context of complex numbers, when you identify real numbers, you're looking for the terms without an \(i\).
In the expression \(7-(3+4i)+6i\), the real numbers are 7 and 3. These illustrate how real numbers can be extracted and subsequently calculated on their own.
Imaginary Numbers
Imaginary numbers add a fascinating twist to mathematics. Defined by the imaginary unit \(i\), which represents \(\sqrt{-1}\), these numbers help address operations that aren't possible with real numbers alone, such as finding square roots of negative numbers.
  • Imaginary numbers exist along a separate imaginary axis, perpendicular to the real number line.
  • Mathematically, multiplying an imaginary number by itself returns a negative real number.
  • In any complex number, imaginary numbers are represented as \(bi\).
In our expression \(7-(3+4i)+6i\), the imaginary parts are \(-4i\) and \(6i\). By grouping these terms, we can isolate and calculate the imaginary components to achieve the final solution.
Standard Form of Complex Numbers
Complex numbers combine both real and imaginary numbers in a specific format known as the standard form. This form is a simple and systematic way to represent the combined effect of both number types. It is written as \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part.
  • Using this format helps in easily identifying and computing with complex numbers.
  • It allows you to separate calculations into real and imaginary components.
  • Consistency in this format aids in unifying complex number calculations and comparisons.
In the problem \(7-(3+4i)+6i\), after simplifying both the real (7 - 3 = 4) and imaginary parts (-4i + 6i = 2i), we arrive at the standard form \(4 + 2i\). This clearly shows both parts, structured neatly in the equation.

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