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Chapter 1: Problem 88

Simplify each expression and write it in the standard form \(a+b i\). \((6-4 i)+(2+5 i)\)

Short Answer

Expert verified
The simplified form is \(8 + i\).

Step by step solution

01

Identify the Real Parts

In the expression \((6-4i)+(2+5i)\), first identify the real parts which are \(6\) and \(2\).
02

Identify the Imaginary Parts

Next, identify the imaginary parts from the expression. These are \(-4i\) and \(5i\).
03

Add the Real Parts

Add the real parts together: \(6 + 2 = 8\).
04

Add the Imaginary Parts

Add the imaginary parts together: \(-4i + 5i = 1i\).
05

Combine the Results

Combine the results from Steps 3 and 4 to form the standard form of the complex number, \(8 + 1i\).

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Key Concepts

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

Addition of Complex Numbers
Adding complex numbers might seem challenging at first, but it's quite similar to adding simple algebraic expressions. Complex numbers have two parts: a real part and an imaginary part. When adding them together, the primary goal is to add these parts separately, as they each follow distinct rules.

Let's look at an example
  • (6 - 4i) + (2 + 5i) where we clearly have two complex numbers.
  • First, we take the real parts of each number, which are 6 and 2.
  • Then, we add the imaginary parts: -4i and 5i.

By adding the real parts, we obtain 6 + 2 = 8 and by adding the imaginary parts, we get -4i + 5i = i. Combine these results to achieve the final sum: 8 + i.
Real and Imaginary Parts
As we delve into complex numbers, it's essential to differentiate between their two components, the real and imaginary parts. A complex number is usually expressed as a + bi, where a represents the real part and bi represents the imaginary part. Both parts play crucial roles in arithmetic operations involving complex numbers.

  • The real part, a, is simply the number that is not multiplied by the imaginary unit, i.
  • The imaginary part, on the other hand, includes the component multiplied by i.

In the expression 6 - 4i, the real part is 6, and the imaginary part is -4i. Recognizing these components allows us to perform operations, such as addition, more efficiently by combining like terms.
Standard Form of Complex Numbers
Expressing a complex number in its standard form is essential for clarity, simplicity, and consistent mathematical communication. The standard form of a complex number is a + bi, where a and b are real numbers. Here, a is the real part, and b is the coefficient of the imaginary part, i.

For example, if we add the complex numbers (6 - 4i) and (2 + 5i), the result should always be written as 8 + i.
  • This format ensures that both components are clearly visible and distinct.
  • It's important to write the imaginary part last, even if it is negative, to maintain consistency.

The standard form offers a straightforward way to represent complex numbers, making further mathematical operations more seamless and intuitive.

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Most popular questions from this chapter

When log \(y\) is graphed as a function of \(x, a\) straight line results. Graph straight lines, each given by two points, on a log-linear plot, and determine the functional relationship. (The original \(x-y\) coordinates are given.) $$ \left(x_{1}, y_{1}\right)=(-1,4),\left(x_{2}, y_{2}\right)=(2,8) $$

Compare a square with side length \(1 \mathrm{~m}\) against a square with side length \(100 \mathrm{~m}\). The area of the larger square is order(s) of magnitude larger than the area of the smaller square.

Use a logarithmic transformation to find a linear relationship between the given quantities and graph the resulting linear relationship on a log-log plot. $$ y=3 x^{2} $$

Use a logarithmic transformation to find a linear relationship between the given quantities and determine whether a log-log or log-linear plot should be used to graph the resulting linear relationship. $$ N(t)=130 \times 2^{1.2 t} $$

Sketch the graph of each function. Do not use a graphing calculator. (Assume the largest possible domain.) $$ y=1-1 / x $$
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