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Chapter 2: Problem 23

Perform the addition or subtraction and write the result in standard form. \((9-i)-(8-i)\)

Short Answer

Expert verified
The result of subtracting \((8-i)\) from \((9-i)\) is \(1+0i\), or more simply, just 1.

Step by step solution

01

Identify the complex numbers

The complex numbers provided in the exercise are \(9-i\) and \(8-i\). Both of these are already in the standard form of a complex number, \((a - bi)\).
02

Apply subtraction operation

\((9-i)-(8-i)\) can be rewritten as \(9-i-8+i\), according to the distributive property of subtraction over addition.
03

Combine like terms

Now, combine the like terms to simplify the expression. This gives \(1+0i\).

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Key Concepts

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

Addition and Subtraction of Complex Numbers
Complex numbers can be a bit tricky at first glance, but once you understand the basic operations involved, they become quite straightforward.
The exercise given focuses on performing subtraction of two complex numbers. A complex number is usually in the form of \((a + bi)\), where \(a\) is the real part and \(b\) is the imaginary part.
To subtract two complex numbers, we perform the operation on their respective real and imaginary parts separately.
  • Subtract the real parts and then subtract the imaginary parts.
  • In the case of our example, we have two complex numbers \(9-i\) and \(8-i\).
  • We perform the subtraction separately:
    • Real parts subtraction: \(9 - 8 = 1\)
    • Imaginary parts subtraction: \(-i - (-i) = 0i\)
  • Finally, we combine these results to get \(1 + 0i\).
These steps ensure you correctly apply the operations and get the desired result when working with complex numbers.
Standard Form of Complex Numbers
The standard form of a complex number is quite significant as it allows for easier manipulation and understanding of these numbers.
Complex numbers are expressed in the form \(a + bi\), where:
  • \(a\) is the real component, which is a real number.
  • \(b\) is the imaginary component, which is also a real number but is multiplied by \(i\), the imaginary unit \(i\) where \(i^2 = -1\).
This standard form is essential for performing calculations such as addition, subtraction, multiplication, and division. For example, in our exercise, both numbers \(9-i\) and \(8-i\) are already in this form, making the arithmetic operation straightforward.
Being comfortable with identifying and using this form is crucial when dealing with complex number operations.
Complex Number Operations
Operations with complex numbers follow a pattern quite similar to operations with real numbers, with some additional considerations due to the presence of the imaginary part.
When performing any arithmetic operation, it's important to:
  • Treat the real and imaginary parts separately.
  • Use proper simplification methods like combining like terms.
  • Be aware of how the imaginary unit \(i\) behaves, especially since \(i^2 = -1\).
In our example, we see subtraction:
  • First, distribute the negative sign across the second complex number, which turns subtraction into the addition of negative terms, resulting in adding the inverse.
  • Finally, combine the like terms, which means adding real parts together and imaginary parts together to simplify into the solution, \(1 + 0i\).
This consistency of approach helps in making operations on complex numbers predictable and easier to handle.

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Most popular questions from this chapter

Solve the inequality and graph the solution on the real number line. \(\frac{1}{x-3} \leq \frac{9}{4 x+3}\)

Solve the inequality and graph the solution on the real number line. \(x^{2}+4 x+4 \geq 9\)

Solve the inequality and graph the solution on the real number line. \(x^{2}<9\)

Solve the inequality and graph the solution on the real number line. \(x^{2}-6 x+9<16\)

(a) find the interval(s) for \(b\) such that the equation has at least one real solution and (b) write a conjecture about the interval(s) based on the values of the coefficients. \(2 x^{2}+b x+5=0\)
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