Problem 43 - Elementary and Intermediate Algebra

Question

Perform the operations. Write all answers in the form \(a+b i .\) See Example 3 $$ (8+\sqrt{-25})-(7+\sqrt{-4}) $$

Step-by-Step Solution

Answer

The result is \(1 + 3i\).

Step 1: Understand the Problem

We have been given two complex numbers and need to subtract the second from the first: \( (8 + \sqrt{-25}) - (7 + \sqrt{-4}) \). We know that \( \sqrt{-1} = i \), thus \( \sqrt{-25} = \sqrt{25} \cdot \sqrt{-1} = 5i \) and \( \sqrt{-4} = \sqrt{4} \cdot \sqrt{-1} = 2i \).

Step 2: Substitute Imaginary Parts

Substitute \( \sqrt{-25} \) with \( 5i \) and \( \sqrt{-4} \) with \( 2i \) in the expression: \( (8 + 5i) - (7 + 2i) \).

Step 3: Subtract the Real Parts

Subtract the real numbers separately: \( 8 - 7 = 1 \).

Step 4: Subtract the Imaginary Parts

Subtract the imaginary numbers separately: \( 5i - 2i = 3i \).

Step 5: Combine the Results

Combine the results from Steps 3 and 4 into the form \( a + bi \): \( 1 + 3i \).

Key Concepts

Imaginary Unit

Complex numbers can be mystifying at first because they involve a special number: the imaginary unit, represented by the symbol \( i \). The imaginary unit \( i \) is defined by the equation \( i^2 = -1 \). This may seem unusual since there is no real number whose square is negative. What \( i \) does is provide a way to work with square roots of negative numbers. By convention, anytime you see \( \sqrt{-1} \), you replace it with \( i \).
For the exercise given, converting square roots of negative numbers like \( \sqrt{-25} \) and \( \sqrt{-4} \) involves the imaginary unit. You split each square root into a product of the square root of a positive number and \( i \). For example:

• \( \sqrt{-25} = \sqrt{25} \times \sqrt{-1} = 5i \)
• \( \sqrt{-4} = \sqrt{4} \times \sqrt{-1} = 2i \)

Thus, using \( i \), you're able to express roots that would otherwise be undefined in the real number system.

Operations with Complex Numbers

Performing operations with complex numbers, such as addition, subtraction, multiplication, or division, follows some straightforward rules. A complex number is usually written in the standard form \( a + bi \), where \( a \) is the real part and \( bi \) is the imaginary part. When performing operations:

• **Addition/Subtraction**: Simply combine like terms, i.e., real parts with real parts, and imaginary parts with imaginary parts.
• **Multiplication**: You use the distributive property, remembering that \( i^2 = -1 \).
• **Division**: This often involves multiplying the numerator and denominator by the conjugate of the denominator to simplify.

In our example, to subtract the complex numbers \( (8 + 5i) - (7 + 2i) \):

• Subtract the real parts: \( 8 - 7 \) yielding \( 1 \).
• Subtract the imaginary parts: \( 5i - 2i \) yielding \( 3i \).

This gives the final result as \( 1 + 3i \). Each operation is treated separately for clarity and accuracy.

Subtraction of Complex Numbers

When you subtract complex numbers, the concept is to subtract each corresponding part—real from real and imaginary from imaginary. This keeps each component defined separately and clearly. To keep it clear:

• Identify each part of the number: for example, in \( 7 + 2i \), the real part is 7, and the imaginary part is \( 2i \).
• Subtract the respective parts. Continuing from our task where \( 8 + 5i \) and \( 7 + 2i \) are the numbers involved:
  • First, the real components: subtract \( 7 \) from \( 8 \), which gives you \( 1 \).
  • Then, the imaginary components: subtract \( 2i \) from \( 5i \), which gives you \( 3i \).

Finally, gather these results into the unified form \( 1 + 3i \). This process ensures you manage both dimensions of a complex number precisely and accurately during subtraction.