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Chapter 2: Problem 71

Explain the error. $$\sqrt{-9}+\sqrt{-16}=\sqrt{-25}=i \sqrt{25}=5 i$$

Short Answer

Expert verified
The error lies in the incorrect simplification of the complex numbers and their addition. The correct sum of \( \sqrt{-9} \) and \( \sqrt{-16} \) is \( 7i \), not \( 5i \) as shown in the equation.

Step by step solution

01

Identify the error

The crucial error in the equation \( \sqrt{-9}+\sqrt{-16}=\sqrt{-25}=i \sqrt{25}=5 i \) is that the square roots haven't been simplified correctly.
02

Simplify the complex numbers

To resolve this error, you must correctly find the square root of negative numbers: \( \sqrt{-9} = 3i \) and \( \sqrt{-16} = 4i \).
03

Correct Addition

You should now add the two complex numbers correctly: \( 3i + 4i = 7i \).
04

Verify the final result

So, after adding the two numbers together, we get 7i, not 5i as shown in the given equation. This verifies that the original equation is incorrect.

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Key Concepts

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

Imaginary Numbers
Imaginary numbers are an essential component of complex numbers. They arise when you attempt to find the square root of a negative number. Typically, the square root of a positive number has a straightforward solution, but negative numbers introduce a twist: their square roots are not real. Thus, imaginary numbers are born.
An imaginary number is represented as a real number multiplied by the imaginary unit, denoted as \(i\). The defining property of \(i\) is \(i^2 = -1\). This property allows us to work with square roots of negative numbers and engage in new realms of mathematical operations.
Remember:
  • The imaginary unit \(i\) helps us explore and calculate the roots of negative numbers.
  • When you see a term like \(3i\), it signifies "three times the imaginary unit."
Square Root
The square root is a mathematical operation that determines what number, multiplied by itself, will yield the original number. For instance, the square root of 9 is 3 because \(3 \times 3 = 9\). Things get interesting with negative numbers, as real number solutions no longer apply.
When finding the square root of a negative number, the result involves imaginary numbers. For example, take \(\sqrt{-9}\). You can break it down into:
  • \(\sqrt{9} = 3\) (the positive square root of 9).
  • Add the imaginary unit: \( \sqrt{-9} = 3i \).
You can always follow a similar process鈥攆ind the square root as you normally would, then introduce \(i\) to indicate the imaginary component.
Addition of Complex Numbers
Adding complex numbers is similar to adding algebraic expressions. Remember, complex numbers have a real part and an imaginary part. When adding them, focus only on like terms: real parts with real parts and imaginary parts with imaginary parts.
Consider two complex numbers: \(3i\) and \(4i\).
Their addition looks like this:
  • Combine the imaginary parts: \(3i + 4i\).
  • The result is \(7i\), since both numbers were purely imaginary.
To sum up, for any complex numbers, simply add the real components together and the imaginary ones together for your final result.

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