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Give all answers in $a+b i$ form. $$(3+\sqrt{-4})-(2+\sqrt{-9})$$

   Give all answers in $a+b i$ form.
$$(3+\sqrt{-4})-(2+\sqrt{-9})$$

College Algebra

College Algebra

R. David Gustafson,… 11th Edition

Chapter 1, Problem 27

Instant Answer

Solved by Expert Amy Jiang

07/07/2020

Step 1

We know that $\sqrt{-1}$ is represented as $i$ in complex numbers. So, $\sqrt{-4}$ becomes $2i$ (since $\sqrt{4}$ is $2$) and $\sqrt{-9}$ becomes $3i$ (since $\sqrt{9}$ is $3$). So, the expression becomes: $$(3+2i)-(2+3i)$$ Show more…

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Give all answers in $a+b i$ form. $$(3+\sqrt{-4})-(2+\sqrt{-9})$$

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Key Concepts

Complex Numbers

Complex numbers extend the real numbers by incorporating an imaginary unit, typically represented as i, where i² = -1. They are expressed in the form a + bi, with a being the real part and b being the imaginary part. This form simplifies operations like addition, subtraction, multiplication, and division of complex numbers while allowing for the representation of solutions to equations that do not have real solutions.

Imaginary Unit

The imaginary unit, denoted as i, is defined by the property that i² = -1. It is used to transform the square roots of negative numbers into an algebraic form, allowing them to be easily manipulated in mathematical expressions. For instance, the square root of a negative number, such as ?(-k), can be rewritten as i?(k), which fits into the standard form of complex numbers.

Arithmetic Operations on Complex Numbers

Arithmetic operations on complex numbers, including addition, subtraction, multiplication, and division, are performed by handling the real and imaginary parts separately. In addition and subtraction, one combines the corresponding real parts and the imaginary parts independently. This method ensures that the result remains in the standard form a + bi, facilitating further analysis and computation.

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