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Chapter 6: Problem 2

Calculate the given expression. \((2+3 i)-(-3-2 i)\)

Short Answer

Expert verified
The expression simplifies to \( 5 + 5i \).

Step by step solution

01

- Identify the expression

The given expression is \( (2 + 3i) - (-3 - 2i) \). This involves subtracting one complex number from another.
02

- Distribute the negative sign

Distribute the negative sign inside the parentheses: \( (2 + 3i) - (-3 - 2i) = (2 + 3i) + (3 + 2i) \). Notice that subtracting a negative is the same as adding the positive counterpart.
03

- Group the real and imaginary parts

Combine the real parts together and the imaginary parts together: \( (2 + 3) + (3i + 2i) \). This simplifies to \( 5 + 5i \).
04

- Simplify

Simplify the expression to combine the real and imaginary parts: \( 5 + 5i \).

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Key Concepts

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

Subtracting Complex Numbers
In this exercise, we are working with complex numbers. A complex number has both a real part and an imaginary part. For example, in the complex number, (2 + 3i), 2 is the real part and 3i is the imaginary part. The key to subtracting complex numbers is to remember that each part (real and imaginary) must be handled separately. When we subtract the complex number (-3 - 2i) from (2 + 3i), we need to distribute the negative sign to both the real and imaginary parts of the number being subtracted. This changes the expression to addition: (2 + 3i) - (-3 - 2i) becomes (2 + 3i) + (3 + 2i).
Distributing Negative Signs
Distributing negative signs correctly is crucial when subtracting complex numbers. When the negative sign is distributed, it affects both elements inside the parentheses. In our exercise, (2 + 3i) - (-3 - 2i) becomes (2 + 3i) + (3 + 2i). This is because subtracting a negative number is equivalent to adding a positive number.
  • The minus before the parentheses changes the sign of both -3 and -2i.
  • So, -(-3) becomes +3 and -(-2i) becomes +2i.
Hence, the expression simplifies from subtraction to addition.
Combining Real and Imaginary Parts
After distributing the negative sign, we need to combine the real and imaginary parts. Treat the real numbers separately from the imaginary ones. Think of it like combining like terms. Here鈥檚 how to do it step-by-step:
  • First, we take the real parts: 2 and 3.
  • Then, we take the imaginary parts: 3i and 2i.
So, we have (2 + 3) + (3i + 2i). Combine these to get 5 + 5i. This is our simplified answer. The real parts add together to make 5, and the imaginary parts add together to make 5i. This final form 5 + 5i is a combination of the real and imaginary parts we started with.

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Most popular questions from this chapter

Write the given equation in polar coordinates. Graph the function \(r=1+2 \cos \theta\) in polar coordinates.

Convert the given point from polar coordinates to Cartesian coordinates. \(\left(6,90^{\circ}\right)\)

Calculate the given expression. \(\overline{(2+3 i)}-\overline{(-3-2 i)}\)

Prove the given identity for all complex numbers. \(\overline{\left(\frac{z_{1}}{z_{2}}\right)}=\frac{\overline{z_{1}}}{\overline{z_{2}}}\)

Convert the given point from polar coordinates to Cartesian coordinates. \(\left(-1,405^{\circ}\right)\)
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