Which Property Allows You To Write The Expression \[$(3+7i)+(8-6i)\$\] As \[$(8-6i)+(3+7i)\$\]?

by ADMIN 97 views

Introduction

In mathematics, the commutative property is a fundamental concept that allows us to rearrange the order of numbers or expressions without changing their value. In this article, we will explore the commutative property of addition in complex numbers, specifically the property that allows us to write the expression {(3+7i)+(8-6i)$}$ as {(8-6i)+(3+7i)$}$.

What are Complex Numbers?

Complex numbers are numbers that can be expressed in the form {a+bi$}$, where {a$}$ and {b$}$ are real numbers, and {i$}$ is the imaginary unit, which satisfies the equation {i^2=-1$}$. Complex numbers can be added, subtracted, multiplied, and divided, just like real numbers.

The Commutative Property of Addition

The commutative property of addition states that the order of the numbers being added does not change the result. In other words, when we add two numbers, we can swap their positions and still get the same result. This property is denoted by the equation {a+b=b+a$}$.

Applying the Commutative Property to Complex Numbers

Now, let's apply the commutative property of addition to the complex numbers {(3+7i)$}$ and {(8-6i)$}$. We can write the expression {(3+7i)+(8-6i)$}$ as {(8-6i)+(3+7i)$}$ because the order of the numbers being added does not change the result.

Why Does the Commutative Property Work?

The commutative property of addition works because of the way we define addition in complex numbers. When we add two complex numbers, we add their real parts and their imaginary parts separately. In the case of {(3+7i)$}$ and {(8-6i)$}$, we have:

{(3+7i)+(8-6i)=(3+8)+(7i-6i)=(11+1i)$}$

and

{(8-6i)+(3+7i)=(8+3)+(-6i+7i)=(11+1i)$}$

As we can see, the order of the numbers being added does not change the result.

Real-World Applications

The commutative property of addition has many real-world applications. For example, in finance, when we add two amounts of money, we can swap their positions and still get the same result. In physics, when we add two velocities, we can swap their positions and still get the same result.

Conclusion

In conclusion, the commutative property of addition in complex numbers allows us to write the expression {(3+7i)+(8-6i)$}$ as {(8-6i)+(3+7i)$}$. This property is a fundamental concept in mathematics that has many real-world applications. By understanding the commutative property of addition, we can simplify complex calculations and make sense of the world around us.

Examples and Exercises

Here are some examples and exercises to help you practice the commutative property of addition in complex numbers:

Example 1

Write the expression {(2+5i)+(3-4i)$}$ as {(3-4i)+(2+5i)$}$.

Solution

{(2+5i)+(3-4i)=(2+3)+(5i-4i)=(5+1i)$}$

and

{(3-4i)+(2+5i)=(3+2)+(-4i+5i)=(5+1i)$}$

Example 2

Write the expression {(4-3i)+(2+6i)$}$ as {(2+6i)+(4-3i)$}$.

Solution

{(4-3i)+(2+6i)=(4+2)+(-3i+6i)=(6+3i)$}$

and

{(2+6i)+(4-3i)=(2+4)+(-3i+6i)=(6+3i)$}$

Exercise 1

Write the expression {(1+2i)+(3-4i)$}$ as {(3-4i)+(1+2i)$}$.

Exercise 2

Write the expression {(2-3i)+(4+5i)$}$ as {(4+5i)+(2-3i)$}$.

Glossary

  • Complex number: A number that can be expressed in the form {a+bi$}$, where {a$}$ and {b$}$ are real numbers, and {i$}$ is the imaginary unit.
  • Imaginary unit: A number that satisfies the equation {i^2=-1$}$.
  • Commutative property of addition: The property that states that the order of the numbers being added does not change the result.

References

  • "Complex Numbers" by Math Open Reference
  • "Commutative Property of Addition" by Khan Academy

Frequently Asked Questions

Q: What is the commutative property of addition in complex numbers?

A: The commutative property of addition in complex numbers is a fundamental concept that states that the order of the numbers being added does not change the result. In other words, when we add two complex numbers, we can swap their positions and still get the same result.

Q: How do I apply the commutative property of addition to complex numbers?

A: To apply the commutative property of addition to complex numbers, simply swap the positions of the two numbers being added. For example, if we have the expression {(3+7i)+(8-6i)$}$, we can write it as {(8-6i)+(3+7i)$}$.

Q: Why does the commutative property of addition work in complex numbers?

A: The commutative property of addition works in complex numbers because of the way we define addition in complex numbers. When we add two complex numbers, we add their real parts and their imaginary parts separately. This means that the order of the numbers being added does not change the result.

Q: What are some real-world applications of the commutative property of addition in complex numbers?

A: The commutative property of addition in complex numbers has many real-world applications. For example, in finance, when we add two amounts of money, we can swap their positions and still get the same result. In physics, when we add two velocities, we can swap their positions and still get the same result.

Q: Can I use the commutative property of addition to simplify complex calculations?

A: Yes, you can use the commutative property of addition to simplify complex calculations. By swapping the positions of the numbers being added, you can make the calculation easier to understand and solve.

Q: Are there any limitations to the commutative property of addition in complex numbers?

A: No, there are no limitations to the commutative property of addition in complex numbers. It works for all complex numbers, regardless of their values.

Q: Can I use the commutative property of addition to solve equations involving complex numbers?

A: Yes, you can use the commutative property of addition to solve equations involving complex numbers. By swapping the positions of the numbers being added, you can make the equation easier to solve.

Q: Are there any other properties of addition in complex numbers that I should know about?

A: Yes, there are other properties of addition in complex numbers that you should know about. For example, the associative property of addition states that the order in which we add three or more complex numbers does not change the result.

Q: Can I use the commutative property of addition to solve problems involving complex numbers in other areas of mathematics?

A: Yes, you can use the commutative property of addition to solve problems involving complex numbers in other areas of mathematics, such as algebra and geometry.

Common Mistakes to Avoid

  • Swapping the positions of the numbers being added incorrectly: Make sure to swap the positions of the numbers being added correctly to avoid getting the wrong result.
  • Not using the commutative property of addition when it is applicable: Make sure to use the commutative property of addition when it is applicable to simplify complex calculations.
  • Not understanding the concept of complex numbers: Make sure to understand the concept of complex numbers before using the commutative property of addition.

Tips and Tricks

  • Use the commutative property of addition to simplify complex calculations: By swapping the positions of the numbers being added, you can make the calculation easier to understand and solve.
  • Use the commutative property of addition to solve equations involving complex numbers: By swapping the positions of the numbers being added, you can make the equation easier to solve.
  • Use the commutative property of addition to solve problems involving complex numbers in other areas of mathematics: By swapping the positions of the numbers being added, you can make the problem easier to solve.

Glossary

  • Complex number: A number that can be expressed in the form {a+bi$}$, where {a$}$ and {b$}$ are real numbers, and {i$}$ is the imaginary unit.
  • Imaginary unit: A number that satisfies the equation {i^2=-1$}$.
  • Commutative property of addition: The property that states that the order of the numbers being added does not change the result.

References

  • "Complex Numbers" by Math Open Reference
  • "Commutative Property of Addition" by Khan Academy