The Correct Application of the Commutative Property

Oct 13, 2025 by miodeya.com 110 views

Understanding the Commutative Property

In mathematics, the commutative property is a fundamental concept that states that the order of the numbers being added or multiplied does not change the result. This property is often represented as:

a + b = b + a

a × b = b × a

Aiko's Mistake

Aiko attempted to rewrite the sum ${$ (4 + 5i) + (-3 + 7i)$}$ as ${$ (-3 + 7)i + (4 + 5)i$}$. However, this is not a correct application of the commutative property.

The Correct Application of the Commutative Property

To correctly apply the commutative property, Aiko should have rearranged the terms as follows:

[$(4 + 5i) + (-3 + 7i) = (4 + (-3)) + (5i + 7i)$

This simplifies to:

[$(4 - 3) + (5i + 7i) = 1 + 12i$

Aiko's Error

Aiko's mistake was in incorrectly applying the commutative property by rearranging the terms in a way that changed the result. The correct application of the commutative property would have resulted in a different sum.

The Correct Statement

The correct statement that explains Aiko's error is:

B. Aiko incorrectly used the commutative property by rearranging the terms in a way that changed the result.

Conclusion

In conclusion, Aiko's error was in incorrectly applying the commutative property by rearranging the terms in a way that changed the result. The correct application of the commutative property would have resulted in a different sum.

Understanding the Distributive Property

The distributive property is another fundamental concept in mathematics that states that a single operation can be distributed over multiple terms. This property is often represented as:

a(b + c) = ab + ac

a(b - c) = ab - ac

Aiko's Error in Using the Distributive Property

Aiko's error was not in using the distributive property, but rather in incorrectly applying the commutative property.

The Distributive Property in Complex Numbers

The distributive property can be applied to complex numbers in a similar way to real numbers. For example:

[$(2 + 3i)(4 + 5i) = 2(4 + 5i) + 3i(4 + 5i)$

This simplifies to:

[$8 + 10i + 12i + 15i^2 = 8 + 22i - 15$

Conclusion

In conclusion, Aiko's error was in incorrectly applying the commutative property by rearranging the terms in a way that changed the result. The distributive property is a separate concept that can be applied to complex numbers in a similar way to real numbers.

Understanding the Associative Property

The associative property is another fundamental concept in mathematics that states that the order in which we perform operations does not change the result. This property is often represented as:

(a + b) + c = a + (b + c)

(a × b) × c = a × (b × c)

Aiko's Error in Using the Associative Property

Aiko's error was not in using the associative property, but rather in incorrectly applying the commutative property.

The Associative Property in Complex Numbers

The associative property can be applied to complex numbers in a similar way to real numbers. For example:

[$(2 + 3i) + (4 + 5i) + (6 + 7i) = (2 + 4 + 6) + (3i + 5i + 7i)$

This simplifies to:

[$12 + 15i$

Conclusion

In conclusion, Aiko's error was in incorrectly applying the commutative property by rearranging the terms in a way that changed the result. The associative property is a separate concept that can be applied to complex numbers in a similar way to real numbers.

Real-World Applications of Mathematical Properties

Mathematical properties such as the commutative, distributive, and associative properties have numerous real-world applications. For example:

In physics, the commutative property is used to describe the behavior of particles in quantum mechanics.
In engineering, the distributive property is used to design and analyze complex systems.
In finance, the associative property is used to calculate compound interest.

Conclusion

Q: What is the commutative property in mathematics?

A: The commutative property is a fundamental concept in mathematics that states that the order of the numbers being added or multiplied does not change the result. This property is often represented as:

a + b = b + a

a × b = b × a

Q: What was Aiko's mistake in the problem?

A: Aiko's mistake was in incorrectly applying the commutative property by rearranging the terms in a way that changed the result.

Q: What is the correct application of the commutative property?

A: The correct application of the commutative property would have resulted in a different sum. For example:

[$(4 + 5i) + (-3 + 7i) = (4 + (-3)) + (5i + 7i)$

This simplifies to:

[$(4 - 3) + (5i + 7i) = 1 + 12i$

Q: What is the distributive property in mathematics?

A: The distributive property is another fundamental concept in mathematics that states that a single operation can be distributed over multiple terms. This property is often represented as:

a(b + c) = ab + ac

a(b - c) = ab - ac

Q: How is the distributive property applied to complex numbers?

A: The distributive property can be applied to complex numbers in a similar way to real numbers. For example:

[$(2 + 3i)(4 + 5i) = 2(4 + 5i) + 3i(4 + 5i)$

This simplifies to:

[$8 + 10i + 12i + 15i^2 = 8 + 22i - 15$

Q: What is the associative property in mathematics?

A: The associative property is another fundamental concept in mathematics that states that the order in which we perform operations does not change the result. This property is often represented as:

(a + b) + c = a + (b + c)

(a × b) × c = a × (b × c)

Q: How is the associative property applied to complex numbers?

A: The associative property can be applied to complex numbers in a similar way to real numbers. For example:

[$(2 + 3i) + (4 + 5i) + (6 + 7i) = (2 + 4 + 6) + (3i + 5i + 7i)$

This simplifies to:

[$12 + 15i$

Q: What are some real-world applications of mathematical properties?

A: Mathematical properties such as the commutative, distributive, and associative properties have numerous real-world applications. For example:

In physics, the commutative property is used to describe the behavior of particles in quantum mechanics.
In engineering, the distributive property is used to design and analyze complex systems.
In finance, the associative property is used to calculate compound interest.

Q: Why is it important to understand mathematical properties?

A: Understanding mathematical properties is essential in various fields of study, including physics, engineering, and finance. Mathematical properties provide a foundation for solving complex problems and making accurate calculations.

Q: How can I practice applying mathematical properties?

A: You can practice applying mathematical properties by working through examples and exercises in your math textbook or online resources. You can also try solving real-world problems that involve mathematical properties.

Q: What are some common mistakes to avoid when applying mathematical properties?

A: Some common mistakes to avoid when applying mathematical properties include:

Incorrectly applying the commutative property by rearranging terms in a way that changes the result.
Failing to distribute a single operation over multiple terms.
Not considering the order in which operations are performed.

Q: How can I improve my understanding of mathematical properties?

A: You can improve your understanding of mathematical properties by:

Practicing regularly and working through examples and exercises.
Seeking help from a teacher or tutor if you are struggling with a concept.
Exploring real-world applications of mathematical properties.