Which Property Of Addition Is Shown Below?$\[ 3 + 4i + 5 + 6i = 3 + 5 + 4i + 6i \\]A. Identity Property B. Associative Property C. Commutative Property D. Inverse Property

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In mathematics, addition is a fundamental operation that combines two or more numbers to produce a sum. However, there are different properties of addition that govern how numbers can be added together. In this article, we will explore the properties of addition and determine which property is shown in the given equation.

What are the Properties of Addition?

There are four main properties of addition: identity property, commutative property, associative property, and inverse property. Each of these properties has a specific rule that governs how numbers can be added together.

Identity Property

The identity property of addition states that when a number is added to zero, the result is the original number. In other words, any number added to zero remains unchanged. This property is denoted by the equation:

a + 0 = a

For example, 5 + 0 = 5.

Commutative Property

The commutative property of addition states that the order of the numbers being added does not change the result. In other words, the numbers can be swapped without affecting the sum. This property is denoted by the equation:

a + b = b + a

For example, 3 + 4 = 4 + 3.

Associative Property

The associative property of addition states that when three or more numbers are added together, the order in which they are grouped does not change the result. In other words, the numbers can be grouped in different ways without affecting the sum. This property is denoted by the equation:

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

For example, (2 + 3) + 4 = 2 + (3 + 4).

Inverse Property

The inverse property of addition states that for every number, there is a corresponding negative number that, when added together, results in zero. In other words, the sum of a number and its negative is always zero. This property is denoted by the equation:

a + (-a) = 0

For example, 5 + (-5) = 0.

Which Property is Shown in the Given Equation?

Now that we have explored the properties of addition, let's examine the given equation:

3+4i+5+6i=3+5+4i+6i{ 3 + 4i + 5 + 6i = 3 + 5 + 4i + 6i }

To determine which property is shown in this equation, let's analyze it step by step.

First, we can combine the like terms on the left-hand side of the equation:

3+4i+5+6i=(3+5)+(4i+6i){ 3 + 4i + 5 + 6i = (3 + 5) + (4i + 6i) }

Using the commutative property of addition, we can rewrite the equation as:

(3+5)+(4i+6i)=3+5+4i+6i{ (3 + 5) + (4i + 6i) = 3 + 5 + 4i + 6i }

Now, we can see that the equation is showing the associative property of addition. The order in which the numbers are grouped does not change the result.

Conclusion

In conclusion, the given equation is showing the associative property of addition. This property states that when three or more numbers are added together, the order in which they are grouped does not change the result.

Key Takeaways

  • The identity property of addition states that when a number is added to zero, the result is the original number.
  • The commutative property of addition states that the order of the numbers being added does not change the result.
  • The associative property of addition states that when three or more numbers are added together, the order in which they are grouped does not change the result.
  • The inverse property of addition states that for every number, there is a corresponding negative number that, when added together, results in zero.

In our previous article, we explored the properties of addition and determined which property is shown in the given equation. However, we know that there are many more questions that readers may have about the properties of addition. In this article, we will answer some of the most frequently asked questions about the properties of addition.

Q: What is the difference between the commutative and associative properties of addition?

A: The commutative property of addition states that the order of the numbers being added does not change the result. For example, 3 + 4 = 4 + 3. On the other hand, the associative property of addition states that when three or more numbers are added together, the order in which they are grouped does not change the result. For example, (2 + 3) + 4 = 2 + (3 + 4).

Q: Can you give an example of the identity property of addition?

A: Yes, the identity property of addition states that when a number is added to zero, the result is the original number. For example, 5 + 0 = 5.

Q: How does the inverse property of addition relate to the concept of negative numbers?

A: The inverse property of addition states that for every number, there is a corresponding negative number that, when added together, results in zero. For example, 5 + (-5) = 0.

Q: Can you explain why the commutative property of addition is important in mathematics?

A: The commutative property of addition is important in mathematics because it allows us to rearrange the order of numbers being added without changing the result. This property is essential in many mathematical operations, such as solving equations and manipulating algebraic expressions.

Q: How does the associative property of addition relate to the concept of grouping numbers?

A: The associative property of addition states that when three or more numbers are added together, the order in which they are grouped does not change the result. This property is essential in many mathematical operations, such as solving equations and manipulating algebraic expressions.

Q: Can you give an example of a real-world application of the properties of addition?

A: Yes, the properties of addition are essential in many real-world applications, such as finance, science, and engineering. For example, in finance, the commutative property of addition is used to calculate the total cost of a purchase, while the associative property of addition is used to calculate the total cost of a series of purchases.

Q: How can I remember the properties of addition?

A: One way to remember the properties of addition is to use a mnemonic device. For example, you can use the phrase "COMA" to remember the commutative, associative, and identity properties of addition.

Conclusion

In conclusion, the properties of addition are essential in mathematics and have many real-world applications. By understanding the commutative, associative, identity, and inverse properties of addition, we can better appreciate the rules that govern how numbers can be added together.

Key Takeaways

  • The commutative property of addition states that the order of the numbers being added does not change the result.
  • The associative property of addition states that when three or more numbers are added together, the order in which they are grouped does not change the result.
  • The identity property of addition states that when a number is added to zero, the result is the original number.
  • The inverse property of addition states that for every number, there is a corresponding negative number that, when added together, results in zero.

By understanding the properties of addition, we can better appreciate the rules that govern how numbers can be added together. Whether you're working with simple arithmetic or complex mathematical operations, the properties of addition are essential to keep in mind.