Problem 7 - Precalculus Mathematics for Calculus

Question

State the property of real numbers being used. $$7+10=10+7$$

Step-by-Step Solution

Answer

This equation demonstrates the commutative property of addition.

Step 1: Identify the operation

Look at the given equation: \(7 + 10 = 10 + 7\). The operation being used here is addition.

Step 2: Recognize the property involving addition

Think about the properties of real numbers related to addition. One such property is the commutative property of addition, which states that changing the order of addends does not change the sum.

Step 3: Confirm the property with the equation

Verify that the given equation matches the commutative property. In the equation \(7 + 10 = 10 + 7\), the first expression \(7 + 10\) and the second expression \(10 + 7\) are the same because of the commutative property of addition, where \(a + b = b + a\).

Key Concepts

Commutative Property

In the realm of mathematics, the commutative property is a fundamental concept that simplifies mathematical operations. When we say an operation is commutative, it means the order in which we perform the operation does not affect the result. For the operation of addition, this is particularly important. Let's consider what this looks like: If you have two numbers, say \(a\) and \(b\), and you're adding them together, it doesn't matter if you write it as \(a + b\) or \(b + a\). Both expressions will yield the same sum. Mathematically, this is written as \(a + b = b + a\).Why is this property so helpful? It allows us to rearrange numbers in arithmetic or algebraic expressions in a way that often makes calculations simpler or more intuitive.

• This property only applies to addition and multiplication, not subtraction or division.
• Recognizing this property can help streamline complex calculations.
• The commutative property is crucial for mental math, enabling easier rearrangement of numbers.

Understanding and using the commutative property can greatly enhance both speed and accuracy when working with numbers.

Addition

Addition is one of the most basic and important operations in mathematics. It is the process of combining two or more numbers to get a new total. When performing addition, you work with terms known as addends and their result known as the sum.Let's break it down with a simple example: Assume you have two numbers, 7 and 10. Adding these numbers together, you would write it as \(7 + 10\), and the result, or sum, would be 17.

• Addition is associative, meaning the grouping of numbers doesn't change the sum; for instance, \((1 + 2) + 3\) is the same as \(1 + (2 + 3)\).
• It is also commutative as highlighted previously, which allows for flexibility in rearranging numbers.
• Knowing basic addition helps form the foundation for understanding more complex mathematical processes such as multiplication, which is essentially repeated addition.

Overall, addition is a fundamental operation that is not only easy to grasp but also forms a stepping stone for more complicated math concepts.

Real Numbers

Real numbers form a vast set of numbers that we use daily and include all the numbers you can find on the number line. This includes whole numbers, fractions, integers, and irrational numbers. Important characteristics of real numbers:

• Real numbers are continuous, meaning between any two real numbers, there's always another real number.
• They can be positive, negative, or zero.
• Real numbers follow specific properties such as commutativity, associativity, and distributivity.

In the context of the given exercise, both 7 and 10 are real numbers. This means they adhere to the commutative property of addition. Whenever you're dealing with real numbers and their operations, these properties ensure that calculations behave predictably. Understanding real numbers is crucial because they serve as the basis for most of foundational mathematics and advanced topics alike.