Problem 2 - Precalculus: Mathematics for Calculus

Question

Complete each statement and name the property of real numbers you have used. (a) \(a b=\) ______;______Property (b) \(a+(b+c)=\) ______;______ Property (c) \(a(b+c)=\) ______;______ Property

Step-by-Step Solution

Answer

(a) \( ab = ba \); Commutative Property. (b) \( a+(b+c)=(a+b)+c \); Associative Property. (c) \( a(b+c)=ab+ac \); Distributive Property.

Step 1: Identify the missing part in (a)

In (a), the statement is \( ab = ba \). This is because multiplication is commutative for real numbers, meaning you can swap the order of the factors without changing the product.

Step 2: Identify the missing part in (b)

In (b), the statement is \( a + (b + c) = (a + b) + c \). This is the Associative Property of Addition, which states that when three or more numbers are added, the grouping of the numbers does not affect the sum.

Step 3: Identify the missing part in (c)

In (c), the statement is \( a(b + c) = ab + ac \). This demonstrates the Distributive Property of real numbers, which involves distributing the multiplication over addition.

Key Concepts

Commutative Property

The commutative property is a fundamental aspect of real numbers. This property applies to both addition and multiplication and explains that the order in which two numbers are added or multiplied does not affect the result. For example, if you have two numbers, say 3 and 5, applying this property means:

• For addition: \( 3 + 5 = 5 + 3 \)
• For multiplication: \( 3 \times 5 = 5 \times 3 \)

This means that both \( ab = ba \) for multiplication and \( a + b = b + a \) for addition. It’s a property that can make calculations more flexible as it allows you to rearrange numbers to simplify the computation process.

Associative Property

The associative property describes how the grouping of numbers affects addition and multiplication operations among real numbers. This property tells us that the way numbers are grouped in parentheses does not change their sum or product. For instance:

• For addition: \( (a + b) + c = a + (b + c) \)
• For multiplication: \( (a \times b) \times c = a \times (b \times c) \)

Think of it as a way to "associate" numbers differently without changing the outcome. This is very useful when dealing with complex calculations, as it allows us to group numbers in a way that might be easier to compute while ensuring we arrive at the same result.

Distributive Property

The distributive property is essential when you want to combine addition and multiplication operations. It states that when a number is multiplied by a sum, you can "distribute" the multiplication over each addend. Mathematically, it is expressed as:

• \( a(b + c) = ab + ac \)

This property allows us to simplify expressions and solve equations effectively. Imagine you have \( 3(4 + 2) \). Instead of adding first, you distribute the multiplication like this:

• \( 3 \times 4 + 3 \times 2 \)

It’s a handy tool for expanding expressions and is widely used in algebra to break down complex problems into simpler ones. With this property, you can confidently tackle equations knowing that breaking them into parts will yield the same result as if they were solved directly.