Problem 30 - Beginning and Intermediate Algebra

Question

Decide whether each statement is an example of a commutative, an associative, an identity, \(a n\) inverse, or the distributive property. $$ 1+(-1)=0 $$

Step-by-Step Solution

Answer

Inverse property.

Step 1 - Understand the Statement

The given statement is: \[ 1 + (-1) = 0 \]. This statement involves the addition of two numbers.

Step 2 - Identify the Property

Look for the mathematical property that describes the operation in the statement. The statement shows that when 1 is added to its negative (-1), the result is 0.

Step 3 - Determine the Property

The property described here is the inverse property of addition. The inverse property states that for any number \(a\), there exists a number \(b\) such that \(a + b = 0\). In this case, \(1\) and \(-1\) are inverses of each other.

Key Concepts

commutative property

The commutative property is a fundamental concept in mathematics. It tells us that the order in which we add or multiply numbers does not change the result.

In mathematical terms, it states:
\( a + b = b + a \)
for addition and
\(ab = ba \)
for multiplication.

Examples make this clearer:

• For addition: 3 + 5 = 5 + 3
• For multiplication: 4 × 7 = 7 × 4

The main idea is that you can switch the places of the numbers, and the answer remains the same. This property is very useful for simplifying calculations.

Notice that the commutative property does not apply to subtraction or division. For instance, 7 - 3 is not the same as 3 - 7.

associative property

The associative property refers to how numbers are grouped in an operation. It states that the way numbers are grouped does not change the sum or product.

In mathematical notation, for addition you have:
\( (a + b) + c = a + (b + c) \)
and for multiplication:
\( (ab)c = a(bc) \).

This property helps us simplify complex calculations by reorganizing terms.

• Take addition: (2 + 3) + 4 = 2 + (3 + 4)
• And multiplication: (2 × 3) × 4 = 2 × (3 × 4)

The associative property does not work with subtraction or division either. For example, (5 - 2) - 1 is not the same as 5 - (2 - 1).

Remember, it's all about how numbers are grouped, not the order they appear.

identity property

The identity property focuses on operations that return the same value you started with.

For addition, the identity property states:
\(a + 0 = a \)
And for multiplication:
\(a \times 1 = a \)
.

• Addition: 8 + 0 = 8
• Multiplication: 7 × 1 = 7

The main idea is that adding zero to a number or multiplying a number by one does not change it.

These properties are very useful, especially in simplifying equations and understanding basic arithmetic.

Imagine you have 5 apples and you add zero more apples; you still have 5 apples. Similarly, if you multiply the number of apples by one, you still have the same amount.

distributive property

The distributive property connects addition and multiplication. It allows us to distribute a multiplication over an addition or subtraction.

Mathematically, it looks like this:
\( a(b + c) = ab + ac \)

• Example: 3(2 + 4) = 3×2 + 3×4

This property is powerful for simplifying expressions.

Think of it like this: if you have 3 groups of (2 + 4), it's the same as having 3 groups of 2 plus 3 groups of 4.

It works the same way with subtraction too:
\( a(b - c) = ab - ac \)

• Example: 5(7 - 3) = 5×7 - 5×3

So, the distributive property helps in breaking down complex problems into simpler parts.