Problem 59 - Algebra 1

Question

Decide whether the statement is true or false . If it is false, give a counterexample. The product \((-a) \cdot(-1)\) is always positive.

Step-by-Step Solution

Answer

The statement is false. The counterexample is when a = 0. The product in this case is \( 0 \cdot -1 = 0 \), which is not positive.

Step 1: Understand the Statement

First, it's crucial to understand what the statement means. The statement is saying \( -a \cdot -1 \) is always positive. Let's check if it's true or false.

Step 2: Test the Statement using an Example

Let's take \( a = 2 \). Now, \( -a = -2 \). Thus, substituting the value of \( -a \) in \( -a \cdot -1 \), we get \( -2 \cdot -1 = 2 \) which is indeed positive.

Step 3: Generalize the Result

By introducing any number in place of \( a \), one can see that \( -a \cdot -1 = a \), which is the absolute value of \( a \) and is always positive when \( a \) is not equal to zero. However, if \( a = 0 \), then \( -a \cdot -1 = 0 \cdot -1 = 0 \), which is neither positive nor negative. Hence the statement is false.

Step 4: Provide Counterexample

A counterexample is a case where the statement doesn't hold true. An example of this is when \( a = 0 \). In this case \( -a \cdot -1 = 0 \), which is not positive, thus making the statement false.

Key Concepts

Negative Numbers

Negative numbers can be a bit confusing, but with a little practice, they become much easier to handle. A negative number is essentially a number with a "minus" sign before it. This signifies that it is less than zero. For example, -3 is three units to the left of zero on the number line.

• Negative numbers are used to represent values below zero, such as debts or temperatures below freezing.
• They are opposite to positive numbers, which are greater than zero.
• Zero is considered neutral, as it is neither positive nor negative.

When multiplying negative numbers, it’s important to remember that multiplying two negative numbers results in a positive number. This is because the two "negatives" effectively cancel each other out.
However, if you multiply a negative number by a positive number, the result will be negative.

Multiplication

Multiplication is a fundamental operation in mathematics that represents repeated addition. For example, multiplying 3 by 4 essentially means adding 3 four times: 3 + 3 + 3 + 3.
Multiplication has a few important properties:

• Commutative Property: This means the order of multiplication doesn't change the result. For instance, 3 x 4 is the same as 4 x 3.
• Associative Property: This allows numbers to be grouped in any manner. For example, (2 x 3) x 4 is the same as 2 x (3 x 4).
• Distributive Property: This states that multiplying a number by a group of numbers added together is the same as doing each multiplication separately. For example, 2 x (3 + 4) is equal to (2 x 3) + (2 x 4).

Especially relevant in the context of multiplying negative numbers, consider that multiplying two negative values results in a positive, such as -2 x -1 = 2. This reflects the rule that a negative times a negative equals a positive.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations (addition, subtraction, multiplication, and division). Variables are letters that represent unknown numbers. In expressions like \(-a \cdot -1\), \(a\) is a variable that can take any value.
Here’s why algebraic expressions are important:

• They allow us to generalize mathematical problems by using variables.
• They help in solving equations and finding unknown values.
• They provide a compact method to express complex mathematical relationships.

When handling algebraic expressions, especially with negative numbers, one must remember that:
- The negative sign affects the whole part of an expression it is placed in front of.
- Simplifying algebraic expressions often involves applying the basic rules of arithmetic, such as the distributive property.