Problem 11 - Elementary Algebra

Question

Find the opposite of each real number. $$ -(-1) $$

Step-by-Step Solution

Answer

Answer: The opposite of -1 is 1.

Step 1: Multiply -1 by the given number

To find the opposite of a number, you need to multiply it by -1. In this case, the given number is -1, and we want to find its opposite. So, we will multiply -1 by -1 to calculate the opposite of the given number: $$ -1 \times -1 $$

Step 2: Perform the multiplication

Now we just have to multiply the two numbers together. Since both numbers are -1, we use the rule that a negative multiplied by a negative gives a positive result: $$ -1 \times -1 = 1 $$

Step 3: Write the result

The multiplication result is the opposite of the given number, which is -1. So, the opposite of -1 is 1.

Key Concepts

Negative Numbers Multiplication

Understanding how to multiply negative numbers is crucial in mathematics, particularly when dealing with real numbers and solving algebraic expressions. When we multiply two negative numbers, the result is always a positive number. This might seem counterintuitive at first, but it follows a logical rule in math known as the 'sign rule' for multiplication.
For instance, let's consider the multiplication of \( -1 \) and \( -1 \) as in our exercise. According to the sign rule, a negative number (\( -1 \) in this case) multiplied by another negative number (\( -1 \) again) results in a positive number. Therefore, \( -1 \times -1 = 1 \). This rule is essential because it helps preserve the structure of real numbers across various operations and ensures consistency in mathematical computations.

• \( - (negative) \times - (negative) = + (positive) \)
• \( - (negative) \times + (positive) = - (negative) \)
• \( + (positive) \times - (negative) = - (negative) \)
• \( + (positive) \times + (positive) = + (positive) \)

It's also worth mentioning that the multiplication of negative numbers is not only important in academic exercises but also has practical applications in various fields like finances, engineering, and physics, where it helps in modeling and solving real-world problems.

Real Numbers

Real numbers encompass a vast category of numbers including natural numbers, whole numbers, integers, rational numbers, and irrational numbers. Essentially, any number that can be found on the number line is referred to as a 'real number.' They are so named because they represent a wide range of values that can be used to denote real-world quantities.
In the context of our exercise, we are dealing with the real number \( -1 \). The operation of finding the opposite — or additive inverse — is a common task with real numbers. The opposite of a real number \( a \) is the real number \( -a \), which, when added to \( a \), yields zero. This idea mirrors the fact that every real number has a 'mirror image' on the number line. If you were to visually represent real numbers on the number line, their opposites would be found at equidistant points from zero, but in opposite directions.

Properties of Real Numbers

Real numbers abide by certain algebraic rules and properties, like the commutative, associative, and distributive properties. These properties make the set of real numbers extremely versatile for carrying out arithmetic operations and solving equations. Applying the concepts of real numbers helps students to become proficient in understanding the mathematical world that surrounds us.

Algebraic Rules

Algebraic rules are the backbone of manipulating expressions and equations in algebra. They form the foundation that allows mathematicians and students alike to move symbols around confidently to simplify expressions or solve equations.
In our exercise, we apply an algebraic rule that helps us find the opposite of a number. Algebra teaches us that to get the opposite, or the additive inverse, of any number, we multiply it by \( -1 \). This rule is grounded in the concept of symmetry with respect to addition and multiplication. The opposite of number \( a \) is \( -a \) because \( a + (-a) = 0 \). We are effectively 'undoing' the value of \( a \) by pairing it with \( -a \).

Key Algebraic Rules to Remember

• The distributive property: \( a(b + c) = ab + ac \)
• The commutative property of addition and multiplication: \( a + b = b + a \), as well as \( ab = ba \)
• The associative property of addition and multiplication: \( (a + b) + c = a + (b + c) \) and \( (ab)c = a(bc) \)
• The additive inverse: \( a + (-a) = 0 \)
• The multiplicative inverse (reciprocal): \( a \times \frac{1}{a} = 1 \), where \( a \) is not zero

Mastering these rules not only helps in academic success but also develops critical thinking skills that are applicable in solving complex problems across various disciplines.