Problem 49 - Algebra 2

Question

NUMBER THEORY For Exercises \(46-49,\) use the properties of real numbers to answer each question. If \(m n=m\) and \(m \neq 0,\) what is the value of \(n ?\)

Step-by-Step Solution

Answer

The value of \( n \) is 1.

Step 1: Analyze the Equation

Start by examining the given equation, which is \( m \cdot n = m \). This signifies that the product of \( m \) and \( n \) is equal to \( m \).

Step 2: Apply Properties of Real Numbers

Use the property of real numbers called the 'multiplicative identity property', which states that any number multiplied by 1 gives the number itself. This implies \( n \) could be 1, since it doesn't change \( m \) when multiplied.

Step 3: Verify Other Possible Solutions

Since \( m eq 0 \) and we must maintain the equality \( m \cdot n = m \), only the value \( n = 1 \) maintains the equality, as \( m \cdot 1 = m \). If \( n \) were any other number except 1, \( m \cdot n eq m \) except only when \( n = 1 \).

Key Concepts

Properties of Real Numbers

Real numbers encompass a broad set of numbers that include whole numbers, fractions, and irrational numbers, like \( \sqrt{2} \). Understanding their properties is crucial when solving mathematical problems, especially equations. Here are some fundamental properties:

• **Commutative Property:** The order doesn't matter in addition or multiplication. So, \( a + b = b + a \) and \( a \cdot b = b \cdot a \).
• **Associative Property:** The way numbers are grouped in parentheses does not affect their sum or product, \( (a + b) + c = a + (b + c) \) and \( (a \cdot b) \cdot c = a \cdot (b \cdot c) \).
• **Distributive Property:** This allows you to multiply a number by a group of numbers added together, \( a \cdot (b + c) = a \cdot b + a \cdot c \).
• **Identity Properties:** The identity for addition is 0, meaning \( a + 0 = a \). For multiplication, as we'll explore further, it is 1.

These properties help in simplifying calculations and solving equations. They ensure consistency and allow for more efficient problem-solving strategies.

Multiplicative Identity Property

The multiplicative identity property is an essential concept in number theory. It states that any real number multiplied by 1 remains unchanged. In symbols, this is expressed as \( a \cdot 1 = a \). This property is fundamental because:

• **Simplification:** It helps when simplifying expressions, as any factor of 1 can be ignored in multiplication.
• **Integrity of Identity:** It preserves the identity of numbers during operations, ensuring they retain their original value.
• **Equation Solving:** As seen in the exercise, knowing that multiplying a real number by 1 keeps it the same allows us to deduce crucial values for variables in equations.

So, whenever you see an expression like \( m \cdot n = m \) with \( m eq 0 \), suspect the presence of a multiplicative identity at play, suggesting \( n = 1 \).

Equations

Equations are mathematical statements that show the equality of two expressions. Solving equations involves finding the value of the unknown variable(s) that make the equation true. The exercise focuses on setting equations using real numbers and their properties:

• **Equating Expressions:** In problems like \( m \cdot n = m \), the goal is to determine what value of \( n \) satisfies the equation.
• **Using Properties:** By applying properties of real numbers, like the multiplicative identity, we simplify the problem, reducing complex expressions to a simpler form where the value of the unknown becomes apparent.
• **Verification:** After finding a proposed solution, substitute it back into the original equation to check if it holds true. This ensures no mistakes were made in the solving process.

Working through equations is a cornerstone of algebra, facilitating our understanding of more complex mathematical concepts later on.