Problem 19 - Prealgebra

Question

Name the property shown by each statement. $$1 \cdot 4=4$$

Step-by-Step Solution

Answer

The property shown is the multiplicative identity property.

Step 1: Identify the Operation

In the given expression, the operation is multiplication. Recognize that the expression involves multiplying the number 1 by the number 4.

Step 2: Recall Multiplicative Identity Property

The multiplicative identity property states that any number multiplied by 1 remains unchanged, or retains its identity. This is a fundamental property of multiplication in mathematics.

Step 3: Apply the Property

Observe that multiplying 1 by any number, in this case 4, leaves the number unchanged. Thus, the equation \(1 \cdot 4 = 4\) is a specific instance of the multiplicative identity property.

Step 4: Conclusion

The statement is an example of the multiplicative identity property, as the number 4 remains the same after being multiplied by 1.

Key Concepts

Properties of Multiplication

Multiplication is a basic math operation that includes several important properties:

• Commutative Property: This property means the order of numbers doesn't change the result. For example, \(a \cdot b = b \cdot a\).
• Associative Property: This property states that when three or more numbers are multiplied, grouping doesn't affect the product. For example, \((a \cdot b) \cdot c = a \cdot (b \cdot c)\).
• Multiplicative Identity Property: This states any number multiplied by 1 remains unchanged. For example, \(a \cdot 1 = a\).
• Distributive Property: This one links addition and multiplication, e.g., \(a \cdot (b + c) = a \cdot b + a \cdot c\).

Understanding these properties helps in simplifying and solving multiplication problems effectively.

Basic Arithmetic Operations

Arithmetic operations form the foundation of mathematics and include:

• Addition: Combining two numbers to get a sum, e.g., \(a + b\).
• Subtraction: Taking one number away from another, e.g., \(a - b\).
• Multiplication: Calculating the total of one number repeated several times, e.g., \(a \times b\).
• Division: Splitting a number into equal parts, e.g., \(a \div b\).

Each operation follows specific rules and properties, making them predictable and reliable for problem-solving. Multiplication's key property, the Multiplicative Identity, is crucial in maintaining the integrity of numbers.

Mathematical Identities

Mathematical identities are equalities that are true regardless of the values of any variables involved, and they serve as tools for simplifying expressions and proofs. Some identities go beyond numbers and involve variables:

• Trigonometric Identities: Such as \(\sin^2(x) + \cos^2(x) = 1\), used to solve trigonometric equations.
• Algebraic Identities: Like \((a+b)^2 = a^2 + 2ab + b^2\), which helps in expanding expressions.
• Multiplicative Identity: In arithmetic, the multiplicative identity is the number 1, as any number multiplied by 1 remains itself, e.g., \(a \cdot 1 = a\).

These identities are foundational and widely used in simplifying complex mathematical expressions, ensuring accuracy in calculations.