Problem 67 - Fundamentals of Mathematics

Question

(Section 2.2) Find the product. \(1,004 \cdot 1,005\).

Step-by-Step Solution

Answer

1,004 multiplied by 1,005 equals 1,009,020.

Step 1: Break Down the Numbers

Recognize that 1,004 and 1,005 are close to 1,000, which represents their base value for simplification. Notice that 1,004 is 4 more than 1,000, and 1,005 is 5 more than 1,000.

Step 2: Use the Distributive Property

Apply the distributive property: \[ (1,000 + 4) imes (1,000 + 5) \]This expands to:\[ = 1,000 imes 1,000 + 1,000 imes 5 + 1,000 imes 4 + 4 imes 5 \]

Step 3: Perform the Multiplication

Multiply each pair:\[ 1,000 imes 1,000 = 1,000,000 \]\[ 1,000 imes 5 = 5,000 \]\[ 1,000 imes 4 = 4,000 \]\[ 4 imes 5 = 20 \]

Step 4: Sum All the Products

Add the results of the multiplication:\[ 1,000,000 + 5,000 + 4,000 + 20 = 1,009,020 \]

Step 5: Conclusion

Thus, the product of 1,004 and 1,005 is 1,009,020.

Key Concepts

Multiplication

Multiplication is one of the core operations in mathematics. It involves combining equal groups, similar to repeated addition. If you have two numbers, say 'a' and 'b', multiplication tells you the total of 'a' groups of 'b' things each. When it comes to larger numbers, multiplication can seem daunting, but understanding it conceptually helps simplify the process.

For example, multiplying 1,004 and 1,005 requires breaking the problem down into manageable parts, using properties like the distributive property. This approach makes even large number multiplication a lot easier. Learning how to multiply effectively is fundamental for problem-solving in math.

Arithmetic Simplification

Arithmetic simplification is a technique used to make complex calculations more manageable. It involves breaking down numbers and using mathematical properties to simplify the operation. This is crucial when dealing with large numbers.

In our problem, breaking down 1,004 and 1,005 to their base value of 1,000 and their respective differences (4 and 5) transforms the multiplication into a simpler arithmetic exercise. By simplifying the calculation, you reduce the complexity and make it easier to process mentally or on paper. Doing this requires an understanding of the numbers you're working with and how they relate to each other.

Expanded Form

Expanded form expresses numbers as a sum of their individual place values. This form is particularly useful when performing multiplication using the distributive property.

Consider 1,004 as (1,000 + 4) and 1,005 as (1,000 + 5). By writing numbers this way, it's easier to apply multiplication. This approach allows for breaking the problem into smaller, more manageable pieces, ensuring that each part is calculated correctly.

• 1,000 times itself is easy to visualize and compute.
• Additional multiplications of smaller numbers (like 4 and 5) become straightforward.

Using expanded form aids in visual clarity and arithmetic accuracy, especially when multiplying larger numbers.

Problem Solving

Problem solving in mathematics often involves applying a systematic approach to reach a solution. With multiplication of large numbers, it involves recognizing patterns and utilizing mathematical properties efficiently.

For our exercise, the task required an understanding of how to simplify numbers and use the distributive property. By breaking down numbers into simpler components (1,000, 4, and 5) and multiplying them step by step, you structure the problem-solving process. This not only results in accuracy but also in efficiency.

• Identify the base and differences of the numbers.
• Apply distributive properties correctly.
• Sum all the partial products to find the solution.

In essence, successful problem solving in math requires logical thinking, step-by-step analysis, and knowledge of arithmetic properties.