Why does it work to break apart a number by place values to multiply
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Breaking apart a number by place values, also known as the distributive property, allows for easier multiplication because it breaks down the calculation into smaller, more manageable parts. Here's a step-by-step explanation:
Step 1: Understand the place value system
To break apart a number, it's important to understand the concept of place values. In our number system (base-10 or decimal system), each digit in a number represents a certain place value. The rightmost digit represents the ones place, the digit to its left represents the tens place, the next digit represents the hundreds place, and so on.
Step 2: Divide the number into place values
Once you've identified the place values, break apart the number by expanding it using these place values. For example, let's break down the number 1234:
- Thousands place: 1 x 1000
- Hundreds place: 2 x 100
- Tens place: 3 x 10
- Ones place: 4 x 1
Step 3: Multiply each place value separately
To calculate the product, multiply each place value separately. Using our example above:
- Thousands place: 1 x 1000 = 1000
- Hundreds place: 2 x 100 = 200
- Tens place: 3 x 10 = 30
- Ones place: 4 x 1 = 4
Step 4: Sum up the products
Finally, sum up the products from each place value to get the final result. In our example, the sum would be: 1000 + 200 + 30 + 4 = 1234.
Breaking apart a number by place values makes multiplication easier because it helps organize the calculation into smaller components, making it easier to multiply mentally or perform the calculations in steps.
Breaking apart a number by place values and multiplying is a technique called the distributive property. It helps simplify the multiplication process by breaking it down into smaller, more manageable parts. This technique is based on the fact that any number can be expressed as the sum of its place values, where each place value represents a power of 10.
Let's take an example to understand this better. Consider the number 345. We can express it as:
345 = 300 + 40 + 5
Now, let's say we want to multiply 345 by 7. We can distribute the multiplication over each place value:
345 × 7 = (300 + 40 + 5) × 7
Using the distributive property, we can now multiply 7 with each place value separately:
(300 × 7) + (40 × 7) + (5 × 7)
Simplifying each multiplication, we get:
2100 + 280 + 35
Finally, by combining these results, we can find the product of 345 and 7:
2100 + 280 + 35 = 2415
Breaking apart the number by its place values and multiplying makes the calculation easier because multiplying single-digit numbers is simpler and faster than multiplying larger numbers. It also helps us focus on one place value at a time, reducing the chances of making mistakes.
So, by breaking apart a number by its place values and multiplying, we can simplify complex multiplications and arrive at the correct answer more effectively.