How Do You Long Division

Mastering Long Division: A Comprehensive Guide

Long division, a fundamental arithmetic operation, might seem daunting at first glance, but with a structured approach and consistent practice, it becomes second nature. This comprehensive guide will demystify the process, breaking it down into manageable steps, explaining the underlying principles, and addressing common queries. Whether you're a student brushing up on your skills or an adult looking to solidify your mathematical foundation, this guide will equip you with the knowledge and confidence to conquer long division.

Understanding the Basics: What is Long Division?

Long division is a method used to divide large numbers into smaller, more manageable parts. It's an extension of the simple division you learned in elementary school, but it allows us to handle much larger dividends (the number being divided) and divisors (the number we're dividing by). The process involves a series of steps that systematically break down the division problem into smaller, easier-to-solve parts. Think of it as a structured way to repeatedly subtract the divisor until you reach a remainder (what's left over after the division). The result is a quotient (the answer), along with the remainder.

Step-by-Step Guide to Long Division

Let's learn long division through a practical example. We'll divide 738 by 6.

1. Setting up the Problem:

First, write the problem in the standard long division format:

     _____
6 | 738

The number 738 (the dividend) goes inside the long division symbol, and the number 6 (the divisor) goes outside.

2. Dividing the First Digit:

Begin by dividing the first digit of the dividend (7) by the divisor (6). 6 goes into 7 one time (1). Write the '1' above the 7:

     1
     _____
6 | 738

3. Multiplying and Subtracting:

Now, multiply the quotient (1) by the divisor (6): 1 x 6 = 6. Write this beneath the 7:

     1
     _____
6 | 738
   - 6

Next, subtract 6 from 7: 7 - 6 = 1. Write the result below the 6:

4. Bringing Down the Next Digit:

Bring down the next digit of the dividend (3) next to the 1:

5. Repeating the Process:

Now, repeat steps 2-4. Divide 13 by 6. 6 goes into 13 two times (2). Write the '2' above the 3:

Multiply 2 by 6 (2 x 6 = 12) and write it below the 13:

Subtract 12 from 13: 13 - 12 = 1. Write the result below the 12:

6. Bringing Down the Last Digit:

Bring down the last digit of the dividend (8) next to the 1:

7. Final Division and Remainder:

Repeat the process one last time. Divide 18 by 6. 6 goes into 18 three times (3). Write the '3' above the 8:

Multiply 3 by 6 (3 x 6 = 18) and write it below the 18. Subtract 18 from 18: 18 - 18 = 0. This means there's no remainder.

8. The Result:

The quotient is 123, and the remainder is 0. Therefore, 738 divided by 6 is 123.

Handling Remainders

Sometimes, you'll have a remainder after completing the long division process. Let's look at an example where we have a remainder:

Divide 857 by 4:

Here, the quotient is 214, and the remainder is 1. This can be expressed as 214 with a remainder of 1, or as a mixed number: 214 1/4.

Long Division with Decimals

Long division can also involve decimals. The process remains similar, but you'll need to add zeros to the dividend after the decimal point if needed to continue the division. For instance, dividing 15 by 4:

Here, we added two zeros after the decimal point in the dividend to continue dividing until we reached a remainder of 0.

The Scientific Explanation: The Distributive Property and Repeated Subtraction

At its core, long division relies on the distributive property of multiplication over addition/subtraction. When we divide 738 by 6, we're essentially finding how many times 6 can be subtracted from 738 until we reach zero (or a remainder). Each step in long division reflects this repeated subtraction. Consider the expanded form of 738: 700 + 30 + 8. We systematically divide each part by 6:

700/6 ≈ 100 (with a remainder)
(700-600)/6 + 30/6 = 100 + 5 = 110 (with a remainder)
(700 + 30 -600 - 30)/6 + 8/6 = 120 (with a remainder)

Long division streamlines this process, making it efficient and organized.

Common Mistakes to Avoid

Incorrect placement of digits: Pay close attention to the placement of digits during each step. A slight misalignment can lead to significant errors in the final answer.
Forgetting to bring down digits: Make sure you consistently bring down the next digit after each subtraction. Omitting this step will disrupt the entire process.
Errors in multiplication or subtraction: Double-check your multiplication and subtraction calculations to ensure accuracy.
Incorrect handling of decimals: When working with decimals, pay close attention to the placement of the decimal point in both the dividend and the quotient.

Frequently Asked Questions (FAQs)

Q: What if the divisor is larger than the first digit of the dividend?
- A: If the divisor is larger than the first digit of the dividend, you need to consider the first two (or more) digits of the dividend for the first step of division.
Q: What does a remainder signify?
- A: A remainder signifies that the division wasn't perfectly even. It represents the amount that is left over after the division is complete.
Q: Can I use a calculator for long division?
- A: While calculators can provide the answer quickly, understanding the process of long division is crucial for building a strong mathematical foundation. Calculators should be used for verification, not as a replacement for learning the process.
Q: How can I improve my speed and accuracy in long division?
- A: Practice is key! Start with simpler problems and gradually increase the complexity. Focus on accuracy over speed initially, then work on improving your speed as you gain confidence.

Conclusion: Mastering the Art of Long Division

Long division, despite its initial complexity, is a fundamental arithmetic skill with wide-ranging applications. By systematically following the steps outlined in this guide, paying close attention to detail, and practicing regularly, you can confidently master this essential mathematical technique. Remember, understanding the underlying principles—repeated subtraction and the distributive property—will solidify your understanding and make the process more intuitive. So, embrace the challenge, persevere through practice, and witness your long division skills blossom! With dedication and practice, you’ll not only conquer long division, but also build a strong foundation for more advanced mathematical concepts.