Long Division Calculator

Understanding Long Division

Long division is a method for dividing large numbers that breaks the division process into smaller, manageable steps. It's one of the fundamental arithmetic operations taught in mathematics and provides a systematic way to find both the quotient and remainder when dividing any two numbers.

This method is particularly useful when dealing with numbers too large for mental math and when you need to show the complete working process. Long division also forms the foundation for understanding polynomial division, decimal division, and other advanced mathematical concepts.

Components of Division

Division Terminology

Dividend ÷ Divisor = Quotient + Remainder/Divisor

Example: 100 ÷ 7 = 14 + 2/7 = 14 R 2

Term	Definition	Example (100 ÷ 7)
Dividend	The number being divided	100
Divisor	The number we divide by	7
Quotient	The result of division (whole part)	14
Remainder	What's left over after division	2

Step-by-Step Long Division Process

		Step 1: Set up the division by writing the dividend under the division symbol and the divisor to the left
	

		Step 2: Divide the first digit(s) of the dividend by the divisor. Write the result above the division line
	

		Step 3: Multiply the quotient digit by the divisor and write the result below the dividend
	

Step 4: Subtract this result from the dividend portion above it

Step 5: Bring down the next digit of the dividend and repeat the process

		Step 6: Continue until all digits have been processed or desired decimal places are reached
	

Long Division Examples

Example 1: 84 ÷ 4

4 goes into 8 exactly 2 times (2 × 4 = 8)

4 goes into 4 exactly 1 time (1 × 4 = 4)

Result: 84 ÷ 4 = 21 (no remainder)

Example 2: 156 ÷ 12

12 goes into 15 once (1 × 12 = 12, remainder 3)

Bring down 6 to make 36

12 goes into 36 exactly 3 times (3 × 12 = 36)

Result: 156 ÷ 12 = 13 (no remainder)

Division with Remainders

Division	Quotient	Remainder	Check (Q×D+R)
17 ÷ 5	3	2	3×5+2 = 17 ✓
25 ÷ 7	3	4	3×7+4 = 25 ✓
100 ÷ 7	14	2	14×7+2 = 100 ✓
89 ÷ 12	7	5	7×12+5 = 89 ✓

Converting to Decimal Form

Decimal Division Process

Continue division by adding decimal point and zeros

When remainder exists, add .000... to dividend and continue dividing

Example: 100 ÷ 7 as decimal

Step	Division	Result
1	100 ÷ 7	14 remainder 2
2	20 ÷ 7 (add decimal)	14.2 remainder 6
3	60 ÷ 7	14.28 remainder 4
4	40 ÷ 7	14.285 remainder 5
5	50 ÷ 7	14.2857 remainder 1

Types of Division Results

Result Type	Description	Example
Exact Division	No remainder, terminates	15 ÷ 3 = 5
Terminating Decimal	Decimal ends after finite steps	7 ÷ 4 = 1.75
Repeating Decimal	Decimal pattern repeats infinitely	1 ÷ 3 = 0.333...
Non-repeating Decimal	Irrational results (rare in basic division)	√2 ÷ 1 = 1.414...

Checking Your Work

Division Check Formula

(Quotient × Divisor) + Remainder = Dividend

This should always equal the original dividend

Example Check for 127 ÷ 9 = 14 R 1:

Check: 14 × 9 + 1 = 126 + 1 = 127 ✓

Common Mistakes and How to Avoid Them

		Mistake 1: Forgetting to bring down the next digit
		
Solution: Always bring down one digit at a time in order

		Mistake 2: Incorrect multiplication or subtraction
		
Solution: Double-check each arithmetic step

		Mistake 3: Placing quotient digits in wrong positions
		
Solution: Align quotient digits carefully above corresponding dividend digits

		Mistake 4: Not checking the final answer
		
Solution: Always verify using the check formula

Applications of Long Division

Application	Use Case	Example
Money Calculations	Splitting bills equally	$127 ÷ 5 people = $25.40 each
Time Calculations	Converting units	3725 seconds ÷ 60 = 62 minutes 5 seconds
Rate Problems	Speed, efficiency calculations	450 miles ÷ 6 hours = 75 mph
Fraction Simplification	Converting improper fractions	22/7 = 3 1/7

Mental Math Shortcuts

Divisibility Rules: Use these to make division easier

Divisor	Rule	Example
2	Last digit is even	124 (ends in 4, even) → divisible by 2
3	Sum of digits divisible by 3	123 (1+2+3=6, divisible by 3)
4	Last two digits divisible by 4	1324 (24 ÷ 4 = 6)
5	Last digit is 0 or 5	125 (ends in 5)
9	Sum of digits divisible by 9	234 (2+3+4=9)
10	Last digit is 0	150 (ends in 0)

Working with Large Numbers

Strategy 1: Break down the problem into smaller, manageable parts.

Strategy 2: Use estimation to check if your answer is reasonable.

Strategy 3: Work systematically, one digit at a time.

Strategy 4: Keep track of your work clearly to avoid errors.

Advanced Applications

Polynomial Long Division

Similar process for dividing polynomials

Example: (x² + 5x + 6) ÷ (x + 2) = x + 3

Base Conversion

Convert numbers between different number systems

Example: Convert 100₁₀ to binary by repeatedly dividing by 2

Practice Problems Strategy

Start Simple: Begin with single-digit divisors and two-digit dividends

Progress Gradually: Increase complexity as you become more comfortable

		Focus on Accuracy: Speed comes naturally with practice; accuracy is more important
	

Use Real-World Problems: Practice with money, measurement, and rate problems

		Master the Basics: Ensure multiplication and subtraction skills are solid before tackling complex division problems
	