Long Division

Arithmetic
process

Also known as: standard division algorithm, multi-digit division

Grade 3-5

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A step-by-step algorithm for dividing a multi-digit number (dividend) by another number (divisor), producing a quotient and possibly a remainder. Long division is essential for working with large numbers and is the basis for dividing decimals and polynomials later.

This concept is covered in depth in our algebraic division algorithm tutorial, with worked examples, practice problems, and common mistakes.

Definition

A step-by-step algorithm for dividing a multi-digit number (dividend) by another number (divisor), producing a quotient and possibly a remainder.

💡 Intuition

Long division is like distributing items into groups one place value at a time. If you have 156 stickers to share among 12 friends, you first figure out how many groups of 12 fit in 156 by working from the biggest place value down: how many 12s in 15? Then bring down the next digit and repeat.

🎯 Core Idea

Divide, multiply, subtract, bring down—repeat this cycle for each digit of the dividend.

Example

156 \div 12 = 13 \quad \text{because } 12 \times 13 = 156

Formula

\text{Dividend} = \text{Divisor} \times \text{Quotient} + \text{Remainder}

Notation

The division bracket \overline{)\phantom{0}} places the quotient on top, the dividend inside, and the divisor outside

🌟 Why It Matters

Long division is essential for working with large numbers and is the basis for dividing decimals and polynomials later.

💭 Hint When Stuck

Write out the long division steps on paper: Divide, Multiply, Subtract, Bring down -- repeat for each digit.

Formal View

a = bq + r where a is the dividend, b is the divisor, q = \lfloor a/b \rfloor is the quotient, and 0 \leq r < b is the remainder (Division Algorithm)

🚧 Common Stuck Point

Estimating how many times the divisor fits into each partial dividend, especially with two-digit divisors.

⚠️ Common Mistakes

  • Forgetting to bring down the next digit before continuing
  • Placing a quotient digit in the wrong position (misalignment)
  • Not writing a zero in the quotient when the divisor doesn't fit into a partial dividend (e.g., 408 \div 4 = 102, not 12)

Frequently Asked Questions

What is Long Division in Math?

A step-by-step algorithm for dividing a multi-digit number (dividend) by another number (divisor), producing a quotient and possibly a remainder.

Why is Long Division important?

Long division is essential for working with large numbers and is the basis for dividing decimals and polynomials later.

What do students usually get wrong about Long Division?

Estimating how many times the divisor fits into each partial dividend, especially with two-digit divisors.

What should I learn before Long Division?

Before studying Long Division, you should understand: division, subtraction, multiplication.

How Long Division Connects to Other Ideas

To understand long division, you should first be comfortable with division, subtraction and multiplication. Once you have a solid grasp of long division, you can move on to dividing decimals.

Want the Full Guide?

This concept is explained step by step in our complete guide:

Polynomial Long Division: Step-by-Step Method with Examples →
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