Long Multiplication

An online tool to perform long multiplication and see the detailed steps.

Enter the multiplicand and multiplier to get the result.

Practical Examples

Click on an example to see how the calculator works with different numbers.

Basic Multiplication

2-digit by 2-digit

Multiplying two double-digit numbers.

Multiplicand: 89

Multiplier: 42

Multiplying Hundreds

3-digit by 2-digit

A three-digit number multiplied by a two-digit number.

Multiplicand: 735

Multiplier: 68

Large Number Multiplication

4-digit by 3-digit

A four-digit number multiplied by a three-digit number.

Multiplicand: 9876

Multiplier: 543

Handling Zeros

Multiplying with Zero

Example involving a zero in the multiplier.

Multiplicand: 505

Multiplier: 25

Other Titles
Understanding Long Multiplication: A Comprehensive Guide
From basic principles to complex calculations, this guide covers everything you need to know about the long multiplication method.

What is Long Multiplication?

  • A systematic method for multiplying multi-digit numbers.
  • Breaks down complex problems into simpler steps.
  • Essential for manual arithmetic and understanding number properties.
Long multiplication is a traditional algorithm used to multiply two numbers with two or more digits. It simplifies the process by breaking it down into a series of smaller, more manageable single-digit multiplications and additions. The method organizes calculations vertically, making it easy to keep track of place values and partial products.
The Core Components
Multiplicand: The first number, typically written on top.
Multiplier: The second number, written below the multiplicand.
Partial Products: The result of multiplying the multiplicand by each digit of the multiplier.
Final Product: The sum of all the partial products.
This calculator automates the entire process, showing you the multiplicand, multiplier, all the intermediate partial products, and the final result, just as you would write it out by hand.

Fundamental Concept

  • For 78 x 34, you first calculate 78 * 4 (the ones digit) and then 78 * 30 (the tens digit).
  • The partial products are 312 and 2340, which sum up to 2652.

Step-by-Step Guide to Using the Long Multiplication Calculator

  • Enter the multiplicand in the first input field.
  • Enter the multiplier in the second input field.
  • Click the 'Calculate' button to generate the solution.
Our calculator is designed to be intuitive. Follow these simple steps to get your answer along with a detailed breakdown of the work involved.
The Calculation Process
Step 1: Input Numbers: Type the number you want to multiply into the 'Multiplicand' field and the number you are multiplying by into the 'Multiplier' field.
Step 2: Multiplication by Ones Digit: The calculator starts by multiplying the entire multiplicand by the rightmost digit (the ones place) of the multiplier. This result is the first partial product.
Step 3: Multiplication by Subsequent Digits: It then multiplies the multiplicand by the next digit to the left in the multiplier (tens, hundreds, etc.). Each new partial product is placed on a new line and shifted to the left to account for its increasing place value.
Step 4: Final Addition: Finally, all the calculated partial products are added together to give you the final product.

Calculation Example

  • To solve 246 × 53:
  • 1. First partial product (246 × 3): 738
  • 2. Second partial product (246 × 5, shifted left): 12300
  • 3. Final Product (738 + 12300): 13038

Real-World Applications of Long Multiplication

  • Budgeting and financial planning.
  • Calculating areas and volumes in construction.
  • Scaling ingredients in recipes or chemicals in labs.
Long multiplication is not just a classroom exercise; it is a fundamental skill with numerous practical applications in daily life and various professions.
Financial Calculations
When planning a budget, you might need to calculate total expenses. For example, if a company has 75 employees who each receive a $450 bonus, long multiplication (75 x 450) is used to find the total bonus payout of $33,750.
Construction and Engineering
To determine the amount of material needed for a project, such as finding the square footage of a large area. If a parking lot has 25 rows and each row can fit 115 cars, you can calculate its capacity as 25 x 115 = 2875 cars.

Practical Scenarios

  • A concert venue has 128 rows of seats, and each row has 56 seats. The total capacity is 128 x 56 = 7168 people.
  • If a book has 432 pages and you want to print 850 copies, you will need to print 432 x 850 = 367,200 pages in total.

Common Misconceptions and Correct Methods

  • Forgetting to shift partial products correctly.
  • Making errors when carrying numbers over.
  • Mistakes in the final addition of partial products.
Accuracy in long multiplication depends on being systematic and careful. A few common pitfalls can lead to incorrect answers.
The Importance of Place Value Alignment
Misconception: A frequent mistake is to align all partial products to the right, without shifting them. This fails to account for the fact that you're multiplying by tens, hundreds, etc.
Correct Method: Each partial product must be shifted one place to the left relative to the one before it. Multiplying by the tens digit requires a leftward shift of one place (or adding one zero). Multiplying by the hundreds digit requires a shift of two places (or adding two zeros), and so on. This correctly reflects the magnitude of each part of the calculation.

Alignment Correction

  • Problem: 52 × 24
  • Incorrect Alignment: 52×4=208 and 52×2=104. Adding 208+104=312. (Wrong)
  • Correct Alignment: 52×4=208. 52×2(0)=1040. Adding 208+1040=1248. (Correct)

Mathematical Derivation and Principles

  • Based on the distributive property of multiplication.
  • Decomposes numbers into their base-10 components.
  • The algorithm is a structured way of applying distributivity.
The long multiplication algorithm is a practical application of the distributive property of multiplication over addition, which states that a × (b + c) = (a × b) + (a × c). It works by breaking down numbers into their place value components (e.g., 345 = 300 + 40 + 5).
Applying the Distributive Property
When we calculate 123 × 45, we are essentially calculating 123 × (40 + 5). Using the distributive property, this expands to (123 × 40) + (123 × 5). These two parts are the 'partial products'. The long multiplication algorithm provides a systematic way to compute these products and add them together. The first partial product is 123 × 5 = 615. The second is 123 × 40 = 4920. Their sum, 615 + 4920, gives the final answer, 5535.

Distributive Property in Action

  • Decomposition of 86 × 21:
  • 86 × (20 + 1)
  • = (86 × 20) + (86 × 1)
  • = 1720 + 86
  • = 1806