\( \definecolor{colordef}{RGB}{249,49,84} \definecolor{colorprop}{RGB}{18,102,241} \)

Long Multiplication

Long multiplication is a method used for multiplying larger numbers. It requires knowledge of the multiplication table for single digits.

Multiplication Tables for Multiples of 10


Imagine starting with a simple one-digit number, such as 3. What happens when we multiply by multiples of 10? Let’s explore step by step:
  • \(\textcolor{colordef}{3}\times \textcolor{colorprop}{1}0\) \(=\) \(\textcolor{colordef}{3}\times\)
    \(=\) \(+\) \(+\)
    \(=\) \(\textcolor{olive}{3}0\)
  • \(\textcolor{colordef}{3}\times \textcolor{colorprop}{2}0\) \(=\) \(\textcolor{colordef}{3}\times\)
    \(=\) \(+\) \(+\)
    \(=\) \(\textcolor{olive}{6}0\)
  • \(\textcolor{colordef}{3}\times \textcolor{colorprop}{3}0\) \(=\) \(\textcolor{colordef}{3}\times\)
    \(=\) \(+\) \(+\)
    \(=\) \(\textcolor{olive}{9}0\)
  • \(\textcolor{colordef}{3}\times \textcolor{colorprop}{4}0=\textcolor{olive}{12}0\)
  • \(\textcolor{colordef}{3}\times \textcolor{colorprop}{5}0=\textcolor{olive}{15}0\)
  • \(\textcolor{colordef}{3}\times \textcolor{colorprop}{6}0=\textcolor{olive}{18}0\)
  • \(\textcolor{colordef}{3}\times \textcolor{colorprop}{7}0=\textcolor{olive}{21}0\)
  • \(\textcolor{colordef}{3}\times \textcolor{colorprop}{8}0=\textcolor{olive}{24}0\)
  • \(\textcolor{colordef}{3}\times \textcolor{colorprop}{9}0=\textcolor{olive}{27}0\)
Notice that the multiplication table for multiples of 10 looks similar to the regular table, but you just add a zero at the end.

Proposition Multiplication Table for Multiple of 10
To multiply by multiples of 10, use the regular multiplication table and add the zeros from the factors to the result:
\(\begin{aligned}\textcolor{colordef}{3}\times \textcolor{colorprop}{1}&=\textcolor{olive}{3}\\\textcolor{colordef}{3}\times \textcolor{colorprop}{2}&=\textcolor{olive}{6 }\\\textcolor{colordef}{3}\times \textcolor{colorprop}{3}&=\textcolor{olive}{9 }\\\textcolor{colordef}{3}\times \textcolor{colorprop}{4}&=\textcolor{olive}{12 }\\\textcolor{colordef}{3}\times \textcolor{colorprop}{5}&=\textcolor{olive}{15 }\\\textcolor{colordef}{3}\times \textcolor{colorprop}{6}&=\textcolor{olive}{18 }\\\textcolor{colordef}{3}\times \textcolor{colorprop}{7}&=\textcolor{olive}{21 }\\\textcolor{colordef}{3}\times \textcolor{colorprop}{8}&=\textcolor{olive}{24 }\\\textcolor{colordef}{3}\times \textcolor{colorprop}{9}&=\textcolor{olive}{27 }\\\end{aligned}\) \(\quad\) \(\begin{aligned}\textcolor{colordef}{3}00\times \textcolor{colorprop}{1}0&=\textcolor{olive}{3}000\\\textcolor{colordef}{3}00\times \textcolor{colorprop}{2}0&=\textcolor{olive}{6}000\\\textcolor{colordef}{3}00\times \textcolor{colorprop}{3}0&=\textcolor{olive}{9}000\\\textcolor{colordef}{3}00\times \textcolor{colorprop}{4}0&=\textcolor{olive}{12}000\\\textcolor{colordef}{3}00\times \textcolor{colorprop}{5}0&=\textcolor{olive}{15}000\\\textcolor{colordef}{3}00\times \textcolor{colorprop}{6}0&=\textcolor{olive}{18}000\\\textcolor{colordef}{3}00\times \textcolor{colorprop}{7}0&=\textcolor{olive}{21}000\\\textcolor{colordef}{3}00\times \textcolor{colorprop}{8}0&=\textcolor{olive}{24}000\\\textcolor{colordef}{3}00\times \textcolor{colorprop}{9}0&=\textcolor{olive}{27}000\\\end{aligned}\)

Long Multiplication by One-Digit Numbers


When multiplying a large number by a single-digit number, it can be tedious and time-consuming to handle each step separately. For example:
To simplify, you can perform the additions step by step using the carry-over method, similar to column addition.

Method Column Multiplication
To calculate \(23 \times 7\), follow these steps:
  1. Step 1: Write the multiplication in columns
    Write the numbers in columns, aligning the digits based on their place value (units, tens, hundreds).
  2. Step 2: Multiply the ones
    $$\textcolor{colordef}{3} \text{ ones} \times \textcolor{colordef}{7} \text{ ones} = 21 \text{ ones} = \textcolor{olive}{2} \text{ tens} + \textcolor{colorprop}{1} \text{ one}$$ Write the carry-over (\(\textcolor{olive}{2}\)) above the tens column.
  3. Step 3: Multiply the tens
    $$\textcolor{colordef}{2} \text{ tens} \times \textcolor{colordef}{7} \text{ ones} + \textcolor{olive}{2} \text{ tens (carry-over)} = \textcolor{colorprop}{16} \text{ tens}$$ Write \(\textcolor{colorprop}{16}\) in the tens and hundreds columns.
  4. Result: \(23 \times 7 = 161\).

Long Multiplication by Multi-Digit Numbers


To calculate \(23 \times 37\), we can use the distributive property:$$\begin{aligned}23 \times 37 &= \textcolor{colorprop}{23 \times 7} + \textcolor{colordef}{23 \times 30} & & \text{(by distributivity: } 37 = 7 + 30) \\&= \textcolor{colorprop}{161} + \textcolor{colordef}{690} & & \text{(intermediate products)} \\&= 851 && \text{(add the intermediate results)} \\\end{aligned}$$This approach forms the basis of column multiplication for multi-digit numbers. The process organizes the steps efficiently:

Method Column Multiplication for Multi-Digit Numbers
To calculate \(23 \times 37\), follow these steps:
  1. Step 1: Write the multiplication in columns
  2. Step 2: Multiply the ones digit (7)
    1. Multiply the ones: \(\textcolor{colordef}{3} \times \textcolor{colordef}{7} = \textcolor{olive}{2}\textcolor{colorprop}{1}\)
    2. Multiply the tens: \(\textcolor{colordef}{2} \times \textcolor{colordef}{7} + \textcolor{colordef}{\scriptstyle 2}\text{\scriptsize(carried)} = 14 + 2 = \textcolor{colorprop}{16}\)
  3. Step 3: Multiply the tens digit (3)
    1. Add a placeholder . (or 0) as a for the multiplication with a tens digit
    2. Multiply the ones: \(\textcolor{colordef}{3} \times \textcolor{colordef}{3}= \textcolor{colorprop}{9}\)
    3. Multiply the tens: \(\textcolor{colordef}{2} \times \textcolor{colordef}{3}= \textcolor{colorprop}{6}\)
  4. Step 4: Add the intermediate results \(\textcolor{colordef}{161} + \textcolor{colordef}{690} = \textcolor{colorprop}{851}\)
  5. Result: \(23 \times 37 = 851\).