Vertical Multiplication

Vertical multiplication is a method used to multiply numbers by aligning them vertically, with one number on top of the other. This layout takes advantage of our place value system, making it easier to multiply digits systematically and keep track of partial products, especially when dealing with multi-digit numbers. This method is particularly useful when mental math becomes difficult or when you need to show your work clearly.

Vertical multiplication works because it systematically applies the distributive property while maintaining proper place values. When we multiply 23 × 45, we're actually calculating (20 + 3) × (40 + 5), which expands to multiple smaller multiplications that are easier to manage when organized vertically.

Correct notation: Ones under ones, tens under tens, and hundreds under hundreds.

• Write the larger number (more digits) on top
• The smaller number goes below, aligned to the right
• This ensures each digit is in its correct place value position

Why this works: Place value is the foundation of our number system. Proper alignment ensures we multiply and add values correctly.

Let's see an example:

Observe the exercise $4\times 32=$
To convert it into a vertical multiplication, we must write the numbers one under the other, ensuring that the ones are under the ones, the tens under the tens, and the hundreds under the hundreds.
Moreover, the longer number, the one that contains more digits, should be written at the top.

Solution:

Now we will multiply the ones digit $2$ by the ones digit $4$. We will write the result and continue.

Now we will multiply the ones digit $2$ by the tens digit $3$ and write the result as follows:

When the result is greater than $9$ it is stored at the top left and must be remembered to add it to the next result. In the result row, only the ones digit is noted.

Let's move on to a more complex exercise.
$36\times 8=$

Solution:
Let's write it in vertical form:

Let's multiply the ones digit $8$
by the ones digit $6$
We will get $48$
$48$ is greater than $9$.
Therefore, we will apply the second rule and note in the result row only the ones digit $8$.
The $4$ will be written at the top left and remembered to add it to the result of the next multiplication.
We store it above the $3$.
Now let's multiply the ones digit $8$ by the tens digit $3$ and let's not forget to add $4$ to the result.
$3\times 8=24$
$24+4=28$
We will note $28$.

Erase "the carried numbers" at the top left before moving on to multiply the next digit, this prevents confusion.

Add $0$ below the result to indicate that you move to the next digit, each row of results starts one place to the left in relation to the previous row.

Now we will see the multiplication of a two-digit number by another two or three-digit number, so we can apply the third and fourth rules.
Observe the exercise:  $358\times 38=$
Solution:

Let's write it in vertical form according to rule $1$.
Multiply the ones digit $8$,
by each of the digits according to rule $2$.

$8\times 8=64$
$8\times 5=40$
$40+6=46$
$8\times 3=24$
$24+4=28$
Now, according to rule $3$ let's erase the "carried numbers" on the top left to avoid confusion.
Furthermore, according to rule $4$ we will add $0$ below the answer to indicate that we have moved to the next digit and we will start writing the row of results one step to the left from the previous row.

That is:

After erasing and moving one step to the left, we can move to the tens digit $3$ and continue multiplying it with the ones, tens, and hundreds, just as we have done so far.
Notice that, the result will be written to the left of the $0$ we added in the following way: 
Make sure to write the digits correctly, each digit below the corresponding one.

$3\times 8=34$
We will keep the $2$ and continue.
$3\times 5=15$
$15+2=17$
We will keep the $1$ and continue.
$3\times 3=9$
$9+1=10$

At this point, all we have left to do is, add all the solutions obtained, in the same way we solve a common addition exercise in vertical form.

Attention: if it were a multiplication of a three-digit number by another three-digit number, when moving to the third digit, we should reserve another place with the $0$. That is, $2$ places and then the answer would be written $2$ steps to the left.

Misalignment: Always check that digits are in correct columns

• Forgotten Carries: Double-check that carries are added correctly
• Place Value Errors: Remember to shift left for each new digit of the multiplier
• Addition Errors: Carefully add partial products at the end

• Estimation: Round numbers and check if your answer is reasonable
• Reverse Check: Use division to verify (13,604 ÷ 38 should equal 358)
• Digital Root: Advanced students can use digital root checking