Multiplying

Multiplying by 10, 100 and 1000

Multiplying by these numbers is very useful in mathematics because you do not need a calculator or complicated calculations. It is based on shifting place values.

When we multiply a whole number (a number without a decimal point), we simply add as many zeros to the end as there are zeros in the multiplier:

This also works for larger numbers – we simply add the zeros after the last digit:

Multiples of tens, hundreds and thousands

The same principle works when we do not multiply directly by 10, 100 or 1000, but by a multiple of them, such as 50 or 700. We can split such numbers into factors to make the multiplication easier to see.

For example, $50 = 5 × 10$ and $700 = 7 × 100$. Therefore:

A quick method is therefore to multiply the numbers without the ending zeros ($5 × 7 = 35$) and then add all the zeros from both factors (one from 50 and two from 700, so three altogether): $35000$.

Hidden zeros

Be careful about one common mistake. Sometimes a zero already appears when we multiply the numbers without their ending zeros – and it is easy to overlook it. Consider this example:

Step 1. Multiply the numbers without the ending zeros: $5 × 6 = 30$. The result already ends in a zero – this is the hidden zero. It does not come from the zeros in the original factors; it appears as a result of the multiplication.

Step 2. Add the zeros from the original factors: 50 has one zero and 60 has one zero, so there are two zeros altogether.

Step 3. Add these two zeros after $30$: $3000$.

If we overlook the hidden zero, we might write $300$ – but that is ten times too small. We can check our answer: $50 × 60$ must be greater than, for example, $50 × 50 = 2500$, so $300$ cannot be correct.

Hidden zeros appear whenever the product of the numbers without their ending zeros ends in zero, for example $2 × 5 = 10$, $4 × 5 = 20$, $5 × 8 = 40$ or $25 × 4 = 100$:

The rule is simple: we always add the zeros from the original factors after the complete product of the numbers without their ending zeros – including any zeros that are already part of that product. We never replace or remove zeros that already appear in the product.

Lattice Multiplication

Lattice (Indian) multiplication is written using a prepared grid.

If we want to calculate $29 × 4$, we write the results of multiplying the individual digits into the diagonally divided cells. In our example, $9 × 4 = 36$ and $2 × 4 = 8$. If the result of multiplying two digits is less than 10, it is useful to write 0 in the tens place for clarity, as shown in the following picture.

Once all the digits have been multiplied, we add the numbers along the diagonals, working from right to left. We write down the ones and carry the tens to the next diagonal. The first diagonal contains only 6, but the next one gives $3 + 8 = 11$. We write 1 in the lower (yellow) cell and carry the other 1, representing one ten, to the next diagonal (white) column, where $0 + 1 = 1$. The digits that appear in the bottom row after adding form the final result of the multiplication. The result is 116.

Lattice Multiplication

Lattice multiplication, also known as Indian multiplication, is written in a prepared grid.

If we want to calculate 38 × 27, we write the results of multiplying the individual digits into the boxes divided by diagonals. In our example, 3 × 2 = 6, 8 × 2 = 16, 3 × 7 = 21, and 8 × 7 = 56. If the result of multiplying two digits is less than 10, it is helpful to write 0 in the tens place to keep the grid clear, as shown in the picture below.

When all the digits have been multiplied, we add the numbers along the diagonals from right to left. We write down the ones and carry the tens to the next diagonal. For example, in our picture, 6 + 5 + 1 = 12, so we write 2 in the bottom box and carry 1 to the next diagonal. The digits we get after adding are then read from top to bottom and to the right, and this gives us the final product. In our case, the result is 1026.

Multiplication in columns

Multiplying vertically (in columns) helps us when counting with higher numbers. As the example below (75×7) shows, we have to write the numbers below each other so that the larger number is above the smaller one. The numbers must be aligned to the right, so the ones are under the ones, the tens under the tens, etc. We will multiply the ones and tens of the larger number (75) with the smaller number (here 7). We start the calculation from the right hand side. In our case, we multiply 7×5, which is 35. Under the ones, we write only 5 as the final digit from the number 35, and we remember the carry over (3) for the next multiplication. Next, we calculate 7×7 and add the carry over 3 from the previous calculation operation to the result 49 (49+3=52). There is no more digit in the hundreds position, so we write 52 at the beginning of the result. The final result is 525.

If we look at the 348×3 example, we will proceed in the same way as in the previous problem. We must multiply all ones, tens and hundreds of the higher number (348) by the lower number (3). We start from the right hand side by multiplying 3×8 and from the result 24 we write only 4 in the ones position and remember to carry over 2 for the next operation. We continue 3×4 and to the result of 12 we carry over and add the 2 we have from the previous round. From the result 14, we again write down only the final number 4 under the tens and keep the rest (1) in our memory until the next calculation operation. This is a 3×3 multiplication, to which we carry over the 1. Then we write 10 at the beginning of the result, because we no longer have another digit in the thousands position that needs to be multiplied. The final result is 1044.

Multiplication by a two-digit number

When multiplying with a two-digit number in writing, we use the same procedure as when using a one-digit number. We write the numbers below each other so that the right is aligned. As we can see in the example below, we start again from the right hand side. First, we multiply all the digits in the upper number by the number 2. Then we multiply the digits of the upper number by the digit 3. We write this result in the next line. We will write it from the right hand side, starting at the tens position, i.e. at the position where 3 is located. If we have both products calculated, we add a plus sign, underline and add both results.

Multiplication in columns

Multiplying vertically (in columns) helps us when counting with higher numbers. As the example below (75×7) shows, we have to write the numbers below each other so that the larger number is above the smaller one. The numbers must be aligned to the right, so the ones are under the ones, the tens under the tens, etc. We will multiply the ones and tens of the larger number (75) with the smaller number (here 7). We start the calculation from the right hand side. In our case, we multiply 7×5, which is 35. Under the ones, we write only 5 as the final digit from the number 35, and we remember the carry over (3) for the next multiplication. Next, we calculate 7×7 and add the carry over 3 from the previous calculation operation to the result 49 (49+3=52). There is no more digit in the hundreds position, so we write 52 at the beginning of the result. The final result is 525.

If we look at the 348×3 example, we will proceed in the same way as in the previous problem. We must multiply all ones, tens and hundreds of the higher number (348) by the lower number (3). We start from the right hand side by multiplying 3×8 and from the result 24 we write only 4 in the ones position and remember to carry over 2 for the next operation. We continue 3×4 and to the result of 12 we carry over and add the 2 we have from the previous round. From the result 14, we again write down only the final number 4 under the tens and keep the rest (1) in our memory until the next calculation operation. This is a 3×3 multiplication, to which we carry over the 1. Then we write 10 at the beginning of the result, because we no longer have another digit in the thousands position that needs to be multiplied. The final result is 1044.

Multiplication by a two-digit number

When multiplying with a two-digit number in writing, we use the same procedure as when using a one-digit number. We write the numbers below each other so that the right is aligned. As we can see in the example below, we start again from the right hand side. First, we multiply all the digits in the upper number by the number 2. Then we multiply the digits of the upper number by the digit 3. We write this result in the next line. We will write it from the right hand side, starting at the tens position, i.e. at the position where 3 is located. If we have both products calculated, we add a plus sign, underline and add both results.

Multiplication in columns

Multiplying vertically (in columns) helps us when counting with higher numbers. As the example below (75×7) shows, we have to write the numbers below each other so that the larger number is above the smaller one. The numbers must be aligned to the right, so the ones are under the ones, the tens under the tens, etc. We will multiply the ones and tens of the larger number (75) with the smaller number (here 7). We start the calculation from the right hand side. In our case, we multiply 7×5, which is 35. Under the ones, we write only 5 as the final digit from the number 35, and we remember the carry over (3) for the next multiplication. Next, we calculate 7×7 and add the carry over 3 from the previous calculation operation to the result 49 (49+3=52). There is no more digit in the hundreds position, so we write 52 at the beginning of the result. The final result is 525.

If we look at the 348×3 example, we will proceed in the same way as in the previous problem. We must multiply all ones, tens and hundreds of the higher number (348) by the lower number (3). We start from the right hand side by multiplying 3×8 and from the result 24 we write only 4 in the ones position and remember to carry over 2 for the next operation. We continue 3×4 and to the result of 12 we carry over and add the 2 we have from the previous round. From the result 14, we again write down only the final number 4 under the tens and keep the rest (1) in our memory until the next calculation operation. This is a 3×3 multiplication, to which we carry over the 1. Then we write 10 at the beginning of the result, because we no longer have another digit in the thousands position that needs to be multiplied. The final result is 1044.

Multiplication by a two-digit number

When multiplying with a two-digit number in writing, we use the same procedure as when using a one-digit number. We write the numbers below each other so that the right is aligned. As we can see in the example below, we start again from the right hand side. First, we multiply all the digits in the upper number by the number 2. Then we multiply the digits of the upper number by the digit 3. We write this result in the next line. We will write it from the right hand side, starting at the tens position, i.e. at the position where 3 is located. If we have both products calculated, we add a plus sign, underline and add both results.