Multiplying Decimal Numbers
Double Abacus Approach
Short Course
This is a short course (slide show) on how to multiply decimal numbers. Just click on the arrow on the right to start the show.
Multiplying Decimal Numbers
Setting up the multiplication calculation that involves decimal numbers is almost exactly the same as it is with integer numbers. We will set the multiplicand and multiplier in exactly the same way as we did in the previous chapters. But in order to do this, there are two new things we need to do first:
First, we must remove any extra zero digits.
Extra zeros can be found on either side of the decimal. They don’t change the value of number if they are removed. For example:
The number 007.46000 has several zeros that can be removed. On the left side of the decimal, the first 2 zeros can be removed. The digit 7 is a significant digit but the leading zeros don’t change the number at all. On the right side of the decimal, the zeros to right of the digit 6 can be removed. The digit 6 is the last significant digit on right so any zeros to the right of it can be removed. So, the number 007.46000 is the same as 7.46.
Another example is the number 050.030. The most significant digit on the left is the digit 5, in the tens column. Any zeros to left of this digit can be removed. However, the zeros to the right of digit are also significant and can’t be removed. To the right of the decimal point, the least significant digit is 3 in the hundredth’s column. Any zeros to the right of this digit can be removed. However, the zero in the tenth’s column is significant and can’t be removed. So the number 050.030 is the same as 50.03.
For our calculations, we can also remove zeros from the right of the decimal point when there are no digits on the left side of the decimal. For example, one of our calculations uses the number 0.07495. In this case, we can remove the zero to the left of the decimal point and the first zero. For our calculation, we can use the digits 7495.
The second task is to remove (temporarily) the decimal point. When we enter both the multiplicand and multiplier, we will set integer numbers just as we did in the previous chapters. In the previous examples, 7.46 will be entered as 746, 50.03 will be entered as 5003 and .07495 will be entered as 7495.
Now, set up the multiplicand. The multiplicand (without the extra zeros and decimal point) is the first of the two numbers in the multiplication calculation and is set on the left most columns of the top abacus. Set the multiplicand, from left to right, starting in column 13.
Then, set up the multiplier. Count the number of digits in the multiplier (without the extra zeros and decimal point). This number is the column you should start to set the multiplier from left to right.
You are ready to begin the multiplication process. The process for calculating the final product is the same process used in the previous chapters with one step added at the end of the process. Each digit of the multiplicand is multiplied by each digit of the multiplier. Then we calculate the position of the decimal point and insert it into the product for the final product. This is the process in more detail:
First, we multiply the leftmost digit of the multiplier and the leftmost digit of the multiplicand. If the temporary multiplicand has 3 digits and the temporary multiplier has 3 digits, set the product in the two columns starting in column 6.
Then, we multiply the same digit of the multiplier to next digit of the multiplicand. Add this product to the partial product after shifting 1 column to the right. The partial product is completed when all digits of the multiplicand have been multiplied by the digit in the multiplier. A new digit will be used to multiply each digit in the multiplicand. This means that the new first position of the partial product has shifted one column to the left.
Continue multiplying each digit of the multiplier and adding the product to the partial product after shifting the position one column to the right. When you have completed multiplying each digit of both numbers and adding the product to the partial product, all the digits of the multiplier will be cleared. The result will be the final product without the decimal point.
Now, you are ready for the last step, inserting the decimal point back into the product. Add the number of digits to the right of the decimal point in the both the multiplicand and the multiplier. This sum is the number of columns in the decimal of the final product. The decimal point will be located between the leftmost digit of the decimal and the rightmost digit of the integer in the final product. For example, if the multiplicand has 1 decimal digit and the multiplier has 3 decimal digits, the sum of the decimal digits is 4. This means that the product will have 4 decimal digits and the decimal point will be located between columns 5 and 4.
Examples
Now, let’s look at some more examples of this process. In these examples, the multiplicand will have many digits, but the multiplier will always be a single digit. Click on the link to show the steps to calculate the answer.
Example: 7.2 times 9.7
Example: 988.0 times 31.3
Example: 230.9 times 13.8
Example: 0.1738 times 1.74
Example: 43.8 times 8.790
Example: 600.0 times 0.0171