Decimals

Entering Decimals

The only difference between adding integers and adding decimals is the location of the decimal point.  When using the Cranmer Abacus, the decimal point is called the unit mart.  You may recall, from previous chapters, we pointed out that the Cranmer Abacus (as purchased from the American Printing House for the Blind) contains 4 raised marks on the bar in the middle of the abacus.  They are located between the columns 3 and 4, between columns 6 and 7, between columns 9 and 10, and between columns 12 and 13. These raised bumps are called unit marks and organize the columns into groups of three.

Up to this point, all the numbers that we have entered into our Cranmer Abacus have been integers.  The implied “decimal point” has been at the right edge of the abacus.  This means that each unit mark acts like a comma to separate each group of three digits.  However, to enter a decimal number, we will “move” the decimal point to one of the unit marks. But, how do we know which unit mark to use?

1. First, move your index finger across the bar from right to left.  Make sure that you can feel each unit mark, every three columns.

2. Then, count the decimal places in each number that you are adding. The number with the largest number of decimal places will determine which unit mark you should use as your decimal point.

3. If the largest number of decimal digits you are adding is 3 or less, the first unit mark will be the point you should use to locate your decimal place.  The decimal portion of the number will be entered to the right of the unit mark, in columns 3, 2, and 1.  The integer portion of the number will be entered to left of this mark, in columns 4 through 13.

4. If the largest number of decimal digits is 4, 5, or 6, then use the second unit mark.

5. For numbers with 7, 8, or 9 decimal digits, you should use the third unit mark.

6. Lastly, you will need to use the fourth unit mark for numbers with 10, 11, or 12 decimal digits.

You will need to be careful that you align each number correctly. Misaligning numbers and decimal points will result in an incorrect answer to your addition problem.

Adding Decimals

Now, let’s add some decimal numbers. You will use the same principles as you used in previous calculations there are some additional questions that you should ask first:

• • • Which decimal number has the largest number of digits to the right of the decimal point?

    • How many digits are there to the right of the decimal point in this number? 

    • Which unit mark will allow you to set all the digits in this decimal number? Use this unit mark to set the first number.

Then, you can add the digits of each of the numbers using grouping or by counting. Follow these links to review the addition techniques using grouping or counting. 

Subtracting Decimals

Now, let’s subtract some decimal numbers. You will use the same principles as you used in previous subtraction calculations there are some additional questions that you should ask first:

• • • Which decimal number has the largest number of digits to the right of the decimal point?

    • How many digits are there to the right of the decimal point in this number? 

    • Which unit mark will allow you to set all the digits in this decimal number? Use this unit mark to set the first number.

Then, you can subtract the digits of each of the numbers directly or using the ten (10) extras principle and the five (5) extras principle for addition and subtraction like you did in previous sections. You would ask these questions:

• Can the digit be subtracted directly?

• If not, can the digit be subtracted using the five (5) extras principle?

• If not, can the digit be subtracted using the ten (10) extras principle and the five (5) extras principle for addition or subtraction?

Follow these links to review the subtraction techniques using grouping or counting. 

Multiplying Decimals

Multiplying  decimals is easier than you think: 

1.  Enter both numbers you are multiplying (multiplicand and multiplier) with the extra zeros and the decimal point removed. 

2. Multiply both numbers using a one or two abaci (abacuses).

3. When you have completed the calculation, reinsert the decimal point. 

To study this technique in more detail, click on either the Single abacus or Double abacus links. 

Dividing Decimals

Performing a division calculation that involves decimal numbers is almost exactly the same as it is with integer numbers. We will set the divisor and dividend in exactly the same way as we did in the previous chapters. But in order to do this, there are two things we need to do first (this is the same technique we used when we multiplied two decimal numbers):

• First, we must remove any extra zero digits.

• 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, 8.54 will be entered as 854, 50.09 will be entered as 5009 and .04695 will be entered as 4695.

The process for calculating the final quotient is the same process used in the previous pages with a few steps added at the end of the process. The first step at the end of the process is to determine the zero point. This is our starting point for locating the final decimal place. 

Then, we need to locate the decimal place. Subtract the number of decimal places in the divisor from the number of decimal places in the dividend. If the number is positive, move the decimal point to the left. If the number is negative, move the decimal point to the right.

To study this technique in more detail, click on either the Single abacus or Double abacus links.