WHAT IS A DECIMAL NUMBER?
A decimal number is a number that has a fractional component. Stated another way, it means that the number does not just represent a given quantity of wholes, but also represents some parts of a whole. For example, we could think of the number 3 as representing three whole pizza pies, regardless of how many slices each one is cut into. We could think of the number 2.5 as representing two whole pizza pies (again, with no regard to how they are cut), and also one-half of an additional pizza pie. You probably know from everyday experience that 0.5 equals ½, and in this chapter the mathematical reasons for that will be made clear.
EXTENDING THE PLACE VALUE CHART TO DECIMAL PLACES
In Chapter Three we worked with a place value chart for whole numbers. In this chapter we’ll extend the place
value chart to the right to accommodate places for decimal numbers. We use a decimal point (a period) to separate the whole number and the decimal place values.
In a general sense, the decimal place values are the
‚mirror image‛ of the whole number place values.
However, some important explanation is required.
Recall that as we move to the left in the whole number place value chart, each place has a value that is 10 times the value of the place on its right. For example, we have ones, tens, hundreds, thousands, and so forth. Converse-ly, as we move to the right, each place is one-tenth the value of the place on its left. This pattern continues on the right side of a the decimal point.
The concept of decimal place value is easiest to think about in the context of money which we deal with each day. We say $2.40 as ‚two dollars and forty cents,‛ but we know that the 40 cents could made up of 4 dimes. We also know that each dime is worth 10 cents, and that 10 cents is one-tenth of a dollar since it takes 10 dimes to equal a dollar.
If we ignore the dollar sign and the 0, we are left with the decimal number 2.4 to examine. In a non-money context, many people pronounce that number as
‚two-point-W O R K I N G ‚two-point-W I T H D E C I M A L S
four‛ which is fine. However, it is important to under-stand that in math, that value represents two wholes, and four tenths of another whole. You could think of it as two whole pizza pies, and 4 slices from another pie which has been cut into 10 equal slices (tenths).
We know that we group our dimes to the right of the decimal point, and that dimes represent tenths of a dollar. The place value immediately to the right of a decimal point is called the tenths place.
We know that in the amount $6.78, the 8 represents 8 pennies (cents). Each penny is one-hundredth of a dollar since it takes 100 of them to make one dollar. It also takes 10 pennies to make one dime, which means a penny is one-tenth of a dime. Converting our money example to the decimal number 6.78, we can see that the number represents 6 wholes (units), 7 tenths, and 8 hundredths.
Study the following place value chart to see how the pattern continues. No matter where we start on the chart, as we move left we multiply the current place value to 10, and as we move right we divide by 10. Note that there is no ‚oneths‛ place, and that the places to the right of the decimal point all end with ‚ths.‛
Thousands Hundreds Tens Ones / Units DECIMAL POINT Tenths Hundredths Thousandths
2 3 4 5
.
6 7 82,000 300 40 5 0.6 0.07 0.008
Place value breakdown of the number 2,345.678 (Two thousand, three hundred forty-five and
six hundred seventy-eight hundredths)
WHY DO DECIMAL PLACE NAMES END WITH “ths”?
The simplest answer to this question is that there must be some way to distinguish if a column in the place value chart refers to a whole number, or to a fractional (decim-al) value. Any place that ends with ‚ths‛ refers to a fractional value. For example, the number 3 in the hundreds place is worth 300. The number 3 in the hundredths place (represented as 0.03) is worth 3/100.
W O R K I N G W I T H D E C I M A L S
WHERE IS THE “ONEths” PLACE?
When students see that the right side of the place value chart is, in a sense, the mirror image of the left side, a common question is, ‚Where is the oneths place?‛ The simple answer is that there is no such place, but it is important to understand why.
As we move to the left on the place value chart, each column is worth 10 times the value of the column on the right. As we move to the right, each column is worth 1/10 as much as the column on the left. If we think of the ones place as dollar bills, what would be 1/10 as big? The answer is a dime, or ten cents, or one-tenth of a dollar.
That is why the tenths place is directly to the right of the ones place, of course with the decimal point in between since we’ve crossed the line between whole and fraction.
ADDING AND SUBTRACTING DECIMAL NUMBERS
As explained in the Introduction, this book does not go into details about doing arithmetic calculations by hand, except when doing so is straightforward, and serves to review important math concepts. As such, it is worth taking a quick look at how to add and subtract decimal numbers by hand.
The simple rule to remember is to line up the decimal points, and line up the place values on top of each other.
This is just like what we did when we added whole numbers such as 123 + 45. The 4 had to go directly under the 2, since each of those numbers are in the tens place.
Let’s add the decimal numbers 12.3 + 45.67 as shown in the example at left. Notice how the decimal points are arranged directly on top of each other. It is important to see that the 3 in the first number has not been
‚pushed over‛ to the right, since it represents 3 tenths. Instead, an
‚imaginary‛ 0 will serve in the hundredths place of the first number so that we’ll have something to add to the 7 hundredths in the second number. That concept is discussed later in this chapter.
Once the problem has been set up properly, the values can be added just like we did with whole numbers.
When necessary, values may be carried to the place on the left. An example of this in the context of money would be converting 13 pennies into 1 dime and 3 pennies. Recall that our imaginary cash register was only able to hold nine bills in each compartment. We could now extend that concept and say that each change