BMCC: Math Tutorials: Decimals:
Introduction to Decimals
Quick links to topics on this page:
Decimal Notation
Writing Decimals in Standard Form
Writing Decimals in Words
Rounding Decimals
A number that includes a decimal point and digits to the right of the decimal point is said to be written in decimal notation. Such a number is often called just a decimal. The decimal point itself is also often referred to as a "decimal," so if someone uses the word "decimal" you need to make sure whether it is the number or the point being referenced.
These tutorials will use the following convention:
The number of positions a digit appears to the right of a decimal point indicates that digit's place value. This is similar to, but not exactly like, the place values for digits in whole numbers. For instance, consider the whole number 5247 . The right-most digit, 7 , indicates how many ones are in the number, the 4 indicates how many tens are in the number, the 2 indicates how many hundreds are in the number, and the 5 indicates how many thousands are in the number.
When digits appear on the right-hand side of a decimal point, instead of indicating whole numbers or multiples of ten, they are indicating how many portions of a whole number are present. Suppose we turn the previous whole number into a decimal by adding a decimal point and some additional digits to the right of the decimal point, such as 5247.395 . The first digit to the right of the decimal point, 3 , indicates how many tenths are present, the 9 indicates how many hundredths are present, and the 5 indicates how many thousandths are present. When we say a "tenth" we mean one-tenth of 1 , or 
, and when we say "hundredth" we mean 1 one-hundredth of 1 , or 
, etc.
There is a relation between fractions and the place values in decimals. If we have a fraction of seven-tenths, which is written as 
, the equivalent decimal is written by placing a 7 in the tenths place after the decimal point, as 0.7 . If our value was seven-hundredths, or 
, the equivalent decimal is obtained by placing a 7 in the hundredths place, like 0.07 . A zero has to be inserted in the tenths place in order for the 7 to appear in the correct place. What if we had twenty-one hundredths, or 
? In this case, we want the last digit of the number 21 to be positioned in the hundredths place, like 0.21 . If we had twenty-one thousandths, or 
, we want the last digit of 21 to be in the thousandths place, which requires the use of a zero, like 0.021.
What if we start with an improper fraction, such as 
? We know that this fraction has a value greater than 1 , because the numerator is larger than the denominator. To express the fraction as a decimal, the exact process used above is employed again. This fraction is expressed as hundredths (i.e., the denominator is 100), so we want the last digit of the numerator value, 247 , to appear in the hundredths place of the decimal. That means the 7 will appear in the hundredths place, two places past the decimal point, like 0.07 , and the 4 will appear in the tenths place, one place past the decimal point, like 0.4 . There's no more room to the right of the decimal point, so any additional digits are added going to the left of the decimal point-- the 2 gets put into the ones place. The complete decimal is 2.47 . The fact that we have a non-zero digit to the left of the decimal point indicates that this decimal has a value greater than 1 , which we expected because the numerator was larger than the denominator in the original fraction.
Example: Write "four thousand three hundred ninety-eight ten-thousandths" as a decimal.
First we want to make sure we're working with the correct digits. Four thousand three hundred ninety-eight is 4398 , so we want 4398 ten-thousandths. Next we need to identify the proper place to the right of the decimal point. This wasn't mentioned specifically above, but it was stated that the thousandths place is the third place to the right of the decimal point, and the progression is systematic by powers of 10 , so the ten-thousandths place is the fourth place to the right of the decimal point (if we needed the hundred-thousandths place it would be one more place to the right, or the fifth place past the decimal point). Thus the ten-thousandths place, represented here by an "X", is 0.000X .
Now that the correct decimal place has been determined, we want to place the last digit of the number into that place. The last digit of the number is 8 , so that gets put into the position like this: 0.0008 . The rest of the digits are filled in sequentially from right to left from that position, just like in the original whole number. That gives us the decimal 0.4398 .
Another example: Write "seventy-one thousandths" as a decimal.
The number is 71 , so we want the last digit, which is 1 , to appear in the thousandths place. The thousandths place is the third place to the right of the decimal point, or 0.00X . The digit 7 is placed immediately to the left of the 1 , in the hundredths place. Finally, in order to keep those digits in the correct place locations, we have to put a 0 in the tenths place, which gives a final decimal of 0.071 .
One more example: Write "two and fourteen hundredths" as a decimal.
The word "and" is very important in this example. It indicates that the number before "and" is actually a whole number, and is not part of the words being expressed as "hundredths". Only the numbers written after the word "and" are expressed as hundredths. That means the 2 will appear to the left of a decimal, and we need to write the "fourteen" as hundredths.
To write the "fourteen" as hundredths, we place the right-most digit in the hundredths place, which gives 0.14 . The 2 is a whole number, so it appears to the left of the decimal point, and the complete decimal is 2.14 .
You can sort of think of the word "and" as a verbal decimal point.
Converting from a decimal to a word equivalent is, as you might expect, essentially the reverse of the process we just carried out in the previous section when we converted from words to a decimal. The first step is to note which place to the right of the decimal contains the last digit. That will determine how the word equivalent of the decimal is expressed, such as in tenths, or hundredths, etc. Once that has been determined, all of the digits up to and including that final digit are expressed in words as the simple whole number, as if no decimal point was involved at all. The full expression of the decimal in words is achieved by appending the place value expression to the end of the whole number.
Example: Express the decimal 0.903 in words.
The last digit of the decimal is the 3 , and it is located in the thousandths place, so in the word equivalent the decimal is going to be expressed as some number of thousandths. The digits in front of the decimal are 903 , which read as the whole number nine hundred three. Note that we ignore any zeros to the left of the left-most non-zero digit in the decimal. Combining the words for the whole number with the correct term for the place past the decimal gives "nine hundred three thousandths".
Another example: Express the decimal 67.38 in words.
First we identify the correct place past the decimal. The right-most digit is 8 , and it is in the hundredths position, so in the word equivalent the decimal will be expressed as some number of hundredths.
Next we want to express the digits in words for the whole number. In this case, because there are also non-zero digits to the left of the decimal point, there are two steps. We could express the original decimal using words for an improper fraction-- that would give us six thousand seven hundred thirty-eight hundredths. Instead, the digits to the left of the decimal point, which represent a whole number, are expressed in words by themselves as that whole number. Only the digits to the right of the decimal point are expressed as some number of (in this example) hundredths.
So, the digits to the left of the decimal point have the word equivalent "sixty-seven" and the digits to the right of the decimal point have the word equivalent "thirty-eight". They are combined using the word "and" like this: "sixty-seven and thirty-eight hundredths". Writing it in this form indicates that the "sixty-seven" is an actual whole number, and that only the "thirty-eight" part of the words is lumped with the "hundredths" modifier.
You can sort of think of this as reading the decimal point out loud as the word "and".
A final example: Express the decimal 4.007 in words.
The final digit, 7 , is in the thousandths place. The other digits to the to left of the 7, between it and the decimal point, are zeros, so they are not used in the word equivalent for the portion of the number expressed as thousandths-- it will be simply "seven thousandths". The only digit to the left of the decimal point is 4 , which is expressed simply as the word "four". Combining these terms gives "four and seven thousandths".
To "round" a number is to replace an exact value with an approximation. Rounding a number is always done to a specified place value. If the digit immediately to the right of the specified place value is less than 5 , the digit that is actually in the place value is not changed during rounding. If the digit to the right of the place value is equal to or greater than 5 , then the digit in the specified place value is increased by 1 . All of the digits to the right of the specified place value are then eliminated, although digits to the left of the decimal point must be replaced with zeros.
Before we look at rounding a decimal, though, consider the whole number 3298 . If we want to round this number to the hundreds, that means that after rounding, the number should not have any digits to the right of the hundreds place except for place-holding zeros. The hundreds place in this example is occupied by the 2 .
To round, we first need to determine whether the digit 2 is going to remain a 2 after rounding, or whether it should be increased by 1 . For that we look at the number immediately to the right of the hundreds place, which is the 9 in the tens place. Since 9 meets the criteria of being equal to or greater than 5 , the 2 in the hundreds place should be increased by 1 so it is now 3 . Then the numbers to the right of the hundreds place are all replaced with zeros, so that the 3 remains in the hundreds place.
The end result of rounding 3298 to the hundreds is that the 3 in the thousands place remains unchanged, the 2 in the hundreds place gets increased by 1 (to 3) because the digit to its right is equal to or greater than 5 , and both the 9 in the tens place and the 8 in the ones place will be replaced with zeros.
Sometimes instead of saying "rounded to hundreds," the phrase "rounded to the nearest hundred" is used. This wording may be more intuitive, especially for understanding why the 2 in the hundreds place in this example was increased to a 3-- You can think of 3298 as being "closer" to 3300 than to 3200 .
Rounding a decimal is basically the same as rounding a whole number, except that if the decimal is being rounded to a place to the right of the decimal point, any digits to the right of that point do not need to be replaced with zeros-- instead, they are simply eliminated.
Example: Round 3.14159 to the nearest thousandths.
It's important to look closely at the spelling-- since this number is to be rounded to the nearest thousandth (not thousand), that means the rounded number will not have any digits to the right of the thousandths place, which is the third place to the right of the decimal point.
The digit in the thousandths place is a 1 , and the digit immediately to its right is a 5 . Since the digit to the right meets the criteria of being 5 or greater, the digit in the thousandths place should be increased by 1 , which changes the original 1 to a 2 . Then any digits to the right of the thousandths place are eliminated. The eliminated digits do not need to be replaced with zeros, because all of the surviving digits will remain in their original places relative to the decimal point.
Another example: Round 23.002 to the nearest hundredth.
The hundredths position is the second place to the right of the decimal point, and is occupied by a 0 in this number. The digit to the immediate right of the 0 is a 2 . Since 2 is less than 5 , the 0 does not get increased by 1 , but instead remains a 0 . The 2 is eliminated.
Now the question is whether you need to keep the two zeros to the right of the decimal point. Common sense would seem to dictate that the zeros could be eliminated, and that 23.00 is represented perfectly well by 23 . However, in many scientific and economic disciplines, leaving the two zeros after the decimal point has meaning as far as the precision of a measurement. Consider also how we represent dollars-- if we have 2 dollars and 1 dime, the value is usually written as $2.10 (with the extraneous zero) rather than $2.1 (without the extraneous zero). So, there really isn't a standard answer as to whether extraneous zeros should be left on the right-hand side of a decimal point. It depends on the circumstances.
Next: Addition of Decimals
[Previous Page] -- [Next Page]
[Top of Page] -- [Tutorials Home Page]
[BMCC Home Page] -- [E-mail the Tutorial Coordinator]
