Lesson 3: Decimal Numbers (I)November 11, 2010
1. Decimal Numbers
Decimal numbers such as 3.762 are used in situations which call for more precision than whole numbers provide.
As with whole numbers, a digit in a decimal number has a value which depends on the place of the digit. The places to the left of the decimal point are ones, tens, hundreds, and so on, just as with whole numbers. This table shows the decimal place value for various positions:
Note that adding extra zeros to the right of the last decimal digit does not change the value of the decimal number.
|Place (underlined)||Name of Position|
|1.234567||Ones (units) position|
Whole Number Portion
The whole number portion of a decimal number are those digits to the left of the decimal place.
The decimal portion of a decimal number are those digits to the right of the decimal place.
In the number 23.65, the whole number portion is 23 and decimal portion is 65.
In the number 0.024, the whole number portion is 0 and the decimal portion is 024.
1.1. Types of decimal numbers
Any number can be written in “decimal form”.
There are three different types of decimal number: exact, recurring and other decimals.
- An exact or terminating decimal is one that does not go on forever, so you can write down all its digits.
- A recurring decimalis a decimal number which does has infinite decimal figures, but where some of the digits are repeated over and over again. We can distinguish two types:
- Pure recurring decimals are those decimals in which all the digits after the decimal point form a pattern that is repeated indefinitely.
Example: 2, 343434……… is a pure recurring decimal, where all the digits after the decimal point, 34, are repeating.
- Mixed recurring decimals are those decimals the first decimal(s) after the decimal point are not repeated, but the subsequent numbers form a pattern, which is repeated indefinitely.
Example: 0.1252525252525252525… is a mixed recurring decimal, where ’25’ is repeated forever and 1 is not repeated.
Sometimes recurring decimals are written with a bar over the digits which are repeated, or with dots over the first and last digits that are repeated. In Spanish we use an arc.
- Other decimals are those which go on forever and don’t have digits which repeat.
√2=1,414213562…, π =3,141592653589….
1.2. Changing fractions to decimals
A fraction can be changed into a decimal by dividing the numerator by the denominator. You can use a calculator, or divide.
The result could be:
1. A whole number, this is the case when the numerator is a multiple of the denominator. For example: 72/9=8.
2. A terminating decimal, this is the case when, after simplifying the fraction, the denominator only has 2 and 5 as a factors. For example: 197/40=4,925.
3. A pure recurring decimal, this is the case when, after simplifying the fraction, the denominator has only factors different from 2 and 5. For example: 4/11=0,36363636…
4. A mixed recurring decimal, this is the case when, after simplifying the fraction, the denominator has 2 or/and 5 as a factors and other factors different from them. For example: 87/66=1.3181818…
Here you are a video on changing fractions to decimals. You will find they have a tricky method to do this without working out the calculator.
And the reverse?
The reverse is also true: exact and recurring decimals can be written as fractions.
1.3. Converting decimals to fractions
CONVERTING AN EXACT DECIMAL TO A FRACTION
This is an easy question. To do this:
- Write as a numerator the number with no decimal point.
- Write as a denominator 1 followed as many zeros as digits has the decimal portion.
Convert 0,175 into a fraction.
175/1000 Work it out if you don’t believe it !
CONVERTING A RECURRING DECIMAL TO A FRACTION
We know that recurring decimals can be written as fractions. The trick is to use a little algebra. But it is difficult to insert equations on the blog so I will give you the trick:
1. Write in the numerator the difference between the number without the decimal point and the whole number part.
2. Write in the denominator as many nines as decimal digits has the pattern
This trick is valid for any pure recurring decimal number. If the number is mixed recurring decimal then the trick is similar. But “the other decimals”, like π, can’t be changed to fractions.
Here you are a video on changing decimal numbers to fractions. It is the easier case: changing exact decimals to fractions.
1.4. Relation between rational numbers and decimal numbers
We have just seen, in the section above, how we can get four different types of numbers when we divide a fraction (a whole number, a terminating decimal or a pure or mixed recurring decimal).
Therefore, we could say that any fraction can be expressed as a whole number, a terminating decimal or a recurring decimal. We can also say that the opposite is true, i.e. that any whole number, terminating decimal or recurring decimal can be expressed as a fraction. From now on we shall refer to these numbers as rational numbers.
We call fractional numbers to that fractions that can not be converted into whole numbers like 1/3, 4/6, 9/10 .
To sum up:
|ANY NUMBER THAT CAN BE EXPRESSED AS A FRACTION IS KNOWN AS A RATIONAL NUMBER|
We have just seen how we can get four different types of numbers when we divide a fraction. All of these numbers are RATIONAL. We are going to refer to this group of numbers with the letter Q.
We can classify the set Q of RATIONAL NUMBERS as follows:
1.5. Comparing Decimal Numbers
Symbols are used to show how the size of one number compares to another. These symbols are < (less than), > (greater than), and = (equals).
To compare the size of decimal numbers:
1. We compare the whole number portions. The larger decimal number is the one with the lager whole number portion. If the whole number parts are both equal, we compare the decimal portions of the numbers.
2. The leftmost decimal digit is the most significant digit. Compare the pairs of digits in each decimal place, starting with the most significant digit until you find a pair that is different. The number with the larger digit is the larger number.
Note that the number with the most digits is not necessarily the largest.
Compare 0.402 and 0.412.
1. The numbers 0.402 and 0.412 have the same number of digits, and their whole number parts are both 0.
2. We compare the next most significant digit of each number, the digit in the tenths place, 4 in each case. Since they are equal, we go on to the hundredths place, and in this case, 0<1, so 0.402<0.412.
Time to work ! Compare decimals on this web.