Introduction to Decimal Numbers

A number written in decimal notation has a decimal point that acts as a separator to segregate the whole number and fractional part of the decimal number. Whole numbers are numbers from 0 to infinity. The fractional part of a decimal number is always less than 1.

The digits in the whole number part of a decimal number are arranged from right to left like this:

.....Ten-Thousands              Thousands             Hundreds              Tens                   Units

The digits in the fractional part of the decimal numbers are arranged in sequence like this:

Tenths                 Hundredths                      Thousandths                    Ten-thousandths ....

These positions of the digits in the decimal number are known as place values of the decimal number. For example, the place values of a decimal number 325.678 are as follows:

The place value of 5 is units

The place value of 2 is tens

The place value of 3 is hundreds

The place value of 6 is tenths

The place value of 7 is hundredths

The place value of 8 is thousandths

325 is the whole number part, whereas 678 is the fractional part of the decimal number. In other words, we can say that the whole number part of a decimal number is before the decimal point, whereas the fractional part is after the decimal point.

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Types of Decimal Numbers

There are two types of decimal numbers which are explained below:

1) Terminating decimal numbers

These decimal numbers have a finite number of digits after the decimal point. These types of decimal numbers are also known as exact decimal numbers. The number of digits after the decimal point of these numbers is countable.

Examples

89.9856

2.5674

3.87543

-1.89546

1.16

2.32

All these decimal numbers are the examples of terminating or exact decimal numbers because the number of digits after the decimal point is finite. These decimal numbers can be written in the form of p/q, hence they are rational numbers. Remember that rational number is that number which can be written in the form of p/q, where q \neq 0.

2) Non - terminating decimal numbers

As the name suggests, the digits after the decimal point of non - terminating decimals repeat endlessly. In other words, we can say that these decimal numbers have an infinite number of digits after the decimal point. These decimal numbers are further divided into recurring and non - recurring decimal numbers.

  • Recurring Decimal Numbers
  • Non- recurring Decimal Numbers

a) Recurring Decimal Numbers

These decimal numbers have an infinite number of digits after the decimal point, however, these digits are repeated at regular intervals.

Examples

120.353535....

2.22222....

5.313131....

7.898989...

9.111111....

The above numbers are recurring decimal numbers because the number of digits at their decimal places are repeated after regular intervals or follow a specific order. We can write these numbers simply by putting a bar sign over them. For example, all the above numbers can be written as:

120. \overline {35}

2. \overline {2}

5.\overline {13}

7. \overline {89}

9. \overline {1}

These decimal numbers can also be written in the fractional form, hence they are rational numbers. For example, \frac {2} {3} = 0.66666.... or 0. \overline {6} or \frac {8} {9} is equal to 0.8888.... or 0.\overline{8}.

Recurring decimals can be pure periodic or ultimately periodic.

  • Pure periodic decimals are those decimal numbers in which the decimal part id repeated endlessly. For example, 1.333..., 0.5555... and 1.9999... are examples of pure periodic decimal numbers. These numbers can also be written with a bar sign over them as 1. \overline {3}, 0.\overline {5} and 1. \overline {9}.
  • Ultimately period decimal numbers are those numbers in which a periodic part is followed by a non - periodic part. For examples, 34.126666...., 6.178888.... and 45.93333... are examples of ultimately periodic decimals. They can also be written as 34. 12 \overline {6} , 6.17 \overline {8} and 45.9 \overline {3}.

 

b) Non -recurring decimal numbers

These are non -terminating, non -repeating decimal numbers. These decimal numbers have not only an infinite number of digits at their decimal places, but their decimal place digits do not follow a specific order.

Examples

The examples of non-terminating, non -recurring decimal numbers are given below:

1.3687493043....

456.789321633894...

1.2376894....

1.76368939....

21.3749940...

We cannot put a bar sign over the decimal numbers because the digits follow no order. These decimal numbers cannot be written in the form of p/q, hence they are irrational numbers.

Activity - Types of Decimal Numbers

Analyze the following decimals and identify their types.

1.323232....

4.5688937....

6.66666...

7.7 \overline {89}

1.2345....

1.10

2.2355

0.\overline {35}

Solution

1.323232....                                   Non - terminating, recurring decimal number

4.5688937....                               Non - terminating, non - recurring decimal number

6.66666...                                    Non terminating, recurring decimal number

7.7 \overline {89}               Ultimately periodic decimal or non terminating recurring decimal number

1.2345....                                      Non -terminating, non recurring decimal number

1.10                                               Terminating or exact decimal number

2.2355                                           Terminating or exact decimal number

0.\overline {35}                    Non- terminating, recurring decimal number

 

Rounding Decimal Numbers

Sometimes in a question, we are asked to write the final answer up to a certain decimal place value. In this case, we usually round off our answer to the required number of decimal place values. Even we are given a non - terminating decimal number, we can easily convert it into a terminating decimal by rounding it off to a certain number of decimal places. The process of rounding the numbers to the nearest integers is given below:

  • First, determine the place value up to which you want to round off your number.
  • After determining the place value up to which you want to round off the decimal number, see the digits at the right of that place value.
  • If the digit is lesser than 5, then do not change the previous decimal digit.
  • If the decimal digit is 5 or greater than 5, then add 1 to the previous digit.

Consider the following example.

Round off the decimal number 567.81456 up to three decimal places.

For rounding this number to 3 decimal places, we need to see the digit at the fourth decimal place. The digit at the fourth decimal place is 5. Hence, we will add 1 to the last digit 4. The new number after rounding off will be 567.815.

How to Convert Decimals to Fractions and Fractions to Decimal Numbers

We can convert the decimals to fractions and fractions to decimal numbers. To convert the decimal numbers to fractions, we remove the decimal point from the decimal number and add the number of zeroes equal to the number of decimal places followed by the unit in the denominator.

Consider the following examples

1. Convert 376.980 to a fraction.

To convert the above decimal number to fraction, we will remove the decimal separator and add a unit followed by three zeroes in the denominator. We add three zeroes in the denominator because the number has three decimal places. The fractional form of the number will be \frac {376980} {1000}.

If we are given a fraction whose denominator has a unit followed by any number of zeroes, we can easily convert it into a decimal number by moving the decimal point to the left equal to the number of zeroes in the denominator.

2. Convert \frac {1259} {100} to a decimal number.

You can see that the denominator has 2 zeroes, so we will move the decimal point 2 units to the left in the numerator. It means that the above fraction is equal to 12.59. This final number is up to two decimal places and is an example of exact or terminating decimal number.

3. Convert \frac {83.450} {10} to a decimal number.

This is a decimal fraction as we have a decimal n.umber in the numerator. The decimal number has one trailing zero. Trailing zeros are those zeroes that are at the right of the decimal number. To convert this decimal fraction to the decimal number, we will simply move the decimal point 1 unit to the left. Hence, this fraction is equal to 8.345. This decimal is exact or terminating decimal.

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Emma

I am passionate about travelling and currently live and work in Paris. I like to spend my time reading, gardening, running, learning languages and exploring new places.