Decimals: Definition, Types, Properties & Examples

Decimals are an alternative way to represent fractions in Mathematics. Instead of writing 1/2, for example, you can express a fraction as the decimal 0.5, where zero is in ones’ place and five is in tenth place. Decimal is derived from the Latin word Decimus, which means tenth. Therefore, the decimal system has 10 as its base and is also referred to as a base-10 system. Each number in the decimal system is called decimal. They are used to express both whole numbers and fractions. We separate the whole number from the fraction by inserting a “.”, called a decimal point. In other words, the decimal point is the period that separates the digit in the ones place from that in the tenth place in called a decimal. So, the numbers that are placed in the left of the decimal point are the integers or whole numbers and the numbers on the right side right of the decimal point are decimal fractions.

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Decimal: Definition

The decimals are the extension of the number system. The decimals can also be considered as fractions only when the denominators are \(10,100,1000\) etc. The numbers expressed in decimal forms are known as decimals.

Example: \(17.235, 0.149, 125.005, 2534.0\) etc. are the decimal numbers or decimals.

So, each decimal number or decimal has two parts, namely

1. Whole numbers
2. Decimal part

Also, these parts are separated by a dot \(\left( \cdot \right),\) known as the decimal point.

In the decimal number \(27.54,\) the whole number part is \(27,\) and the decimal part is \(54.\)

Decimal Places: The number of digits accommodated in the decimal part of a decimal number is known as decimal places.

Examples: \(3.75\) has two decimal places, and \(85.325\) has three decimal places.

Types of Decimals

There are two types of decimals which are mentioned below, along with the examples:

1. Terminating Decimals
2. Non-Terminating Decimals

Terminating Decimals

These decimals numbers are the decimals that have a finite number of digits after the decimal point. These are also called exact decimal numbers, as the digits after the decimal point is countable.

Example: \(89.9856, 2.32, -1.89546, 3.37543\) etc., are examples of the terminating decimal numbers. We can also write them in the form of  \(\frac{p}{q},\) as they are rational numbers.

Non-Terminating Decimals

These decimals are decimals that have an infinite number of digits after the decimal point. They are non-terminating decimals that repeat endlessly. These are further divided into two more numbers, and they are:

1. Recurring Decimal Numbers
2. Non-Recurring Decimal Numbers

Recurring Decimal Numbers

These decimals have an infinite number of digits after the decimal point, but these digits are repeatedly occurring.

Examples: \(5.313131…., 7.89898989…., 9.1111111….,\) etc. these are recurring decimals as the number of digits at the decimal places are repeated. So, to make it simple, it can be written by putting assign over them like \(5.\overline {31} ,7.\overline {89} ,9\overline {.1}.\)

They can also be written in the fractional form as these are rational numbers. Example: \(\frac{2}{3} = 0.6666 \ldots \) or \(0.\bar 6,\,\frac{8}{9}\) is equal to \(0.88888\) or \(0.\bar 8\).

This recurring decimal can also be pure periodic or ultimately periodic.

Pure Periodic Decimals

These decimals are those numbers in which a part is repeated endlessly. For example, the numbers \(1.3333…., 0.55555….,\) and \(1.99999…\) are some examples of pure periodic decimals. They can also be written using sign over them like \(1.\bar 3,0.\bar 5\) and \(1.\bar 9.\)

Ultimately Periodic Decimal Numbers

These decimals are those numbers in which the non-periodic part follows the periodic part. For example, \(34.126666…, 6.1788888\) and \(45.9333333\) are a few examples of ultimately periodic decimals. These can also be written like \(34.12\bar 6,6.17\bar 8\) and \(45.9\bar 3\).

Non-recurring Decimal Numbers

These non-recurring decimal numbers are non-terminating and non-repeating decimal numbers. They not only have an infinite number of digits at the decimal places, but their decimal place digits don’t follow a specific order.

Examples: \(1.3687493043…..,1.2376894…….,21.3749940……,1.76368939 \ldots \) are a few non-terminating, non-repeating decimal numbers.

Decimals: Properties

The main properties of the decimal numbers under multiplication and division operations are given below:

1. Commutative property for multiplication: If any two of the decimal numbers are multiplied in any order, the product remains the same.
2. If we multiply any whole number and the decimal number in any order, the product remains the same.
3. Multiplicative identity for decimal numbers: If the decimal fraction is multiplied by the number \(1,\) then the product is in the decimal fraction itself. \(1\) is called the multiplicative identity for the multiplication of decimal numbers.
4. Property of zero: If you multiply the decimal fraction by \(0,\) the product is zero \((0).\)
5. If you divide any decimal number by \(1,\) the quotient is the decimal number itself.
6. If you divide the decimal number by the same number, the quotient is \(1.\) 
7. If \(0\) is divided by any of the decimal numbers, then the quotient is \(0.\)
8. The division of any decimal number by \(0\) is not possible, as the reciprocal of \(0\) does not exist.

Decimals as an Extension of Place Value Table

In case of decimals, the place value system is the same as the whole number for the whole number part. However, after the decimal point, we use decimal fractions to represent the place value. As we are going towards the left, each place/digit in the place value table has a value that is equal to ten times the value of the next digit on its right. For instance, the value of the tenth place is ten times that the one’s place, the value of hundreds place is ten times that of the tenth place and so on. For more clarity on the subject, let us look at some examples of decimal place values.

In other words, the value of one place is one-tenth that of the tenth place, the value of tenths place is one-tenth that of the hundreds place and so on.

Example: consider the place value of six in the following numbers:

Number      Place value
\(6543→\)       \(6000\)
\(5643→\)       \(600\)
\(5463→\)       \(60\)
\(5436→\)       \(6\)

You can observe that the digit six moves one place from left to right its value becomes one-tenth \(\left( {\frac{1}{{10}}} \right)\) of its previous value, when it moves two places from left to right, its value becomes one-hundredths \(\frac{1}{{100}}\) of its previous value and so on. Therefore, if we wish to move towards right beyond one’s place (after the decimal point), we will have to extend the place value table by introducing the places for tenths \(\frac{1}{{10}}\) hundredths \(\frac{1}{{100}}\) thousandths \(\frac{1}{{1000}}\) and so on. In that case, the place value table will take the following shape:

Example: By using the above table, if we write the number \(257 + \frac{3}{{10}} + \frac{2}{{100}}\) then it would be:

The number in the above table will be written as \(257.32\) and is a decimal or decimal number. It is read as “Two hundred fifty-seven point three two” or “Two hundred fifty-seven and thirty-two hundredths”.

A decimal or a decimal number may contain a whole number and a decimal part. The given table exhibits the whole number part and decimal part of some decimals:

Remark: The decimals consisting of only one part, i.e., decimal part or whole number part, are written by using zero in whole or decimal parts.

To read a decimal, we may use the following steps:

1. Read the whole number part.
2. Read the decimal part as ‘point’.
3. Read the numbers to the right side of the decimal point.
   Example: \(17.25\) is read as seventeen point two five.
   OR
4. Read the whole number part.
5. Read the decimal point as ‘and’.
6. Read the number to the right of the decimal point and name the place name of the last digit.
7. Example: \(8.527\) is read as eight and five hundred twenty-seven thousandths

What are Like and Unlike Decimals?

Decimals having the same number of decimal places are known as the like decimals, i.e., decimals having the same number of digits on the right of the decimal point are known as like decimals and other decimals are unlike decimals.

Example: \(5.25, 15.04, 273.89\) are like decimals as there are the same number of digits \((2)\) after the decimal point.

Now, \(9.5, 18.235, 20.0254\) etc. are unlike decimals because they have a different number of digits after the decimal point.

It is important to note that annexing the zeros on the right side of the extreme right digit of the decimal part of a number does not alter the value of the number. Here, unlike decimals, they can be converted into decimals by annexing the required number of zeros on the right side of the extreme right digit in the decimal part.

Example: \(7.4, 15.35, 49.105\) are unlike decimals. These decimals can be rewritten as \(7.400, 15.350. 49.105.\) Now, these are like decimals.

Comparing Decimals

To compare the decimal numbers, we may follow the given steps:

1. Obtain the decimal numbers.
2. Compare the whole parts of the numbers. The number with the greater whole part will be the greater. If the whole parts are equal, go to the next step.
3. Compare the extreme left digits of the decimal parts of the two numbers. The number with a greater extreme left digit will be greater. If the extreme digits of decimal parts are equal, then compare the next digits and so on.

Representation of Decimal Number on the Number Line

Here we shall discuss how the decimals are represented on the number line. For example, let us represent \(0.6\) on the number line.

Now, you know that \(0.6\) is more than zero but less than the number one. There are \(6\) tenths in it. Now, divide the unit length between the numbers \(0\) and \(1\) into \(10\) equal parts and take \(6\) parts as shown below:

Reading Decimal Numbers

We can read a decimal number in two ways. The first way is to read the whole number followed by “point”, then to read the digits in the fractional part separately. This is the more common way to read decimals. For instance, we read 89.65 as eighty-nine point six five. The next way is to read the whole number part followed by “and”, then read the fractional part in the same way as we read the whole numbers but followed by the place value of the last digit. For instance, we can also read 89.65 as eighty-nine and sixty-five hundredths.

Arithmetic Operations on Decimals

Addition and Subtraction of decimals

To add or subtract the decimals, we will follow the following steps:

1. Convert the given decimals to like decimals.
2. Write the decimals in the columns with their decimal points directly below each other so that the tenths come under the tenths, the hundredths come under the hundredths and so on.
3. Add or subtract in the same way as we add or subtract whole numbers.
4. Place the decimal point, in the answer, directly below the other decimal points.
5. Example: Addition

6. Example: Subtraction

Multiplication of a Decimal by a Whole Number

To multiply a decimal by a whole number, follow the given steps:

1. You need to multiply the number without the decimal point by the given whole number.
2. Then later, mark the decimal point in the product to have as many places of decimals are there in the given decimal.

Multiplication of a Decimal by Another Decimal

To multiply a decimal by another decimal, follow the given steps:

1. You have to multiply the two numbers without the decimal point, just like whole numbers.
2. Then, insert the decimal point in the product by counting as many places from right to left as the sum of the number of decimal places of the given decimals.

Dividing a Decimal by \(10,100,1000\) etc..

To divide the decimal by \(10, 100, 1000\) etc., follow the given steps:

1. When you divide a decimal by \(10,\) the decimal point is shifted to the left by one place.
2. When you divide a decimal by \(100,\) the decimal point is shifted to the left by two places.
3. When you divide a decimal by \(1000,\) the decimal point is shifted to the left by three places.

Dividing a Decimal by a Whole Number

To divide a decimal number by a whole number, the division is performed in the same way as in the whole numbers. We first divide the two numbers ignoring the decimal point, and then place the decimal point in the quotient in the same position as in the dividend.

Dividing a Decimal by a Decimal

To divide the decimal by another decimal, follow the steps given below:

1. Multiply the dividend and divisor by or \(1000\) etc, to convert the divisor into a whole number.
2. Then you have to divide the new dividend by the whole number acquired in step one.

Decimals in Expanded Form

Decimals can be written in an expanded form similar to whole numbers. To write any number in expanded form, we have to write the face value multiplied by the place value of all the digits in the number and then combine them together with a ‘plus’ sign in between. For writing decimals in expanded form, we will be doing the same. For example, let us write the expanded form of 32.745. Let us begin by writing the digits of the given number in the place value chart of decimals, as shown below.

As we can observe, the place values are clearly marked along with the face values of each of the digits of the number 32.745. So, the expanded form of 23.758 can be expressed in the following way:

32.745 = 3 × 10 + 2 × 1 + 7 × 1/10 + 4 × 1/100 + 5 × 1/1000

OR

32.745 = 30 + 2 + 0.7 + 0.04 + 0.005

Rounding Decimals to the Nearest Tenths

TO round a decimal number to the nearest tenths, we should take the digit at the hundredth place into consideration. The digit in this hundredth place can be written in two variations. First, if that number is less than or equal to 4, remove all the digits to the right of the digit in the tenth place. The remaining portion is our desired result. But if the digit at the hundredths place is greater than or equal to 5, we must increment the tenths place digit by 1, and then remove all the digits on the right of the digit at the tenths place.

For example, let us try to round 765.27446 to the tenth place. The hundredth place digit in 765.27446 is 7. Since 7>5, to round this number to the nearest tenth place, we must add 1 to the tenth place digit and ignore the rest. Hence, when we round off 765.27446 to the nearest tenth place, the result will be 765.3.

Decimals: Solved Examples

Q.1. Write the given decimals numbers in the place value table: \(20.5\) and \(4.2.\)
Ans: Let us make a commonplace value table, assigning appropriate place value to the digits in the given numbers. We have,

Q.2. Write the following decimals in ascending order: \(5.64, 2.54, 3.05, 0.259\) and \(8.32.\)
Ans: The first step is to convert the given decimals into like decimals; we get
\(5.640, 2.540, 3.050, 0.259\) and \(8.320.\)
Clearly, \(0.259<2.540<3.050<5.640<8.320\)
Hence, given decimals in the ascending order are:
\(0.259, 2.54, 3.05, 5.64\) and \(8.32.\)

Q.3. Find the product \(0.008×0.74.\)
Ans: To find the product, we first multiply \(8\) by \(74.\)
We have, \(8×74=592\)
Now, \(0.008\) has \(3\) decimal places, and \(0.74\) has two decimal places.
The sum of the decimal places\(=3+2=5\)
So, the product must contain \(5\) places of decimals.
Hence, \(0.008×0.74=0.00592\)

Q.4. Which is greater of \(48.23\) and \(39.35\)?
Ans: The given decimals have distinct whole decimal number parts, so compare the whole parts only.
In \(48.23,\) the whole number part is \(48.\)
In \(39.35,\) the whole number part is \(39.\)
\(∵48>39\)
\(∴48.23>39.35.\)

Q.5. Divide \(42.8\) by \(0.02.\)
Ans: We have,
\(\frac{{42.8}}{{0.02}} = \frac{{42.8 \times 100}}{{0.02 \times 100}} = \frac{{4280}}{2} = 2140\)
Hence, \(42.8 \div 0.02 = 2140\)

Summary

In this article, we discussed that Decimals are an extension of our number system. We also discussed types of decimal numbers, properties, place value table, Like and Unlike Decimals, comparison of decimals, number line representation, and operations on decimals with some examples at the end.

Frequently Asked Questions (FAQs) – Decimal

Frequently asked questions related to the chapter decimals is addressed here:

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