Decimals

Decimals are used to express the whole number and fraction together. Here, we will separate the whole number from the fraction by inserting a ".", which is called a decimal point. For example, let's say you are going to take a cone of ice cream. The vendor tells you that the price of ice cream is $2 and 50 cents. Now, if you want to express this amount in one figure, you will say that the price of the ice cream cone is $2.50. There are many such real-life situations in which you might be using decimals without even realizing it.

Decimals are a set of numbers lying between integers on a number line. They are just another way to represent fractions in mathematics. With the help of decimals, we can write more precise values of measurable quantities like length, weight, distance, money, etc. The numbers to the left of the decimal point are the integers or whole numbers and the numbers to the right of the decimal point are decimal fractions. If we go right from ones place, the next place will be (1/10) times smaller, which will be (1/10)^th or tenth place value. For an instance, observe the place value chart of decimals given below for the number 12.45.

Decimals Place Value Chart

In the case of decimals, for the whole number part, the place value system is the same as the whole number. But after the decimal point, there is a different world of numbers going on in which we use decimal fractions to represent the value. When we are going towards the left, each place is ten times greater than the previous place value. So, to the right of ones place, we have tenths (1/10) and to the right of tenths, we have hundredths (1/100), and so on. Let us look at some examples of decimal place values for more clarity. The place values of each digit of the numbers 73.789, 8.350, and 45.08 are shown.

Reading Decimal Numbers: There are two ways to read a decimal number. The first way is to simply read the whole number followed by "point", then to read the digits in the fractional part separately. It is a more casual way to read decimals. For example, we read 85.64 as eighty-five point six-four. The second way is to read the whole number part followed by "and", then to read the fractional part in the same way as we read whole numbers but followed by the place value of the last digit. For example, we can also read 85.64 as eighty-five and sixty-four hundredths.

Decimals can also be written in expanded form just like whole numbers. As you know that 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 combined together with an addition sign in between. For writing decimals in expanded form, we will be doing the same. For example, let us write the expanded form of 23.758. The first step is to write 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 23.758. So, the expanded form of 23.758 can be expressed in the following way:

23.758 = 2 × 10 + 3 × 1 + 7 × 1/10 + 5 × 1/100 + 8 × 1/1000

OR

23.758 = 20 + 3 + 0.7 + 0.05 + 0.008

Rounding a decimal number to the nearest tenths is done by taking the digit at the hundredth place into consideration. The digit in this hundredth place can have two variations. First, if that number is 4 or less, just remove all the digits to the right of the tenth place digit and the remaining portion is our desired result. But if the digit at the hundredths place is 5 or greater, we need to increase the tenths place digit by 1, and then remove all the digits on the right of the tenths place digit.

For example, let us try rounding 765.27446 to the tenth place. As we can see, the hundredth place digit in 765.27446 is 7. Now, since 7>5, therefore, to round this number to the nearest tenths place, we add 1 to the tenths place digit and ignore the rest. Hence, when we round off 765.27446 to the nearest tenth place, the result will be 765.3.

Check this article to know more about "Rounding Decimals".

In order to compare the decimals, keep the following two things in mind. First, compare the whole number part of the given decimals. The decimal number with the greater whole number will be greater than the other number, and vice-versa. Second, if the digits before the decimal point are equal to each other then we compare the first digit after the decimal point which is the tenth place digit, and examine which is greater or smaller. We repeat this process and keep on comparing digits to the right until we get the unequal digits.

For example, let's compare 23.789 with 23.759. Here we see digits before the decimal point are equal which is 23 = 23. Now moving on to the tenth place digits to compare. i.e., 23.789 with 23.759, we get 7 = 7. Both of them are again equal. Now we move to the next term to the right of the tenth place digit which is the hundredth digit. i.e., 23.789 with 23.759. Now, since 8>5, then we can say that 23.789 > 23.759. ∴ 23.789 > 23.759. This is how we compare decimal numbers.

Decimals can be divided into different categories depending upon what type of digits occur after the decimal point. It will depend upon whether the digits are repeating, non-repeating, or terminating. Let us have a look at how the decimals are categorized based on their type here.

• Terminating decimals: Terminating decimals mean it does not reoccur and end after a finite number of decimal places. For example: 543.534234, 27.2, etc.
• Non-terminating decimals: It means that the decimal numbers have infinite digits after the decimal point. For example, 54543.23774632439473747..., 827.79734394723... etc. The non-terminating decimal numbers can be further divided into 2 parts:
  • Recurring decimal numbers: In recurring decimal numbers, digits repeat after a fixed interval. For example, 94346.374374374..., 573.636363... etc.
  • Non-recurring decimal numbers: In non-recurring decimal numbers, digits never repeat after a fixed interval. For example 743.872367346.., 7043927.78687564... and so on.

► Interesting Facts and Notes on Decimals

Below given are some interesting facts and notes on decimals. This will help you in understanding the topic faster.

• The prefix "deci" in the word "decimal" means ten.
• Always remember to add the decimal point after the ones place so that we know from where the fraction begins.
• To compare the decimal or fractional part of any decimal number, consider one digit at a time from the right of the decimal point. For example, 4.109 < 4.2, as at the tenths place 1 < 2.

► Related Articles

Check these interesting articles related to the concept of decimals in math.

• Adding Decimals
• Dividing Decimals
• Addition and Subtraction of Decimals
• Relationship Between Fractions and Decimals

What is a Non-Decimal Number?

All integers without any fractional part are non-decimal numbers. For representing these we do not use a decimal point. Also, there is no tenths place or hundredths place in these numbers. For example, 34, 5637, 8765 etc.

How to Read Decimals?

Read the whole number part followed by "and", then read the fractional part in the same way as we read whole numbers but followed by the place value of the last digit. For example, 12.87 is "twelve and eighty-seven hundredth".

Are Decimals Integers?

Every integer can be represented in the form of decimals. For example, 12 is an integer that can be represented as 12.00. But decimals are not integers as integers are not in the form of p/q. By default, an integer can be considered as a decimal. For performing arithmetic operations involving integers and decimals, the integers are to be converted as decimals. For the addition of 4 to 3.36, we convert 4 to 4.00.

Why do We Use Decimals?

Decimals are used to make our calculations easy. We use them to quickly compare fractions without doing many calculations. For example, comparing (675/63) with (463/77) can be very time-consuming. Instead, we write them as decimals 10.71 and 6.01 and compare easily and determine which one is greater than the other. We also use them to shorten the complicated fractional equations. We use decimals every day while dealing with money, weight, length, etc. Decimal numbers are also used in situations where more precision is required which whole numbers or fractions cannot provide.

How to Round Decimals?

To round a decimal number look at the next digit to the right side of the place up to which we want to round off. If the digit is less than 5, round down and if the digit is 5 or more than 5, round up the digit at its left. For example, the decimal 3.284 can be rounded to the nearest hundredths as 3.28.

What are Repeating and Non-Repeating Decimals?

Repeating decimals are those in which one or more digits repeat again and again, for example, 1/3 = 0.3333333, while non-repeating decimals are those in which the decimal digits do not form any repeating pattern. For example, 2.09374359827492..... is a non-terminating non-repeating decimal number.

How do you Write in Decimals?

Decimals are used to express the whole number and fraction in a single notation. In decimals, whole numbers and fractions are separated by a point known as a decimal point. For example, in 65.4, 65 is a whole number and 4 is the fractional part (4/10).

What is 1/4 in Decimals?

Let us see how to represent 1/4 in decimals. Here, multiply the numerator and the denominator with 25, to obtain 100 in the denominator. Further, we need to convert this fraction with a denominator of 100 to a decimal by shifting the decimal point to two places towards the left.

1/4 x 25/25 = 25/100 = 0.25