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Decimals derived from the word ‘deci’ meaning $10$ are used in our daily life. Be it money, weight, or length – the decimals find their place in everyone’s life. They are used to express the whole number and fraction together. The main goal of the usage of decimal numbers is to acquire more precision.
The decimals use a symbol ‘.’, commonly known as a decimal point to express all types of numbers – whole numbers and fractions. The decimal point separates the whole part from the fractional part of a number.
There are many real-life situations in which you might be using decimals without even realizing it. For example, if a shopkeeper tells you that the price of a pen is $10$ rupees and $50$ paise, then mathematically, it can be represented as ₹$10.50$.
In this article, you’ll learn what are like decimals and unlike decimals and how they are placed in a decimal place value chart.
What are Decimals?
Decimals are numbers that fall between two whole numbers or integers. For example, $7.5$ is a decimal number lying between $7$ and $8$. It is greater than $7$, and less than $8$ but is not a whole number.
Decimal numbers are the same as fractions but they’re expressed differently. In the above example, $7.5$ is the same as the mixed fraction $7 \frac {1}{2}$ or the improper fraction $\frac {15}{2}$.
With the help of decimals, you 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.
As you might be knowing that if you move towards the left, the place value of a digit increases 10 times. For example, in $5238$, $8$ is placed at ones, $3$ is placed at tens, $2$ at hundreds, and $5$ at thousands.
Similarly, in the case of digits to the right of the decimal point, the place value of a digit decreases 10 times. For example, in $0.5238$, $5$ is placed at tenths $\left( \frac {1}{10} \right)$, $2$ is placed at hundredths $\left( \frac {1}{100} \right)$, $3$ is is placed at thousandths $\left( \frac {1}{1000} \right)$, and $5$ is placed at tenth-thousandths $\left( \frac {1}{10000} \right)$.
Decimal Place Value Chart
As seen above a decimal number consists of two parts
- the whole part: lies to the left of the decimal point
- the decimal part (fractional part): lies to the left of the decimal point
The place values for the whole part are (moving from right to left after the decimal point):
- ones $\left( 1 \right)$
- tens $\left( 10 \right)$
- hundreds $\left( 100 \right)$
- thousands $\left( 1000 \right)$
The place values for the decimal part are (moving from left to right after the decimal point):
- tenths $\left( \frac {1}{10} \right)$
- hundredths $\left(\frac {1}{100} \right)$
- thousandths $\left(\frac {1}{1000} \right)$
Note: There is NO ‘oneths‘ or ‘uniths‘ before a decimal point. Can you guess “Why?“.
Examples – Decimal Place Value Chart
Let’s consider some examples to understand the decimal place value chart.
Ex 1: $402.874$
Ex 2: $29.905$
Ex 3: $7.04$
Ex 4: $19.003$
Reading Decimals
You can read a decimal number in two ways. Let’s consider a decimal number $56.27$.
- In a first way, it is read as eighty-five point two-seven. The whole number is read in the normal way whereas for the decimal part each digit is read separately.
- In a second way, it is read as eighty-five and twenty-seven hundredths. The whole number is read in the normal way and the decimal part is read by its place value in the decimal place value chart.
Expanded Form of Decimals
You know digits of any whole number can be expanded in terms of their place value. We use a place value chart to write the whole numbers in expanded form.
The number $52864$ can be expanded as $5 \times 10000 + 2 \times 1000 + 8 \times 100 + 6 \times 10 + 4 \times 1$ or $5 \times 10^{4} + 2 \times 10^{3} + 8 \times 10^{2} + 6 \times 10^{1} + 4 \times 10^{0}$
Similarly, digits of a decimal number can be expanded in terms of their place value.
The number $9247.613$ can be expanded as $9 \times 1000 + 2 \times 100 + 4 \times 10 + 7 \times 1 + 6 \times \frac {1}{10} + 1 \times \frac {1}{100} + 3 \times \frac {1}{1000}$ or $9 \times 10^{3} + 2 \times 10^{2} + 4 \times 10^{1} + 7 \times 10^{0} + 6 \times \frac {1}{10^{1}} + 1 \times \frac {1}{10^{2}} + 3 \times \frac {1}{10^{3}}$ or $9 \times 10^{3} + 2 \times 10^{2} + 4 \times 10^{1} + 7 \times 10^{0} + 6 \times 10^{-1} + 1 \times 10^{-2} + 3 \times 10^{-3}$.
Types of Decimals
Decimals can be divided into different categories depending upon the type of digits occurring after the decimal point. It will depend upon whether the digits are repeating, non-repeating, or terminating. The four types of decimal numbers include
- Terminating Decimals
- Non-Terminating Decimals
- Recurring Decimals
- Non-Recurring Decimals
1. Terminating Decimals
The number of digits directly after the decimal point in terminating decimal numbers is limited. The number of digits following the decimal point of the ending decimal numbers can be counted so it is finite.
Examples of terminating decimals are $126.543$, $14.7$, $-12.9843$.
All of these decimal numbers are ending decimal numbers or precise decimal numbers because the number of digits following the decimal point is finite.
2. Non-Terminating Decimals
Non-terminating decimal numbers are those in which the digits following the decimal point repeat indefinitely. In other words, decimal numbers can have an endless number of digits following the decimal point. Non-terminating decimals are classified as recurring and non-recurring decimal numbers.
2 a. Recurring Decimals
Recurring decimal numbers have an unlimited number of digits following the decimal point. These numerals, however, are repeated at regular intervals.
Examples of recurring decimals are $1.33333…$, $7.21212121…$, $56.123123123…$.
These are examples of recurring decimal numbers, in which the number of digits following the decimal point is repeated at regular intervals or in a predefined order.
These numbers can also be written by placing a bar sign over the number that is repeated after the decimal point. $1.\overline{3}$, $7.\overline{21}$, $56.\overline{123}$.
These numbers can also be represented in fractional form, making them rational numbers.
2 b. Non-Recurring Decimals
Non-recurring decimal numbers are decimals that do not terminate and do not repeat. Non-recurring decimal numbers have an infinite number of digits at their decimal places, and their digits do not follow a fixed order.
Examples of non-recurring decimals: $3.14159265359…$ (value of $\pi$), $32.564321786…$, $-54.73429030281…$
Note: Non-recurring decimal numbers cannot be represented by a bar sign because the digits after the decimal point do not repeat in a predictable order.
What Are Like Decimals?
Like decimals are the decimal numbers that have the same number of digits after the decimal point. For example, $3.92$ and $5.68$ are like decimals, because both the numbers have $2$ decimal places after the decimal point.
What Are Unlike Decimals?
Unlike decimals are the decimal numbers that have different number of digits after the decimal point. For example, $6.103$ and $5.23$ are unlike decimals, because $6.103$ has $3$ digits after the decimal point whereas $5.23$ has $2$ digits after the decimal point.
Conclusion
Decimal numbers are the most commonly used form to represent fractional quantities. We use these numbers to represent money, weight, capacity, etc. in our day-to-day life. Depending on whether the number of digits after the decimal point is countable or uncountable, the decimals are broadly classified as terminating or non-terminating.
Practice Problems
- Write the following numbers in the expanded form
- $5643.231$
- $56.095$
- $92.6750$
- $729.7865$
- $2.0008$
- Write the following numbers in words (Use both the forms)
- $6754.7865$
- $87.005$
- $321.9880$
- $78.00008$
- $309.40405$
