Types of Decimal Representation with Terminating and Recurring Examples
What are Decimal Numbers?
In Mathematics, a decimal number is considered to be a more precise number when compared to whole numbers. It is an integral part of Mathematics and thus students must have an understanding of the question “What are decimal numbers?” A decimal number is divided into two parts, the whole number and the remaining fractional of the concerned number. All the decimal numbers are separated by a decimal point. For example, the number 1.25 where 1 is an integral part and 25 is the fractional part. It is also possible to have decimal numbers on number lines and not just whole numbers.
What is Decimal Representation?
After understanding the fundamental question of “What are decimal numbers?” Let us now dissect the question of “what is decimal representation?”
All the numbers can be represented in decimal format. A non-negative real number “r” can be used as a decimal representation by expressing in the form of a series, it is written as a sum.
r = \[\sum_{i=0}^{∞}\] bᵢ/10ᵢ
b0= a non-negative integer
b1, b2, b3…= digits of decimal representation which satisfies 0 ≤ bi ≤ 9.
The digits in the above-given sequence are finite where bi is assumed to be 0. Some people may make it infinite whereas they forbid the decimal representation and add a sequence of 9. This makes a distinctive representation where the limitation still permits the decimal representation of a non-negative real number.
Decimal representation defines a number and can be written as:
r = b0. b1 b2 b3…
Here, b0 is the integer part of r (not obligatory between 0 and 9)
b1, b2, b3... are r’s fractional part.
This is the answer to the question of “what is decimal representation?”
How to represent the decimals on the number line?
Representation of decimals on the number line is different as compared to a whole number.
A line is divided into two parts between 0 and 1 like shown below
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Here, the line has 0 at the start and 1 at the end. If we divide this into two parts, we obtain a decimal in between the numbers as seen in the line below
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When divided, 0.5 is obtained which is a decimal. 0 is an integral part whereas 5 is the fractional part, separated by a dot represented as (1-0)/2 = 0.5.
Decimal numbers on the number line can also be shown by dividing the line into 10 parts.
If you want to show the decimals between 8 and 9, then divide the line into 10 parts like 8.1, 8.2, 8.3, 8.4, 8.5, 8.6, 8.7, 8.8, and 8.9.
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In the above-given picture analyzing decimals numbers on number line, after 3 lines, decimal 8.4 is shown.
In the same way, to show the decimal 8.45, divide the portion between 8.4 and 8.5 into 10 parts.
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Here, after 4 consecutive lines 8.45 is denoted.
Similarly, as you move forward for decimals with more fractional value, keep dividing the numbers into 10 equal parts. Also, can be negative as well which we will see in the examples given below.
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Decimal Representation on Number Line Example
Decimal numbers on number lines can be represented in a positive as well as a negative value.
1. Show 0.3, 0.7, -0.4 and -1.2 decimal numbers on number lines.
Because, 0.3 = 3/10, 0.7 = 7/10, -0.4 = -4/10 and -1.2 = -12/10
As the above-mentioned representation of decimals on the number line, divide your line into ten equal parts that are the space between consecutive integers. The fraction 1/10 is represented by each part that is obtained.
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To point 0.3; shift three parts on the right-side of zero.
To point 0.7; shift seven parts on the right-side of zero.
To point -0.4; shift four parts on the left-side of zero
To point -1.2; shift twelve parts on the left-side of zero.
In the below diagram, 0.3, 0.7, -0.4, and -1.2 are denoted on a number line.
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Hence, it is possible to denote fractions and decimals on the number line if you understood how it must be done and how to divide the numbers properly. While plotting a decimal number make sure you count the lines properly so there are no errors.
FAQs on Decimal Representation of Numbers Explained
1. What is decimal representation in maths?
Decimal representation is the way of expressing numbers using the base-10 number system with digits from 0 to 9 and a decimal point. In this system, each digit has a place value such as ones, tenths, hundredths, and so on.
- Example: In 45.67, 4 is in the tens place, 5 in the ones place.
- 6 is in the tenths place and 7 is in the hundredths place.
2. How do you write a fraction in decimal form?
To write a fraction in decimal form, divide the numerator by the denominator. This division converts the fraction into its decimal representation.
- Example: Convert 1/4 to decimal.
- 1 ÷ 4 = 0.25
3. What is a terminating decimal?
A terminating decimal is a decimal number that ends after a finite number of digits. It does not continue infinitely.
- Examples: 0.5, 0.75, 2.125
- Fractions like 1/2, 3/4, and 1/8 produce terminating decimals.
4. What is a repeating decimal?
A repeating decimal is a decimal number in which one or more digits repeat infinitely in a pattern. The repeating part is shown using a bar notation.
- Example: 1/3 = 0.333… = 0.3
- Example: 2/11 = 0.18
5. How do you convert a repeating decimal into a fraction?
To convert a repeating decimal into a fraction, use algebra to eliminate the repeating part. For example, let x = 0.3.
- x = 0.333…
- 10x = 3.333…
- 10x − x = 3.333… − 0.333…
- 9x = 3
- x = 3/9 = 1/3
6. What is the place value in decimal representation?
Place value in decimal representation refers to the value of a digit based on its position relative to the decimal point. Each place represents a power of 10.
- To the left: ones (10⁰), tens (10¹), hundreds (10²)
- To the right: tenths (10⁻¹), hundredths (10⁻²), thousandths (10⁻³)
7. What is the difference between terminating and non-terminating decimals?
The difference is that a terminating decimal ends after a finite number of digits, while a non-terminating decimal continues infinitely. Non-terminating decimals can be repeating or non-repeating.
- Terminating example: 0.8
- Non-terminating repeating: 0.6
- Non-terminating non-repeating: π = 3.141592…
8. How do you round a decimal number?
To round a decimal number, look at the digit immediately to the right of the required place value and apply the rounding rule. If the digit is 5 or more, increase the previous digit by 1; if less than 5, keep it unchanged.
- Example: Round 4.678 to two decimal places.
- The third decimal digit is 8 (≥ 5).
- So, the result is 4.68.
9. Can every fraction be written as a decimal?
Yes, every fraction can be written as a decimal, either as a terminating or repeating decimal. This is because all fractions represent rational numbers.
- Example: 1/5 = 0.2 (terminating)
- Example: 1/7 = 0.142857 (repeating)
10. What is the decimal representation of irrational numbers?
The decimal representation of irrational numbers is non-terminating and non-repeating. Their digits continue infinitely without forming a repeating pattern.
- Example: π = 3.1415926535…
- Example: √2 = 1.41421356…
