Types of Number Systems with Definitions Properties and Solved Examples
The basics for mathematics begin with the study of Number Systems for every class. It is included as the first very chapter so that you can get familiar with the various kinds of numbers which are possible and known to us. They are used to deal with various concepts of mathematics.
Therefore, getting acquainted with international number system becomes a necessity for you to ace in Mathematics. In this pursuit, having quality notes by your side can aid to your experience, and you will be able to progress in this chapter with ease.
Introduction to Number System
There can be several kinds of numbers as per the mathematical Number System definition. The various kinds of numbers are ass discussed below –
Natural Numbers – It is represented by ‘N’ and all positive numbers starting from 1 to infinity are known as natural numbers. For example, 1, 2, 3, 4, 5, . . . . . . and so on.
Whole Numbers – It is represented by ‘W’, and all the natural numbers starting from ‘0’ makes into this list. It is to note that whole numbers do not comprise of negative numbers, fractions, or decimals, etc. For example, 0, 1, 2, 3, 4, 5, . . . . . . and so on.
Integers – All whole numbers, in addition to the negative numbers, make integers. For example, 0, 1, 2, -3, -5. Etc.
Real Numbers – A real number is said to be one which can be represented in the number line and is represented by ‘R’. The number line has in-numerous points, and each one represents a unique real number. It can also include rational or irrational numbers.
Rational Numbers – The numbers which are represented in p/q form are known as rational numbers. Here, q and p both are integers, and the value of q is never equal to zero.
Irrational Numbers – Those numbers which are not in the p/q form where the value of q is zero are in the list of irrational numbers. For instance, 1.010024563…, π, e, √2, etc. are examples of an irrational number.
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Representation of Real Numbers on Number Line
As from the information about number system, the successive magnification process is used to represent real numbers on number line. For example, consider this question.
Mark 4. \[\overline{26}\] on the number line up to 4 decimal places.
You can mark it on the number line using the steps mentioned as follows.
Step 1. Since the given number 4.26 is in between 4 and 5, divide the number line into ten equal parts beginning from 4 till 5.
Step 2. Since the given number 4.26 is in between 4.2 and 4.3, divide the number line into ten equal parts beginning from 4.20 till 4..30.
Step 3. Following the same process, divide it further into ten equal parts starting from 4.260 and ending at 4.270.
Step 4. Follow the same process again and, divide it further into ten equal parts starting from 4.2620 and ending at 4.263.
Here, look at the picture to get a clearer idea about the steps.
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Now that you are aware and have learnt the concept of Number System in Mathematics, you will be able to improve your knowledge. In case you are inquisitive and want to dig deep in the mathematical concepts, Vedantu can help you with that.
You can refer to our Number System examples with solutions prepared by qualified tutors and prepare better for exams. Download our Vedantu app to learn more on the go.
FAQs on Number Systems Complete Guide with Concepts and Applications
1. What is a number system in mathematics?
A number system is a way of representing and organizing numbers using specific symbols and rules. It defines how numbers are written and how arithmetic operations are performed. The main number systems in mathematics include:
- Natural numbers (N): 1, 2, 3, ...
- Whole numbers (W): 0, 1, 2, 3, ...
- Integers (Z): ..., −2, −1, 0, 1, 2, ...
- Rational numbers (Q): Numbers in the form p/q where q ≠ 0
- Real numbers (R): All rational and irrational numbers
2. What are the different types of number systems?
The main types of number systems are Natural, Whole, Integers, Rational, Irrational, and Real numbers. These are classified as follows:
- Natural Numbers (N): Counting numbers starting from 1
- Whole Numbers (W): Natural numbers including 0
- Integers (Z): Positive and negative whole numbers including 0
- Rational Numbers (Q): Numbers expressible as p/q
- Irrational Numbers: Non-terminating, non-repeating decimals (e.g., √2)
- Real Numbers (R): All rational and irrational numbers together
3. What is the difference between rational and irrational numbers?
The key difference is that a rational number can be written as a fraction p/q, while an irrational number cannot. Specifically:
- Rational numbers: Decimal form is terminating or repeating (e.g., 1/2 = 0.5, 1/3 = 0.333...)
- Irrational numbers: Decimal form is non-terminating and non-repeating (e.g., √2, π)
4. How do you convert a decimal into a rational number?
A terminating decimal can be converted into a rational number by writing it over a power of 10 and simplifying. For example:
- 0.75 = 75/100
- Simplify: 75/100 = 3/4
- Let x = 0.333...
- 10x = 3.333...
- 10x − x = 3
- 9x = 3 → x = 1/3
5. What is the difference between natural numbers and whole numbers?
The difference is that whole numbers include 0, while natural numbers do not. Specifically:
- Natural Numbers (N): 1, 2, 3, ...
- Whole Numbers (W): 0, 1, 2, 3, ...
6. What are integers with examples?
Integers are whole numbers that can be positive, negative, or zero. The set of integers (Z) includes:
- Negative integers: −3, −2, −1
- Zero: 0
- Positive integers: 1, 2, 3
7. What is a real number in the number system?
A real number is any number that can be represented on the number line, including both rational and irrational numbers. Real numbers include:
- Rational numbers: Fractions and terminating/repeating decimals
- Irrational numbers: Non-terminating, non-repeating decimals like √3 and π
8. How do you represent real numbers on a number line?
Real numbers are represented on a number line by marking their distance from zero. Steps:
- Draw a straight horizontal line.
- Mark 0 as the origin.
- Mark equal intervals for integers (1, 2, −1, −2).
- Place fractions or decimals between integers according to their value.
9. What is the closure property in number systems?
The closure property states that performing an operation on numbers in a set results in a number from the same set. For example:
- Integers are closed under addition: 2 + (−3) = −1 (an integer)
- Whole numbers are not closed under subtraction: 3 − 5 = −2 (not a whole number)
10. Is zero a rational number?
Yes, zero is a rational number because it can be written in the form p/q where q ≠ 0. For example:
- 0 = 0/1
