Number Systems – A Brief Overview
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
1. What is a number system in mathematics and what are its main types?
In mathematics, a number system is a structured way of representing numbers using digits or other symbols. It provides a unique representation for every number and defines the rules for arithmetic operations. For the CBSE Class 9 syllabus, the primary types of numbers you will study are:
- Natural Numbers (N): Counting numbers starting from 1 (1, 2, 3, ...).
- Whole Numbers (W): Natural numbers including zero (0, 1, 2, 3, ...).
- Integers (Z): All whole numbers and their negative counterparts (..., -3, -2, -1, 0, 1, 2, 3, ...).
- Rational Numbers (Q): Numbers that can be expressed in the form p/q, where p and q are integers and q ≠ 0 (e.g., 1/2, -5, 0).
- Irrational Numbers: Numbers that cannot be expressed as a simple fraction p/q, and their decimal representation is non-terminating and non-repeating (e.g., √2, π).
- Real Numbers (R): The set of all rational and irrational numbers combined.
2. What is the key difference between rational and irrational numbers with examples?
The key difference lies in their decimal representation. Rational numbers can be written as a fraction (p/q) and their decimal form either terminates or repeats. For example, 3/4 = 0.75 (terminating) and 1/3 = 0.333... (repeating). In contrast, irrational numbers cannot be written as a simple fraction, and their decimal representation is non-terminating and non-repeating. For example, the value of π (Pi) is approximately 3.14159265... and the square root of 2 (√2) is approximately 1.41421356..., with both having digits that go on forever without a repeating pattern.
3. How can you represent an irrational number like √3 on the number line?
You can represent √3 on the number line using the Pythagorean theorem. First, you need to locate √2. To do this, construct a right-angled triangle with a base of 1 unit and a height of 1 unit starting from 0 on the number line. The hypotenuse will have a length of √2. Now, from the point √2, draw a perpendicular line of 1 unit height. The hypotenuse of this new right-angled triangle will have a length of √( (√2)² + 1² ) = √(2+1) = √3. You can then use a compass to transfer this length onto the number line to mark the exact position of √3.
4. Why is the concept of 'real numbers' important?
The concept of real numbers is fundamental because it encompasses every possible point on a continuous number line. This is crucial for several reasons:
- Measurement: Real numbers allow us to measure continuous quantities in the real world, such as length, area, time, and temperature, which may not always be simple whole numbers or fractions.
- Geometry: Concepts like the diagonal of a square (√2) or the circumference of a circle (π) involve irrational numbers, which are part of the real number system.
- Calculus and Advanced Maths: The entire field of calculus is built upon the properties of the real number line, including concepts of limits and continuity.
5. How does the decimal expansion of a number definitively tell us if it's rational or irrational?
The decimal expansion is a key identifier. A number is rational if its decimal form is either terminating (e.g., 0.5) or non-terminating but recurring (e.g., 0.1666...). A number is irrational if its decimal form is both non-terminating and non-recurring, meaning the digits continue forever without any repeating pattern. This property is the definitive test for classifying a real number based on its decimal form.
6. Is zero considered a rational number? Why?
Yes, zero is a rational number. A number is defined as rational if it can be expressed in the p/q form, where 'p' and 'q' are integers and 'q' is not equal to zero. Zero perfectly fits this definition because it can be written as 0/1, 0/2, 0/10, or any other fraction where zero is the numerator and the denominator is any non-zero integer. Since it meets the condition, it is classified as a rational number.
7. What does it mean to 'rationalise the denominator' of a fraction?
Rationalising the denominator is the process of converting a fraction that has an irrational number in its denominator into an equivalent fraction with a rational denominator. For instance, in the fraction 1/√2, the denominator is irrational. To rationalise it, we multiply both the numerator and the denominator by √2, resulting in √2/2. This is done to simplify the expression and write it in a standard form, which makes further calculations easier.
8. What is the importance of understanding the laws of exponents for real numbers?
Understanding the laws of exponents is crucial for simplifying complex expressions involving powers and roots. These laws, such as (a^m)(a^n) = a^(m+n) or (a^m)^n = a^(mn), apply to all real numbers, including rational and irrational bases. Mastering these rules helps in solving algebraic equations, simplifying scientific notations, and provides a foundation for more advanced mathematical topics like logarithms and exponential functions, as per the CBSE 2025-26 syllabus.
