These Class 9 Maths Notes and Chapter 3 The World of Numbers Class 9 Ganita Manjari Notes are designed according to the latest CBSE syllabus.
Class 9 Maths Chapter 3 The World of Numbers Notes
Class 9 Maths Ganita Manjari Chapter 3 Notes
Class 9 The World of Numbers Notes
→ Introduction: Mathematics did not begin in a classroom with equations on a board; it began in the dirt, on the bark of trees, and on bones. Our history is full of struggle to get Mathematical facts out in general. Our Number System is most probably the oldest and prominent amongst them. Number system plays an important role in organising our works properly and knowing the world efficiently.
→ Number System: Number system which we know these days starts with the counting numbers and ends at the question mark ‘?’ what is coming up next ? There may be many to come ….
→ Natural Numbers (N): Natural Numbers are simply the counting numbers, what we normally and usually use for counting of objects or anything. It emerged at least tens of thousands of years ago due to humanity’s need to count. As a group (set) we can write the group of natural numbers as N = {1, 2, 3, 4, 5, …………}.
→ Concept of Zero (Shunyata): The concept of zero was formalised in India by philosophers and then brought into mathematics formally by Brahmagupta (629 CE), who transformed the philosophical state of ‘nothingness’ (Shunyata) into an actual number on which one could perform arithmetic operations. Later zero was represented by ‘O’.
→ Whole Numbers (W): After the introduction of zero into the number system, set of natural numbers expanded to give us the set of numbers known as the whole numbers. As a group (set) we can write the group of whole numbers as W = (0, 1, 2, 3, 4, 5, …………}.
→ Concept of Negative Numbers (rina): Brahmagupta introduced the concept of negative numbers, i.e., numbers less than zero. The group of negative numbers can be represented as {…, -4, -3, -2, -1}.
→ Integers (Z): Brahmagupta categorised negative numbers as ‘debts’(rina) represented by — ve sign these days and positive numbers as ‘fortunes’ (dliana) represented by + ve sign or more often with no signs before the number symbols. By joining these two kinds of numbers and ‘zero’ we get a set of new numbers known as integers. So, the set of integers are represented by {…, —3, —2, —1, 1,2, 3,…}
As this set of numbers suggests us that there is no integer which can be regarded as the smallest negative integer and also there is no integer which can be regarded as the largest positive integer.
→ The Arithmetic of Integers: Brahmagupta gave explicit rules for adding and multiplying these integers, which we still use exactly as he wrote them over 1,300 years ago:
(i) A fortune plus a fortune is a fortune:
Sum of two positive integers is a positive integer.
For example, 5 + 7 = 12.
(ii) A debt plus a debt is a debt:
Sum of two negative integers is a negative integer.
For example, (-23) + (- 7) = — 30, if you owe ₹ 23 and borrow ₹ 7 more, you now owe ₹ 30.
(iii) A fortune minus zero is a fortune and a debt minus zero is a debt:
If you subtract a zero from a positive number you get a positive number and if you subtract a zero from a negative number you vet a negative number.
For example, (12) – 0 = 12 and (-12) – 0 = – 12.
(iv) The product of a debt and a fortune is a debt:
If you multiply a negative number and a positive number, you get a negative number.
For example, (-10) × 4 = -40, if you take on 4 debts of ₹ 10, your total debt is ₹ 40.
(v) The product of two debts is a fortune:
If you multiply two negative integers, you get a positive integer.
For example, (-6) × (-4) = 24, if positive integer indicates the number of times we have taken a debt, then a negative integer must represents returning that debt by that number of times.
So, we get a positive integer at the end of the process.
→ Rational Numbers (Q):
Rational numbers are defined as any number that can be expressed as a ratio \(\frac{p}{q}\) (where p and q are integers and q ≠ 0). For example, \(\frac{2}{3}, \frac{-5}{4}, \frac{10}{3}\),… etc are Rational Numbers. It must be noted that any integer can also be written in this form of \(\frac{p}{q}\) (where p and q are integers and q ≠ 0), so, any integer is also a rational number. For example, we can write 3 = \(\frac{3}{1}\), – 7 = \(\frac{-7}{1}\), 0 = \(\frac{0}{1}\), etc.
Note: It may be noted that there exists a rational number between any two given rational numbers. This phenomenon on numbers may be termed as the numbers being dense. So, rational numbers are dense.
→ Operations on Rational Numbers:
We can add, subtract, multiply and divide any two rational numbers.
For example, \(\frac{1}{3}+\frac{1}{6}=\frac{1}{2}, \frac{1}{2}-\frac{1}{4}=\frac{1}{4}, \frac{2}{3} \times \frac{5}{7}=\frac{10}{21}, \frac{4}{5} \div \frac{2}{3}=\frac{4}{5} \times \frac{3}{2}=\frac{12}{10}=\frac{6}{5}\)
→ bsolute Value of a Rational Number:
The absolute value of a rational number x, represents its distance from 0 on the number line and is represented by |x|, read as ‘mod For example, \(\left|\frac{2}{3}\right|=\frac{2}{3}\) and \(\left|\frac{-2}{3}\right|=\frac{2}{3}\).
→ Distance between two Rational numbers on the Number Line:
For two rational numbers a and b, the distance between them on the number line is given by |a – b|.
→ Irrational Numbers (I): There are many numbers which can not be written as in the form of \(\frac{p}{q}\) (where p and q are integers and q ≠ 0). These numbers are called irrational numbers. For example, √2, √3, π, etc. cannot be written in the form (where p and q are integers and q ≠ 0). So, these numbers are irrational numbers.
Note: Irrational numbers given non-repeating and non-terminating decimal representations which is contrary to the rational numbers which give either terminating decimal representations or a non-terminating but repeating decimal representations. Irrational numbers are dense like rational numbers.
→ Real Numbers (R): Real numbers are the total union of all rational and irrational numbers. It forms a perfectly continuous and unbroken line where every real physical measurement has a corresponding point. This means that any real number can be represented on a number line.
→ Cyclic Numbers: The digits found in the repeating block of \(\frac{1}{7}\) = 0.142857…, is such that if we multiply it with 2, 3, 4, 5, … etc, we get 0.285714, 0.42857?, 0.571428, 0.714285,…, which is arrangements-rearrangements of the digits in it. This reveals the elegant and symmetrical internal patterns hidden within the rational numbers.
→ Imaginary Numbers:
There are many numbers whose square are negative numbers. For example, \(\sqrt{(-2)^2}\) = -2, \(\sqrt{(-1)^2}\) = – 1, …, etc. These numbers are called Imaginary Numbers. By representing \(\sqrt{(-1)\) = i, we can represent any imaginary number in terms of 7’ (‘iota’).
For example, \(\sqrt{(-7)}\) = i√7, \(\sqrt{(-10)}\) = i\(\sqrt{10}\), \(\sqrt{(-25)}\) = 5i, etc.