These notes for CBSE Class 9 Maths Chapter 1, the Number System, provide a comprehensive overview of fundamental concepts. They define and differentiate between natural numbers (N), whole numbers (W), and integers (Z). The notes clearly explain rational numbers, which can be expressed as p/q (where q ? 0), and irrational numbers, which cannot be expressed in this form. Key characteristics of their decimal expansions are detailed: rational numbers have terminating or non-terminating recurring expansions, while irrational numbers have non-terminating, non-recurring expansions. The collection of both rational and irrational numbers forms the real number system. The notes also discuss the properties of sums, differences, and products involving rational and irrational numbers, and introduce the concept of rationalizing the denominator. Examples and questions are included to help students understand and practice these concepts, making them ideal for exam revision.
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• Natural numbers are - 1, 2, 3, …………….. denoted by N.
• Whole numbers are - 0, 1, 2, 3, ……………… denoted by W.
• Integers - ……. -3, -2, -1, 0, 1, 2, 3, ……………… denoted by Z.
• Rational numbers - All the numbers which can be written in the form p/q,q ?0 are called rational numbers where p and q are integers.
• Irrational numbers - A number s is called irrational, if it cannot be written in the form p/q where p and q are integers and q ?0
• The decimal expansion of a rational number is either terminating or non terminating recurring. Thus we say that a number whose decimal expansion is either terminating or non terminating recurring is a rational number.
• The decimal expansion of a irrational number is non terminating non recurring.
• All the rational numbers and irrational numbers taken together.
• Make a collection of real number.
• A real no is either rational or irrational.
• If r is rational and s is irrational then r+s, r-s, r.s are always irrational numbers but r/s may be rational or irrational.
• Every irrational number can be represented on a number line using Pythagoras theorem.
• Rationalization means to remove square root from the denominator.
Section - A
Q.1 Is zero a rational number? Can you write in the form p/q, where p and q are integer and ?
Q.2 Find five rational numbers between
Q.3 State whether the following statements are true or false give reasons for your answers.
(i) Every natural no. is whole number.
(ii) Every integer is a whole number.
(iii) Every rational number is a whole number.
(iv) Every irrational number is a real number.
(v) Every real number is an irrational number.
(vi) Every point on the number line is of the form where is a natural no’s.
Q.4 Show how can be represented on the number line?
Q.5 Find the decimal expansion of ? What kind of decimal expansion each has.
Q.6 Show that 1.272727 = can be expressed in the form p/q, where p and q are integers and
Q.7 Write three numbers whose decimal expressions are non-terminating & non recurring?
Q.8 Find three different rational between 3/5 and 4/7.
Q.9 Classify the following numbers as rational or irrational.
Section - C
Q.13 Rationalize the denominator of
Section - D
Q.1 Represent on number line.
Q.2 Recall, ? is defined as the ratio of the circumference (say c) of a circle to its
diameter (say d). That is . This seems to contradict the fact that is irrational. How will you resolve this contradiction?
Q.3 Simplify
Q.1 Write the value of
Q.3 If a & b are rational number, find the value of a & b in each of the following equalities.
Q.4 Prove that is an irrational number using long division method?
Natural numbers are the positive counting numbers starting from 1, denoted by N (1, 2, 3,...).
Rational numbers can be written as p/q (q?0), while irrational numbers cannot be written in this form.
Rational numbers have decimal expansions that are either terminating or non-terminating recurring.
The decimal expansion of an irrational number is non-terminating and non-recurring.
Real numbers are the collection of all rational and irrational numbers.
Rationalizing the denominator means removing any square roots from the denominator of a fraction.
Yes, the sum of a rational number and an irrational number is always an irrational number.
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