Discover the fundamentals of real numbers with the latest Class
9 Mathematics syllabus by Punjab Curriculum and Textbook Board (PCTB), Lahore. This comprehensive guide to Exercise 1.1 offers an easy-to-follow introduction for students and educators alike.
A real number is any number that can be represented on a number line. This includes both rational and irrational numbers.
Example: Numbers like \(2\), \(-3\), \(0.75\), \(\sqrt{2}\), and \(\pi\) are all real numbers.
Combination of Rational and Irrational Numbers
Rational Numbers: Numbers that can be expressed as a fraction \(\frac{p}{q}\), where \(p\) and \(q\) are integers, and \(q \neq 0\).
Example:
\[ \frac{1}{2}, -3, 0.5 \]
Formula:
\[ \text{Rational number} =\{ \frac{p}{q}, \, q \neq 0\} \]
Irrational Numbers: Numbers that cannot be expressed as a simple fraction and have non-terminating, non-repeating decimal expansions.
Example:
\[\sqrt{2}, \pi\]
Decimal Representation of Rational Numbers
(i) Terminating Decimal Numbers
A decimal number that ends after a finite number of digits.
Example:
\[0.5, 0.75, 1.25 \]
Formula:
\[ \frac{p}{q}\]
Where \(q\) is a factor of \(10^n\).
(ii) Non-Terminating and Recurring Decimal Numbers
A decimal number that repeats a pattern indefinitely.
Example: \(0.333…\) (repeating as \(0.\overline{3}\)), \(0.142857…\) (repeating as \(0.\overline{142857}\))
Formula:
\[ x = 0.\overline{d_1d_2…d_n} \]
Decimal Representation of Irrational Numbers
Irrational numbers have non-terminating and non-repeating decimal expansions.
Example:
\[\sqrt{2} = 1.414213…, \pi = 3.141592…\]
Representation of Rational and Irrational Numbers on a Number Line
Definition: Rational numbers can be precisely located on a number line as fractions or integers. Irrational numbers can be approximated on a number line using their decimal values.
Example:
Rational: \(\frac{1}{2}\) is halfway between \(0\) and \(1\).
Irrational: \(\sqrt{2}\) lies between \(1.4\) and \(1.5\).
Properties of Real Numbers
Properties of Real Numbers with Respect to Addition \((+)\), Multiplication \((\times)\), and Inequality
Here are the detailed properties of real numbers:
Properties of Addition \((+)\)
Closure Property
The sum of two real numbers is always a real number.
Formula:
\[a + b \in \mathbb{R}\]
Example:
\[3 + 5 = 8\]
Commutative Property
The order of addition does not affect the result.
Formula:
\[a + b = b + a\]
Example:
\[2 + 4 = 4 + 2 = 6\]
Associative Property
Changing the grouping of numbers does not affect the sum.
Formula:
\[(a + b) + c = a + (b + c)\]
Example:
\[(1 + 2) + 3 = 1 + (2 + 3) = 6\]
Identity Property
Adding \(0\) to any real number gives the same number.
Formula:
\[a + 0 = a\]
Example:
\[7 + 0 = 7\]
Inverse Property
Adding the opposite (additive inverse) of a number results in \(0\).
Formula:
\[a + (-a) = 0\]
Example:
\[5 + (-5) = 0\]
Properties of Multiplication \((\times)\)
Closure Property
The product of two real numbers is always a real number.
Formula:
\[a \times b \in \mathbb{R}\]
Example:
\[4 \times 3 = 12\]
Commutative Property
The order of multiplication does not affect the result.
Formula:
\[a \times b = b \times a\]
Example:
\[3 \times 5 = 5 \times 3 = 15\]
Associative Property
Changing the grouping of numbers does not affect the product.
Formula:
\[(a \times b) \times c = a \times (b \times c)\]
Example:
\[(2 \times 3) \times 4 = 2 \times (3 \times 4) = 24\]
Identity Property
Multiplying any real number by \(1\) gives the same number.
Formula:
\[a \times 1 = a\]
Example:
\[7 \times 1 = 7\]
Inverse Property
Multiplying a number by its reciprocal (multiplicative inverse) results in \(1\).
Formula:
\[a \times \frac{1}{a} = 1, \, a \neq 0\]
Example:
\[5 \times \frac{1}{5} = 1\]
Distributive Property
Multiplication distributes over addition or subtraction.
Formula:
\[a \times (b + c) = a \times b + a \times c\]
Example:
\[2 \times (3 + 4) = 2 \times 3 + 2 \times 4 = 6 + 8 = 14\]
Properties with Respect to Inequality
Transitive Property
If \(a < b\) and \(b < c\), then \(a < c\).
Example: If \(2 < 5\) and \(5 < 8\), then \(2 < 8\).
Addition Property of Inequality
Adding the same number to both sides of an inequality does not change the inequality.
Formula: If \(a < b\), then \(a + c < b + c\).
Example: If \(3 < 5\), then \(3 + 2 < 5 + 2\).
Multiplication Property of Inequality
If \(a < b\) and \(c > 0\), then \(a \times c < b \times c\).
If \(a < b\) and \(c < 0\), then \(a \times c > b \times c\) (the inequality reverses).
Example:
If \(2 < 4\) and \(c = 3\), then \(2 \times 3 < 4 \times 3\).
If \(2 < 4\) and \(c = -3\), then \(2 \times (-3) > 4 \times (-3)\).
Division Property of Inequality
If \(a < b\) and \(c > 0\), then \(\frac{a}{c} < \frac{b}{c}\).
If \(a < b\) and \(c < 0\), then \(\frac{a}{c} > \frac{b}{c}\) (the inequality reverses).
Example:
If \(6 < 12\) and \(c = 2\), then \(\frac{6}{2} < \frac{12}{2}\).
If \(6 < 12\) and \(c = -2\), then \(\frac{6}{-2} > \frac{12}{-2}\).
Substitution Property
If \(a = b\), then \(a\) can be replaced by \(b\) in any inequality.
Example: If \(x = 5\), then in \(x < 7\), we can substitute to get \(5 < 7\).
This comprehensive overview explains the key properties with formulas and examples for better understanding.
