Comprehensive study notes for Real Numbers (Chapter ) Mathematics Matric 9th. Read detailed explanations, solve MCQs, practice questions with answers. Free online education Pakistan.
Rational Numbers ($Q$): Numbers that can be put in the form $\frac{p}{q}$ where $p, q \in Z$ and $q \neq 0$. Examples: $\frac{2}{3}$, $0.6$ (terminating), $2.3535...$ (recurring).
Irrational Numbers ($Q'$): Numbers that cannot be written as quotient of integers. Their decimal representation is non-terminating and non-recurring. Examples: $\sqrt{2}$, $\sqrt{3}$, $\pi$, $e$.
Radical: An expression of the form $\sqrt[n]{a}$. $n$ is index, $a$ is radicand.
Surd: An irrational radical with a rational radicand (e.g., $\sqrt{3}$). Note: $\sqrt{\pi}$ is not a surd because $\pi$ is irrational.
Process of removing radical from denominator. Multiply numerator and denominator by the Conjugate (change sign of surd part).
Example: Conjugate of $2+\sqrt{3}$ is $2-\sqrt{3}$.
Real number properties are used to solve word problems involving integers, geometry, and measurements.
Example (Integers): Sum of three consecutive integers is 42. Let integers be $x, x+1, x+2$.
Equation: $x+(x+1)+(x+2)=42 \implies 3x+3=42 \implies 3x=39 \implies x=13$. Integers: 13, 14, 15.
Surds are frequently used in geometry problems involving finding sides or areas of shapes.
Example (Triangle Area): If Area $= \frac{1}{2} \times \text{base} \times \text{height}$, finding unknown sides often requires rationalization and solving linear equations involving surds like $\sqrt{3}$ or $\sqrt{5}$.
When simplifying expressions with powers, use the Laws of Exponents:
A number of the form $a + bi$ where $a, b \in R$ and $i = \sqrt{-1}$ is called a Complex Number.
$a$ is the real part, $b$ is the imaginary part.
Powers of $i$:
$i^2 = -1, i^3 = -i, i^4 = 1$.