Class 9: Rational and Irrational Numbers

$\displaystyle \textbf{Chapter 1: Rational and Irrational Numbers - Quick Revision Notes}$
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$\displaystyle \textbf{1.\ Number\ System}$
$\displaystyle \bullet\ \text{Real Numbers }(R)=\text{Rational Numbers }(Q)\cup\text{Irrational Numbers}.$
$\displaystyle \bullet\ \text{Every point on the number line represents a real number.}$
$\displaystyle \bullet\ \text{Every rational and every irrational number is a real number.}$
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$\displaystyle \textbf{2.\ Rational\ Numbers}$
$\displaystyle \bullet\ \text{A rational number can be written in the form }\frac{a}{b},\text{ where }a,b\in\mathbb{Z} \\ \text{ and }b\neq0.$
$\displaystyle \bullet\ \text{Every integer is a rational number.}$
$\displaystyle \bullet\ \text{Every terminating decimal is a rational number.}$
$\displaystyle \bullet\ \text{Every recurring (repeating) decimal is a rational number.}$
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$\displaystyle \textbf{3.\ Irrational\ Numbers}$
$\displaystyle \bullet\ \text{An irrational number cannot be expressed in the form }\frac{a}{b},\text{ where }a,b\in\mathbb{Z} \\ \text{ and }b\neq0.$
$\displaystyle \bullet\ \text{Its decimal expansion is non-terminating and non-recurring.}$
$\displaystyle \bullet\ \text{Examples: }\sqrt2,\sqrt3,\sqrt5,\pi.$
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$\displaystyle \textbf{4.\ Decimal\ Representation}$
$\displaystyle \bullet\ \text{Rational numbers have either terminating or recurring decimal expansions.}$
$\displaystyle \bullet\ \text{Irrational numbers always have non-terminating and non-recurring decimal expansions.}$
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$\displaystyle \textbf{5.\ When\ is\ a\ Decimal\ Terminating?}$
$\displaystyle \bullet\ \text{Write the fraction in its simplest form.}$
$\displaystyle \bullet\ \text{If the denominator contains only the prime factors }2\text{ and/or }5, \\ \text{ the decimal expansion terminates.}$
$\displaystyle \bullet\ \text{Otherwise, the decimal expansion is non-terminating recurring.}$
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$\displaystyle \textbf{6.\ Properties\ of\ Rational\ Numbers}$
$\displaystyle \bullet\ \text{Rational numbers are closed under Addition, Subtraction, Multiplication and} \\ \text{Division (except by }0).$
$\displaystyle \bullet\ \text{The result of these operations is always a rational number.}$
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$\displaystyle \textbf{7.\ Rational\ Numbers\ Between\ Two\ Numbers}$
$\displaystyle \bullet\ \text{There are infinitely many rational numbers between any two rational numbers.}$
$\displaystyle \bullet\ \text{One rational number between }a\text{ and }b\text{ is }\frac{a+b}{2}.$
$\displaystyle \bullet\ \text{To obtain more rational numbers, use equivalent fractions or divide the interval into equal parts.}$
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$\displaystyle \textbf{8.\ Square\ Roots}$
$\displaystyle \bullet\ \sqrt{\text{Perfect Square}}=\text{Rational Number}.$
$\displaystyle \bullet\ \sqrt{\text{Non-perfect Square}}=\text{Irrational Number}.$
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$\displaystyle \textbf{9.\ Important\ Results}$
$\displaystyle \bullet\ \text{Rational}+\text{Rational}=\text{Rational}.$
$\displaystyle \bullet\ \text{Rational}-\text{Rational}=\text{Rational}.$
$\displaystyle \bullet\ \text{Rational}\times\text{Rational}=\text{Rational}.$
$\displaystyle \bullet\ \text{Rational}\div\text{Rational}=\text{Rational (divisor }\neq0).$
$\displaystyle \bullet\ \text{Rational}+\text{Irrational}=\text{Irrational}.$
$\displaystyle \bullet\ \text{Rational}-\text{Irrational}=\text{Irrational}.$
$\displaystyle \bullet\ \text{Non-zero Rational}\times\text{Irrational}=\text{Irrational}.$
$\displaystyle \bullet\ \text{The product of two irrational numbers may be rational or irrational.}$
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$\displaystyle \textbf{10.\ Real\ Numbers}$
$\displaystyle \bullet\ R=Q\cup\overline{Q}.$
$\displaystyle \bullet\ \text{Every rational number is a real number.}$
$\displaystyle \bullet\ \text{Every irrational number is also a real number.}$
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$\displaystyle \textbf{11.\ Surds\ (Radicals)}$
$\displaystyle \bullet\ \text{A surd is an irrational root.}$
$\displaystyle \bullet\ \text{Examples: }\sqrt5,\sqrt7,\sqrt[3]{10}.$
$\displaystyle \bullet\ \text{Roots of perfect powers are not surds, e.g. }\sqrt4=2,\ \sqrt{625}=25.$
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$\displaystyle \textbf{12.\ Rationalising\ Factor}$
$\displaystyle \bullet\ \text{A rationalising factor is a number which, when multiplied by a surd, gives a rational product.}$
$\displaystyle \bullet\ \text{Example: Rationalising factor of }\sqrt3\text{ is }\sqrt3.$
$\displaystyle \bullet\ \text{Rationalising factor of }2\sqrt5\text{ is }\sqrt5.$
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$\displaystyle \textbf{13.\ Rationalising\ the\ Denominator}$
$\displaystyle \bullet\ \text{For a denominator containing a single surd, multiply the numerator and denominator} \\ \text{by the same surd.}$
$\displaystyle \bullet\ \text{For a binomial denominator, multiply by its conjugate.}$
$\displaystyle \bullet\ \text{Conjugate of }(a+b\sqrt c)\text{ is }(a-b\sqrt c).$
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$\displaystyle \textbf{14.\ Conjugate\ Surds}$
$\displaystyle \bullet\ \text{Conjugates differ only in the sign between the two terms.}$
$\displaystyle \bullet\ (3+\sqrt5)\text{ and }(3-\sqrt5)\text{ are conjugates.}$
$\displaystyle \bullet\ \text{The product of conjugates is always rational.}$
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$\displaystyle \textbf{15.\ Comparing\ Surds}$
$\displaystyle \bullet\ \text{Compare surds by making their indices equal or by squaring both sides (when positive).}$
$\displaystyle \bullet\ \text{Example: }\sqrt5>\sqrt4\text{ because }5>4.$
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$\displaystyle \textbf{Exam\ Points\ to\ Remember}$
$\displaystyle \checkmark\ \text{A rational number is of the form }\frac{a}{b},\ b\neq0.$
$\displaystyle \checkmark\ \text{Irrational numbers are non-terminating and non-recurring.}$
$\displaystyle \checkmark\ \text{Every terminating decimal is rational.}$
$\displaystyle \checkmark\ \text{Every recurring decimal is rational.}$
$\displaystyle \checkmark\ \text{A denominator containing only }2\text{ and }5\text{ gives a terminating decimal.}$
$\displaystyle \checkmark\ \text{There are infinitely many rational numbers between any two rational numbers.}$
$\displaystyle \checkmark\ \sqrt{\text{Perfect Square}}=\text{Rational},\qquad \sqrt{\text{Non-perfect Square}}=\text{Irrational}.$
$\displaystyle \checkmark\ \text{Rational}+\text{Irrational}=\text{Irrational.}$
$\displaystyle \checkmark\ \text{Non-zero Rational}\times\text{Irrational}=\text{Irrational.}$
$\displaystyle \checkmark\ \text{The product of two irrational numbers may be rational or irrational.}$
$\displaystyle \checkmark\ \text{Use conjugates while rationalising a binomial denominator.}$
$\displaystyle \checkmark\ \text{Real Numbers}=\text{Rational Numbers}\cup\text{Irrational Numbers}.$