Chapter 1 : Number System

In Chapter 1 of Mathematics Class 9, we cover number comparison, notation systems, and number evaluations. All questions are crucial from an exam perspective.

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1. Which of the following is irrational?

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2. The product $\sqrt[3]{2} \cdot \sqrt[4]{2} \cdot \sqrt[12]{32}$ equals

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3. $\sqrt{10} \times \sqrt{15}$ is equal to

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4. The number obtained on rationalising the denominator of $\frac{1}{\sqrt{7}-2}$ is

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5. If $\sqrt{2}=1.4142$, then $\sqrt{\frac{\sqrt{2}-1}{\sqrt{2}+1}}$ is equal to

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6. The value of $\frac{\sqrt{32}+\sqrt{48}}{\sqrt{8}+\sqrt{12}}$ is equal to

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7. Which of the following is equal to $x$ ?

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8. $\frac{1}{\sqrt{9}-\sqrt{8}}$ is equal to

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9. A rational number between $\sqrt{2}$ and $\sqrt{3}$ is

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10. Between two rational numbers

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11. $\sqrt[4]{\sqrt[3]{2^{2}}}$ equals

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12. The decimal expansion of the number $\sqrt{2}$ is

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13. Which of the following is not equal to $\left[\left(\frac{5}{6}\right)^{\frac{1}{5}}\right]^{-\frac{1}{6}}$ ?

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14. After rationalising the denominator of $\frac{7}{3 \sqrt{3}-2 \sqrt{2}}$, we get the denominator as

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15. Decimal representation of a rational number cannot be

Important Definitions and Rules

Rational Number:
A number is called a rational number if it can be expressed in the form of $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$.

Irrational Number:
A number that cannot be written in the form of $\frac{p}{q}$ (where $p$ and $q$ are integers and $q \neq 0$) is called an irrational number.

Real Numbers:
The collection of all rational and irrational numbers is called real numbers.

Decimal Expansion of Real Numbers:
The decimal expansion of a rational number is either terminating or non-terminating recurring, while the decimal expansion of an irrational number is non-terminating and non-recurring.

Note: If $r$ is a rational number and $s$ is an irrational number then-
1. $r+s$ and $r-s$ are irrational numbers.
2. Additionally, if $r$ is a non-zero rational number then $rs$ and $\frac{r}{s}$ are irrational numbers.


For positive real numbers $a$ and $b$:
(i) $\sqrt{ab}=\sqrt{a}\sqrt{b}$
(ii) $\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$
(iii) $(\sqrt{a}+\sqrt{b})(\sqrt{a}-\sqrt{b})=a-b$
(iv) $(a+\sqrt{b})(a-\sqrt{b})=a^2-b$
(v) $(\sqrt{a}+\sqrt{b})^2=a+2\sqrt{ab}+b$

If $p$ and $q$ are rational numbers and $a$ is a positive real number, then-
(i) $a^pa^q=a^{p+q}$
(ii) ${\left(a^p\right)}^q=a^{pq}$
(iii) $\frac{a^p}{a^q}=a^{p-q}$
(iv) $a^pb^p=(ab)^p$

Conclusion

Chapter 1: Number System in Class 9 Mathematics strengthens students' foundational knowledge in math. It includes all basic concepts and important questions useful for exams.