Number Systems Class 9 Maths Chapter 1 Notes

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Last Updated on December 13, 2024 by XAM CONTENT

Master the fundamentals of the Number System with our Class 9 Maths Chapter 1 notes, covering essential concepts, solved examples, and key points to help you excel in your exams

Chapter	Number Systems
Type of Questions	Notes
Nature of Articles	Notes with examples
Board	CBSE
Class	9
Subject	Maths
Useful for	Class 9 Studying Students
Important Link	Class 9 Maths Chapterwise Notes

Number Systems Notes

➤ Natural Numbers (N): Set of counting numbers.

$$
N=\{1,2,3,4,5, \ldots .\}
$$

➤ Whole Numbers (W): Set of natural numbers together with zero.

$$
W=\{0,1,2,3,4,5, \ldots .\}
$$

➤ Integers (Z): Set of all whole numbers and negative of natural numbers.

$$
Z=\{\ldots .,-5,-4,-3,-2,-1,0,1,2, \ldots .\}
$$

➤ Rational Numbers (Q): Numbers which can be expressed in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$.

1. Every integer, natural and whole number is a rational number.
2. There are infinite rational numbers between two rational numbers.
3. The sum, difference or product of two rational numbers is always a rational number. The quotient of a division of one rational number by a non-zero rational number is a rational number.
4. The decimal expansion of a rational number is either terminating or non-terminating repeating (recurring). Let $\frac{P}{q}$ be the simplest form of a given number where $p$ and $q$ are integers and $q \neq 0$.
(i) If $q=2^m \times 5^n$ for some non-negative integers $m$ and $n$, then $\frac{p}{q}$ is a terminating decimal.
(ii) If $q \neq\left(2^m \times 5^n\right)$, then $\frac{p}{q}$ is a non-terminating repeating (recurring) decimal.

➤ Equivalent Rational Numbers: Two rational numbers are said to be equivalent, if numerator and denominators of both rational numbers are in proportion or they are reducible to be equal.

➤ Irrational Numbers ( $\overline{\mathbf{Q}}$ ): Numbers which cannot be expressed in the form $\frac{P}{q}$, where $\rho$ and $q$ are integers and $q \neq 0$.

1. The sum, difference, multiplication or division of two irrational numbers are not always irrational.
2. The decimal expansion of an irrational number is non-terminating non-repeating (non-recurring).

➤ Real Numbers (R): Set of all rational and irrational numbers.
Every real number is represented by a unique point on the number line. Also, every point on the number line represents a unique real number.

➤ Radical Sign: Suppose $a>0$ be a real number and $n$ be a positive integer. Then nth root of $a$ is defined as $\sqrt[n]{a}=b$, if $b^n=a$ and $b>0$. The symbol ‘ $\sqrt{ }$ ‘ used in $\sqrt[n]{a}$, i.e., $a^{\frac{1}{n}}$ is called the radical sign, where $n$ and $a$ are known as index and radicand respectively.

Things to Remember:

1. The sum or difference of a rational number and an irrational number is irrational.
2. The product or quotient of a non-zero rational number with an irrational number is irrational.
3. A rational number between two rational numbers $a$ and $b$ is $\frac{a+b}{2}$.
4. Between two rational numbers $a$ and $b, n$ rational numbers are given by:
$
(a+d),(a+2 d), \ldots,(a+n d)
$
$
\text {where}, d=\frac{b-a}{n+1} .
$

➤ Laws of Radicals: Let $a$ and $b$ be positive real numbers. Then,

1. $\sqrt{a b}=\sqrt{a} \sqrt{b}$
2. $\sqrt{\frac{a}{b}}=\frac{\sqrt{a}}{\sqrt{b}}$
3. $(\sqrt{a}+\sqrt{b})(\sqrt{a}-\sqrt{b})=a-b$
4. $(a+\sqrt{b})(a-\sqrt{b})=a^2-b$
or $(\sqrt{a}+b)(\sqrt{a}-b)=a-b^2$
5. $(\sqrt{a}+\sqrt{b})^2=a+2 \sqrt{a b}+b$
6. $(\sqrt{a}+\sqrt{b})(\sqrt{c}+\sqrt{d})$
$=\sqrt{a c}+\sqrt{a d}+\sqrt{b c}+\sqrt{b d}$

➤ Rationalising Factors: If a and b are positive numbers, then

1. Rationalising factor of $\frac{1}{\sqrt{a}}$ is $\sqrt{a}$.
2. Rationalising factor of $\frac{1}{a \pm \sqrt{b}}$ is $a \mp \sqrt{b}$.
3. Rationalising factor of $\frac{}{\sqrt{a} \pm \sqrt{b}}$ is $\sqrt{a} \mp \sqrt{b}$.

➤ Laws of Exponents: If a and b are positive real numbers and m and n are rational numbers, then

1. $a^m \times a^n=a^{m+n}$
2. $a^m \div a^n=a^{m-n}, m>n$
3. $\left(a^m\right)^n=a^{m n}$
4. $a^m b^m=(a b)^m$
5. $\frac{a^m}{b^m}=\left(\frac{a}{b}\right)^m$
6. $\left(\frac{a}{b}\right)^{-m}=\left(\frac{b}{a}\right)^m$
7. $\left(a^m\right)^{\frac{1}{n}}=a^{\frac{m}{n}}$
8. $a^{-m}=\frac{1}{a^m}$
9. $a^0=1$
10. $(\sqrt[n]{a})^m=a^{\frac{m}{n}}=\sqrt[n]{a^m}$

Also check

• Number System Case Study Questions
• Number System Assertion Reasoning Questions

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