CBSE Notes Class 9 Maths Chapter 1 - Number System

The number system forms the core of Mathematics and is vital for understanding advanced topics. From counting objects to solving complex problems in Physics, engineering, and computer science, numbers are everywhere. This blog will provide a comprehensive overview of the Class 9 Maths chapter 1 number system: exploring number system, its types, properties, and significance in Mathematics.

1.0Types of Numbers

Number systems are fundamental structures in mathematics that provide a way to represent and manipulate numbers. Here are the key number systems:

1. Natural Numbers (N): Natural numbers are positive integers beginning from 1 and extending infinitely (1, 2, 3, 4, ...)
2. Whole Numbers (W):  Whole numbers are similar to natural numbers but include zero (0, 1, 2, 3, ...).
3. Integers (Z): Integers are whole numbers, including positive numbers, negative numbers, and zero. They are denoted by the symbol "Z" in mathematics. Examples of integers include -3, -2, -1, 0, 1, 2, 3, and so on.
4. Rational Numbers (Q): Rational numbers are those that can be expressed as fractions of two integers where the denominator is not equal to zero (e.g., 1/2, -3/4, 5, -7, 0).
5. Real Numbers (R): Real numbers encompass both irrational and rational numbers. They are numbers that can be located on the number line and include integers, fractions, decimals, and square roots of non-negative numbers. Examples of real numbers are -3, 0, 1.5, $\sqrt{2}$, and π.
6. Complex Numbers (C): Complex numbers are mathematical entities of the form a + bi, where "a" and "b" are real numbers and "i" represents the imaginary unit ($\sqrt{(-1)}$). Complex numbers can be represented on a complex plane. It may be noted that N ⊂ Z ⊂ Q ⊂ R ⊂ C.

7. Irrational Number: An irrational number is a real number that cannot be expressed as a fraction of two integers or in $\frac{\mathrm{p}}{\mathrm{q}}$ form. Its decimal expansion is non-terminating and non-repeating, Examples of irrational numbers are √2 , e, or π.

2.0Properties of Irrational Numbers

1. Non-Rational: Irrational numbers cannot be expressed as a ratio of two integers.
2. Non-Terminating, Non-Repeating Decimals: Their decimal expansion neither terminates nor repeats (e.g., π = 3.14159...).
3. Closure: Irrational numbers are not closed under addition, subtraction, multiplication, or division. (e.g., √2 + √2 = 2√2, which is irrational, but √2 × √2 = 2, which is rational).
4. Subset of Real Numbers: Irrational numbers are part of the set of real numbers (ℝ).

3.0Representation of Rational Number on the Number Line  

Irrational numbers can be challenging to represent due to their non-terminating, non-repeating decimal expansions, but they still have precise locations on the number line. Here's how you can visualize and represent them:

Steps to Represent an Irrational Number (e.g., √2) on the Number Line:

• Draw a Number Line: Start with a simple number line with integers like 0, 1, –1, 2, –2, and so on.
• To locate √2, draw a right-angled triangle where both legs are of length 1 unit (based on the Pythagorean theorem: a² + b² = c²).
• For this draw a base OA measuring 1 unit. And on ‘A’ draw a height of 1 unit long. Name it as AB.
• Now join line OB>
• So by using Pythagoras Theorem 

OA = 1, AB = 1 

OB² = OA² + OC²

OB² = 1² + 1²

OB² = 2

Taking square root,

OB = $\sqrt{2}$

The hypotenuse of this triangle will have a length of √2 units.

• Place one end of the hypotenuse at O on the number line. Using a compass or ruler, mark the point where the hypotenuse intersects the number line and call them C. This point represents √2.

This process can be used for other square roots, like √3 or √5, and even for more complex irrational numbers by refining their decimal approximation.

4.0Operations on Real Numbers

1. The addition or subtraction of a rational number with an irrational number always leads to an irrational number.
2. The product or division of a non-zero rational number and an irrational number always leads to an irrational number.
3. When adding, subtracting, multiplying, or dividing two irrational numbers, the result can be either rational or irrational.

5.0Identities involving square roots for Real Numbers

1. $\sqrt{mn}=\sqrt{m}\cdot \sqrt{n}$
2. $(\sqrt{m}+\sqrt{n})(\sqrt{m}-\sqrt{n})=m-n$
3. $(p+\sqrt{n})(p-\sqrt{n})=p^2-n$
4. $(\sqrt{m}+\sqrt{n})(\sqrt{q}+\sqrt{r})=\sqrt{mq}+\sqrt{mr}+\sqrt{nq}+\sqrt{nr}$
5. $(\sqrt{m}+\sqrt{n})(\sqrt{q}-\sqrt{r})=\sqrt{mq}-\sqrt{mr}+\sqrt{nq}-\sqrt{nr}$
6. $(\sqrt{m}+\sqrt{n}{)}^2=m+2\sqrt{mn}+n$

6.0Solved Examples on Number System

Example 1: Determine whether the following numbers are rational or irrational:

1. $\sqrt{16}$

2. $\sqrt{7}$

3. $\frac{5}{3}$

4. $\frac{7}{0}$

Solutions:

1. $\sqrt{16}:$ $\sqrt{16}=4$ is a rational number because it can be represented as $\frac{4}{1}.$
2. $\sqrt{7}:$ Since 7 is not a perfect square, $\sqrt{7}$ is irrational.
3. $\frac{5}{3}$: This is a ratio of two integers where the denominator is not zero, so 5/3 is a rational number.
4. $\frac{7}{0}$: Division by zero is undefined, so $\frac{7}{0}$ is not a valid number.

Example 2: Simplify $\sqrt{72}$ and express it in the form $a\sqrt{b}$, where a and b are integers and b is not a perfect square.

Solution:

Factorize 72:

$72=36\times 2=6^2\times 2$

Simplify:

$\sqrt{72}=\sqrt{36\times 2}=\sqrt{36}\times \sqrt{2}=6\sqrt{2}$

So, $\sqrt{72}$ simplifies to $6\sqrt{2}$.

Example 3: Rationalize the denominator of \frac{1}{\sqrt{2}}.

Solution:

Multiply by $\frac{\sqrt{2}}{\sqrt{2}}$:

$\frac{1}{\sqrt{2}}\times \frac{\sqrt{2}}{\sqrt{2}}=\frac{\sqrt{2}}{2}$

So, the rationalized form of $\frac{1}{\sqrt{2}}is\frac{\sqrt{2}}{2}$.

Example 4: Simplify $(3\sqrt{5}+2\sqrt{3})-(\sqrt{5}-\sqrt{3})$

Solution:

Distribute and combine like terms:

$(3\sqrt{5}+2\sqrt{3})-(\sqrt{5}-\sqrt{3})$

$=3\sqrt{5}+2\sqrt{3}-\sqrt{5}+\sqrt{3}$

$=2\sqrt{5}+3\sqrt{3}$

So, the simplified form is $2\sqrt{5}+3\sqrt{3}.$

Example 5: Find five rational numbers between $\frac{5}{7}\text{ and }\frac{6}{7}$.

Solution:

Multiply and divide $\frac{5}{7}\text{ and }\frac{6}{7}$ by 6.

$\frac{5\times 6}{7\times 6}=\frac{30}{42}\text{ and }\frac{6\times 6}{7\times 6}=\frac{36}{42}$

So, now 5 rational number between $\frac{5}{7}\text{ and }\frac{6}{7}$ are equivalent to $\frac{30}{42}\text{ and }\frac{36}{42}$ are:

$\frac{31}{42},\frac{32}{42},\frac{33}{42},\frac{34}{42},\frac{35}{42}$

Example 5: Express $0.\overline{257}$ in the form $\frac{\mathrm{p}}{\mathrm{q}}$, where p and q are integers and q ≠ 0.

Solution:

Let $x=0.2\overline{57}$= 0.2575757….

Multiply x by 100

100x = 25.7575…

100x = 25.2 + 0.25757…

100x = 25.2 + x

100x – x = 25.2

99x = 25.2

99 x = $\frac{252}{10}$

Which gives, $x=\frac{252}{990}$

7.0Practice Questions on Number System

1. Find five rational numbers between $\frac{3}{8}\text{ and }\frac{5}{8}$_.
2. Express $\overline{37}$ in the form p/q, where p and q are integers and q ≠ 0.
3. Determine whether the following numbers are rational or irrational:

• $\sqrt{25}$
• $\sqrt{50}$
• $\frac{7}{4}$
• $\sqrt{2}+\sqrt{3}$

4. Explain why $\frac{8}{\sqrt{3}}$ is an irrational number. Rationalize the denominator of $\frac{8}{\sqrt{3}}$.
5. Express $\sqrt{27}+\sqrt{12}$ in the simplest form.
6. Simplify the expression after rationalizing the denominator: $\frac{2\sqrt{3}}{\sqrt{2}-1}$
7. Simplify the following expressions:

• $(\sqrt{7}+2\sqrt{5})-(\sqrt{7}-\sqrt{5})$
• $(\sqrt{6}\times \sqrt{3})+(\sqrt{12}-\sqrt{2})$