Class 9 Maths Number Systems Solutions

Is article mein hum Class 9 Ganit ke chapter Number Systems ke solutions discuss karenge. Ye NCERT CBSE pattern par based hai aur students ko concepts clear karne mein help karega.

Kya Seekhenge

• Natural Numbers (Prakritik Sankhya) aur Whole Numbers
• Integers (Purnank) aur Rational Numbers
• Irrational Numbers aur Real Numbers

Links for Chapter-wise Download NCERT Solution for Class 9 गणित in hindi Language

Here we have provided NCERT Solution for Class 9 गणित in hindi Language, Just select the chapters below to get solution of the same:

बहुपद

निर्देशांक ज्यामिति

दो चरों वाले रैखिक समीकरण

यूक्लिड की ज्यामिति का परिचय

रेखाएँ और कोण

त्रिभुज

चतुर्भुज

समांतर चतुर्भुजों और त्रिभुजों के क्षेत्रफल

वृत्त

रचनाएँ

हीरोन का सूत्र

पृष्ठीय क्षेत्रफल और आयतन

सांख्यिकी

प्रायिकता

Class 9 Maths Number Systems: Detailed Solutions

Number systems maths ka basic foundation hai. Is chapter mein hum different types of numbers aur unke properties ko examples ke through samjhenge.

1. Natural Numbers (Prakritik Sankhya)

Natural numbers counting numbers hote hain, jaise 1, 2, 3, ... Ye positive integers hain. Example: 5 natural number hai.

2. Whole Numbers (Poorn Sankhya)

Whole numbers natural numbers plus zero hote hain. Example: 0, 1, 2, 3, ... Zero include karna important hai.

3. Integers (Purnank)

Integers whole numbers aur unke negatives hote hain. Example: -3, -2, -1, 0, 1, 2, 3. Ye without fraction waale numbers hain.

4. Rational Numbers (Parimey Sankhya)

Rational numbers wo numbers hain jo p/q form mein likhe ja sakte hain, jahan p aur q integers hain aur q ≠ 0. Example: 1/2, -4/5, 7 (kyunki 7 = 7/1).

5. Irrational Numbers (Aparimey Sankhya)

Irrational numbers ko p/q form mein express nahi kiya ja sakta. Inka decimal expansion non-terminating aur non-repeating hota hai. Example: √2, π (pi).

6. Real Numbers (Vastvik Sankhya)

Real numbers rational aur irrational numbers ka combination hain. Ye number line par represent kiye ja sakte hain.

Solved Examples with Steps

Example 1: Rational Numbers between Two Numbers

Problem: Find two rational numbers between 1 and 2.

Solution: Pehle, average nikalte hain. (1+2)/2 = 1.5, jo ek rational number hai. Phir, 1 aur 1.5 ka average: (1+1.5)/2 = 1.25. So, 1.25 aur 1.5 do rational numbers hain 1 aur 2 ke beech.

Example 2: Irrational Number Proof

Problem: Prove that √3 is irrational.

Solution: Maan lete hain √3 rational hai. To √3 = a/b, where a aur b coprime integers hain aur b ≠ 0. Dono side square karenge: 3 = a²/b², so a² = 3b². Isse pata chalta hai ki a 3 se divisible hai. Let a = 3c. Tab, (3c)² = 3b² → 9c² = 3b² → b² = 3c². Isse b bhi 3 se divisible hai, jo contradiction hai kyunki a aur coprime the. So, √3 irrational hai.

Example 3: Decimal Expansion

Problem: Show that 0.666... is a rational number.

Solution: Let x = 0.666... To 10x = 6.666... Subtract karenge: 10x - x = 6.666... - 0.666... → 9x = 6 → x = 6/9 = 2/3. So, 0.666... rational number hai kyunki ye 2/3 ke equal hai.

Important Points to Remember

• Every integer ek rational number hai.
• Square root of a prime number hamesha irrational hota hai.
• Real numbers number line par represent kiye ja sakte hain.
• Rational numbers ke decimal expansion terminating ya repeating hote hain.

Practice Problems for Self-Study

1. Find three rational numbers between -1 and 0.

2. Prove that 5 - √2 is irrational.

3. Represent √5 on the number line.

In problems ko solve karke apni understanding improve karein. Agar stuck ho, toh solutions ke liye teachers se consult karein.