Class 9 Ganit: Vastavik Sankhyayein (Real Numbers) - Notes

Is chapter mein hum real numbers ke baare mein padhenge. Real numbers do prakar ke hote hain: Rational numbers aur Irrational numbers. Yeh notes NCERT Class 9 maths ke syllabus par based hain aur aapko number system ko achhe se samajhne mein help karenge.

Key Topics Covered

• Rational numbers kya hote hain?
• Irrational numbers ki pehchaan
• Decimal expansions (terminating, non-terminating recurring, non-terminating non-recurring)
• Number line par representation
• Real numbers ke operations

Har concept ko simple examples ke saath explain kiya gaya hai. Aage padhein detailed notes ke liye.

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

संख्या पद्धति

बहुपद

निर्देशांक ज्यामिति

दो चरों में रैखिक समीकरण

युक्लिड के ज्यामिति का परिचय

रेखाएँ और कोण

त्रिभुज

चतुर्भुज

समान्तर चतुर्भुज और त्रिभुजों के क्षेत्रफल

वृत्त

हीरोन सूत्र

पृष्ठीय क्षेत्रफल एवं आयतन

1. Rational Numbers (परिमेय संख्याएँ)

Rational numbers wo numbers hote hain jinhe p/q form mein likha ja sakta hai, jahan p aur q integers hain aur q ≠ 0. Inki decimal expansion ya to terminating hoti hai ya non-terminating recurring. Jaise:

• ½ = 0.5 (terminating)
• 1/3 = 0.333... (non-terminating recurring)
• 7/8 = 0.875 (terminating)

Har terminating decimal ko rational number mein convert kiya ja sakta hai. Jaise 0.125 = 125/1000 = 1/8

2. Irrational Numbers (अपरिमेय संख्याएँ)

Irrational numbers wo hote hain jo p/q form mein nahi likhe ja sakte. Inki decimal expansion non-terminating aur non-recurring hoti hai. Jaise:

• √2 = 1.4142135623... (kabhi pattern repeat nahi hota)
• √3 = 1.73205080...
• π (pi) = 3.1415926535...

Ek important property: Agar kisi number ka decimal expansion non-terminating aur non-recurring hai, to wo irrational number hai.

3. Real Numbers (वास्तविक संख्याएँ)

Rational aur irrational numbers ka union real numbers kehlata hai. Yani R set mein sab rational aur irrational numbers aate hain. Real numbers ko number line par dikhaya ja sakta hai. Har real number ka number line par ek unique point hota hai aur har point ek real number ko represent karta hai.

Number Line par Representation

Real numbers ko number line par exactly locate kiya ja sakta hai. Jaise √2 ko number line par kaise dikhaye? Pythagoras theorem ka use karte hain:

• 1 unit length ka ek right angle triangle banao (base = 1, altitude = 1).
• Hypotenuse = √2.
• Is hypotenuse ko compass se number line par zero se mark karo.

Aise hi √3, √5 etc. represent kar sakte hain.

4. Decimal Expansions ke Types

• Terminating: Division process end ho jata hai. Denominator ke prime factors sirf 2 aur 5 hote hain.
• Non-terminating recurring: Division mein ek block of digits repeat hota hai. Jaise 1/7 = 0.142857142857...
• Non-terminating non-recurring: Kabhi pattern repeat nahi hota. Ye irrational numbers hote hain.

Kisi bhi rational number ka decimal expansion ya to terminating hota hai ya non-terminating recurring. Irrational ka hamesha non-terminating non-recurring hota hai.

5. Operations on Real Numbers

Real numbers par addition, subtraction, multiplication, division – sab operations applied kar sakte hain. Lekin dhyan rakhein:

• Do rational numbers ka sum/ product hamesha rational hota hai.
• Do irrational numbers ka sum/ product rational ya irrational ho sakta hai. Jaise √2 + (-√2) = 0 (rational), but √2 + √3 irrational.
• Ek rational aur ek irrational ka sum hamesha irrational hota hai.

Important Properties

• Closure: Real numbers closed hote hain addition, subtraction, multiplication ke liye (except division by zero).
• Commutative: a + b = b + a, a × b = b × a
• Associative: (a+b)+c = a+(b+c)
• Distributive: a(b+c) = ab + ac

6. Important Points to Remember

• Har integer ek rational number hai, par har rational integer nahi hai.
• Natural numbers, whole numbers, integers – sab real numbers ke subsets hain.
• Agar kisi number ka decimal expansion barabar repeat ho raha hai (recurring), to wo rational hai.
• √2 irrational hai – ye proof class 9 mein important hai.

In notes ko dhyan se padhiye aur NCERT book ke exercises ko solve karein. Real numbers ki yeh foundation aage ke chapters ke liye bahut useful hogi.