REAL NUMBERS

Euclid’s Division Lemma (EDL) ek fundamental statement hai jo divisibility of integers ke baare mein batati hai. Iska use Euclid’s Division Algorithm (EDA) mein hota hai HCF nikalne ke liye.

Euclid’s Division Lemma

• Statement: Given positive integers \(a\) and \(b\), there exist unique integers \(q\) and \(r\) such that \(a = bq + r\), jahan \(0 \le r < b\).
• Yahan, \(a\) = dividend, \(b\) = divisor, \(q\) = quotient, \(r\) = remainder.
• Ye basically long division ka formal statement hai.

Euclid’s Division Algorithm (EDA)

• Purpose: Two positive integers ke HCF (Highest Common Factor) ko find karne ke liye.
• Method: Repeated application of Euclid’s Division Lemma.

Steps to find HCF of two positive integers \(c\) and \(d\) (where \(c > d\)):

1. Step 1: Apply Euclid’s Division Lemma to \(c\) and \(d\). Find whole numbers \(q\) and \(r\) such that \(c = dq + r\), jahan \(0 \le r < d\).
2. Step 2:

• Agar \(r = 0\) hai, toh \(d\) hi HCF hai \(c\) aur \(d\) ka.
• Agar \(r \ne 0\) hai, toh divisor \(d\) aur remainder \(r\) par Euclid’s Division Lemma apply karo.

1. Step 3: Ye process tab tak continue karo jab tak remainder zero na ho jaye. Jis stage par remainder zero hota hai, us stage ka divisor hi required HCF hota hai.

• Key Idea: HCF(c, d) = HCF(d, r).

Fundamental Theorem of Arithmetic (FTA) number theory ka ek bahut important theorem hai. Ye prime factorisation ki uniqueness batata hai.

Statement of FTA

• Every composite number can be expressed (factorised) as a product of primes, and this factorisation is unique, apart from the order in which the prime factors occur.
• Matlab, har composite number ko primes ke product ke roop mein likha ja sakta hai, aur ye tareeka unique hota hai, chahe primes ka order kuch bhi ho.
• Example: \(12 = 2 \times 2 \times 3 = 2^2 \times 3\). Isko \(2 \times 3 \times 2\) ya \(3 \times 2 \times 2\) bhi likh sakte hain, but factors \(2, 2, 3\) hi rahenge.

Applications of FTA

1. HCF aur LCM find karna: Prime factorisation method se HCF aur LCM nikalne mein FTA ka use hota hai.
2. Irrationality prove karna: \(\sqrt{2}, \sqrt{3}\) jaise numbers ki irrationality prove karne mein FTA ka concept use hota hai.
3. Decimal expansions ki nature determine karna: Rational numbers ke decimal expansions terminating hain ya non-terminating repeating, ye pata lagane mein FTA ka use hota hai.

Prime Factorisation Method

• Steps:

1. Given numbers ko unke prime factors mein factorise karo.
2. Har prime factor ko uski highest power ke saath likho.

• HCF (Highest Common Factor):
• Numbers mein common prime factors ki smallest power ka product.
• Example: \(12 = 2^2 \times 3\), \(18 = 2 \times 3^2\)
• Common factors: \(2, 3\)
• Smallest power of \(2\) is \(2^1\).
• Smallest power of \(3\) is \(3^1\).
• HCF(12, 18) = \(2^1 \times 3^1 = 6\).

• LCM (Least Common Multiple):
• Numbers mein involved sabhi prime factors ki greatest power ka product.
• Example: \(12 = 2^2 \times 3\), \(18 = 2 \times 3^2\)
• All prime factors: \(2, 3\)
• Greatest power of \(2\) is \(2^2\).
• Greatest power of \(3\) is \(3^2\).
• LCM(12, 18) = \(2^2 \times 3^2 = 4 \times 9 = 36\).

Relation between HCF and LCM of two numbers

• For any two positive integers \(a\) and \(b\):

HCF\((a, b)\) \(\times\) LCM\((a, b)\) = \(a \times b\)

• Important: Ye relation sirf do numbers ke liye valid hai, teen ya usse zyada numbers ke liye nahi.

Prime Factorisation Method HCF aur LCM nikalne ka ek systematic tareeka hai, especially jab numbers bade hon.

Steps for HCF and LCM

1. Prime Factorise: Har number ko uske prime factors ke product ke roop mein express karo. Example: \(N = p_1^{e_1} \times p_2^{e_2} \times ... \times p_k^{e_k}\).
2. HCF Calculation:

• Sabhi common prime factors ko identify karo.
• Har common prime factor ki smallest power lo.
• In smallest powers ka product HCF hoga.

1. LCM Calculation:

• Sabhi prime factors (common aur uncommon dono) ko identify karo jo numbers mein appear hote hain.
• Har prime factor ki greatest power lo.
• In greatest powers ka product LCM hoga.

Example: HCF and LCM of 96 and 404

1. Prime Factorisation:

• \(96 = 2 \times 2 \times 2 \times 2 \times 2 \times 3 = 2^5 \times 3^1\)
• \(404 = 2 \times 2 \times 101 = 2^2 \times 101^1\)

1. HCF:

• Common prime factor: \(2\)
• Smallest power of \(2\) is \(2^2\).
• HCF(96, 404) = \(2^2 = 4\).

1. LCM:

• All prime factors: \(2, 3, 101\)
• Greatest power of \(2\) is \(2^5\).
• Greatest power of \(3\) is \(3^1\).
• Greatest power of \(101\) is \(101^1\).
• LCM(96, 404) = \(2^5 \times 3^1 \times 101^1 = 32 \times 3 \times 101 = 96 \times 101 = 9696\).

Verification (for two numbers)

• HCF \(\times\) LCM = Product of numbers
• \(4 \times 9696 = 38784\)
• \(96 \times 404 = 38784\)
• Since LHS = RHS, the verification is correct.

Class 9 mein humne irrational numbers ke baare mein padha tha. Ab hum unki irrationality ko prove karna sikhenge, specifically \(\sqrt{2}, \sqrt{3}, \sqrt{5}\) jaise numbers ki.

What are Irrational Numbers?

• A number \(s\) is called irrational if it cannot be written in the form \(p/q\), jahan \(p\) aur \(q\) integers hain aur \(q \ne 0\).
• Examples: \(\sqrt{2}, \sqrt{3}, \pi, 0.101101110...\)

Proof by Contradiction Method

• Ye ek common technique hai maths mein proofs ke liye.
• Steps:

1. Assume karo ki jo statement prove karna hai, uska opposite true hai.
2. Is assumption ko use karke logical steps follow karo.
3. Agar ye steps ek contradiction (aisa result jo impossible ho ya known fact ke against ho) par pahunchte hain, toh hamari initial assumption galat thi.
4. Iska matlab hai ki original statement true hai.

Theorem 1.3 (Key for Irrationality Proofs)

• Statement: Let \(p\) be a prime number. If \(p\) divides \(a^2\), then \(p\) divides \(a\), jahan \(a\) ek positive integer hai.
• Explanation: Agar koi prime number kisi number ke square ko divide karta hai, toh woh prime number us number ko bhi divide karega.
• Example: Agar \(3\) divides \(6^2 = 36\), toh \(3\) divides \(6\) also.
• Example: Agar \(5\) divides \(10^2 = 100\), toh \(5\) divides \(10\) also.

Is section mein hum dekhenge ki kab ek rational number ka decimal expansion terminating hota hai aur kab non-terminating repeating.

Terminating vs. Non-Terminating Repeating Decimals

• Terminating Decimal Expansion: Wo decimal expansion jo kuch digits ke baad end ho jata hai. Example: \(0.5, 0.25, 1.234\).
• Non-Terminating Repeating (Recurring) Decimal Expansion: Wo decimal expansion jo kabhi end nahi hota aur digits ka ek block repeat hota rehta hai. Example: \(0.333..., 0.142857142857...\).

Theorem 1.5 (Terminating Decimal Condition)

• Statement: Let \(x\) be a rational number whose decimal expansion terminates. Then \(x\) can be expressed in the form \(p/q\), jahan \(p\) aur \(q\) coprime integers hain, aur \(q\) ka prime factorisation \(2^n 5^m\) ke form ka hai, jahan \(n\) aur \(m\) non-negative integers hain.
• Simplified: Agar ek rational number ka decimal expansion terminate karta hai, toh uske denominator \(q\) (jab fraction simplest form mein ho) ke prime factors sirf \(2\) ya \(5\) ya dono honge.

Theorem 1.6 (Checking for Terminating Decimals)

• Statement: Let \(x = p/q\) be a rational number, jahan \(p\) aur \(q\) coprime integers hain. Agar \(q\) ka prime factorisation \(2^n 5^m\) ke form ka hai, jahan \(n\) aur \(m\) non-negative integers hain, toh \(x\) ka decimal expansion terminate karega.
• Simplified: Agar denominator \(q\) ke prime factors sirf \(2\) ya \(5\) ya dono hain, toh decimal expansion terminate karega.

Theorem 1.7 (Non-Terminating Repeating Decimal Condition)

• Statement: Let \(x = p/q\) be a rational number, jahan \(p\) aur \(q\) coprime integers hain. Agar \(q\) ka prime factorisation \(2^n 5^m\) ke form ka nahi hai, toh \(x\) ka decimal expansion non-terminating repeating hoga.
• Simplified: Agar denominator \(q\) ke prime factors mein \(2\) aur \(5\) ke alawa koi aur prime factor bhi hai, toh decimal expansion non-terminating repeating hoga.

Summary for Decimal Expansions

• Step 1: Given rational number \(p/q\) ko simplest form mein reduce karo (agar required ho).
• Step 2: Denominator \(q\) ka prime factorisation karo.
• Step 3:
• Agar \(q\) ke prime factors sirf \(2\) ya \(5\) ya dono hain (i.e., \(2^n 5^m\) form mein), toh decimal expansion terminating hai.
• Agar \(q\) ke prime factors mein \(2\) aur \(5\) ke alawa koi aur prime factor bhi hai, toh decimal expansion non-terminating repeating hai.