Rational numbers — definition, decimal expansion (terminating - repeating)

2.5.3 · Maths › Number Theory (Intermediate)

Definition

Integers ko kyun include karein? Har integer $n$ ko $\frac{n}{1}$ ke roop mein likha ja sakta hai, isliye integers rationals ka ek subset hain: $Z \subset Q$ .

$q = 0$ ko kyun exclude karein? Mathematics mein zero se division undefined hai. Agar hum ise allow karte, to hum field axioms ko tod dete (har non-zero element ka ek multiplicative inverse hona chahiye).

Decimal Expansion: The Two Types

Jab hum $\frac{p}{q}$ ko long division se decimal form mein convert karte hain, to theek do cheezein ho sakti hain:

1. Terminating Decimals

Ye condition kyun? Hamara decimal system base 10 mein hai, aur $10 = 2 \times 5$ . Jab hum long division karte hain, hum remainder ko baar baar 10 se multiply karte hain. Agar denominator $q$ kisi power of 10 ko exactly divide kare, to division terminate ho jaati hai.

Derivation:

$\frac{p}{q}$ ko lowest terms mein lo
Maano decimal $n$ digits ke baad terminate ho jaata hai
Tab $\frac{p}{q} = \frac{k}{1 0 ^{n}}$ kisi integer $k$ ke liye
Cross-multiply karo: $p \cdot 1 0^{n} = k \cdot q$
Kyunki gcd $(p, q) = 1$ , humein chahiye $q ∣ 1 0^{n}$
Kyunki $1 0^{n} = 2^{n} \cdot 5^{n}$ , $1 0^{n}$ ke sirf prime factors 2 aur 5 hain
Isliye $q$ mein sirf prime factors 2 aur 5 ho sakte hain

2. Non-Terminating Repeating Decimals

Ye repeat kyun karte hain? Long division karte waqt, possible remainders limited hain: $0, 1, 2, .., q - 1$ . Kyunki hamare paas sirf $q$ possible remainders hain, zyada se zyada $q$ steps ke baad, humein zaroor koi aisa remainder milega jo pehle aa chuka hai. Jaise hi ek remainder repeat hota hai, poori division pattern repeat ho jaati hai.

Worked example Example 2: Repeating Decimal

$\frac{5}{6}$ ko decimal mein convert karo. Step 1: Denominator check karo $6 = 2 \cdot 3$ 3 hai (jo 2 ya 5 nahi hai) → Repeat karega ✓

Step 2: Long division

5 ÷ 6:
 5.0000...
 remainder 5 → 50 ÷ 6 = 8 remainder 2
 remainder 2 → 20 ÷ 6 = 3 remainder 2  ← remainder 2 phir se!
 remainder 2 → 20 ÷ 6 = 3 remainder 2  ← pattern repeat ho raha hai

$\frac{5}{6} = 0.8333... = 0.8 \overline{3}$ 3 kyun repeat karta hai? Jaise hi remainder 2 dobara aata hai (step 2 par), hum waapas 20 ko 6 se divide karne par aa jaate hain, jo hamesha ke liye same result deta hai.

Worked example Example 3: Longer Repetend

$\frac{1}{7}$ ko decimal mein convert karo. Check: $7 = 7^{1}$ (2, 5 ke alawa prime) → Repeat karega

Long division:

1 ÷ 7:
remainder 1 → 10 ÷ 7 = 1 remainder 3
remainder 3 → 30 ÷ 7 = 4 remainder 2
remainder 2 → 20 ÷ 7 = 2 remainder 6
remainder 6 → 60 ÷ 7 = 8 remainder 4
remainder 4 → 40 ÷ 7 = 5 remainder 5
remainder 5 → 50 ÷ 7 = 7 remainder 1  ← remainder 1 phir se!

$\frac{1}{7} = 0. \overline{142857}$

Repetend ki length 6 hai. Note karo: $ϕ (7) = 6$ , jo theory confirm karta hai!

Converting Repeating Decimals Back to Fractions

Why This Matters

Number Types Distinguish Karna: Decimal behavior turant bata deta hai ki koi number rational hai ya irrational
Computer Representation: Computers rationals ko exactly represent karte hain lekin irrationals ko approximate karte hain — terminating vs. repeating samajhna precision issues predict karne mein help karta hai
Real Numbers ki Foundation: Rationals, reals mein dense hain (koi bhi do real numbers ke beech ek rational hota hai), lekin inme "gaps" hain (irrationals)
Practical Computation: Repeating decimals ko fractions mein convert karna engineering calculations mein zaruri hai jahan exact values matter karti hain

Recall Ek 12-saal ke bacche ko samjhao

Socho tumhare paas ek pizza hai. Ek rational number aise hai jaise kehna "maine 8 mein se 3 slices khaaye" — tum ise fraction mein likh sakte ho: $\frac{3}{8}$ .

Ab, jab tum calculator ya long division se 3 ko 8 se divide karo, to milta hai $0.375$ . Decimal ruk jaata hai! Ise terminating decimal kehte hain. Ye tab hota hai jab bottom number (8) sirf 2's aur 5's ko multiply karke bana ho. (8 = 2 × 2 × 2).

Lekin agar tum 1 ko 3 se divide karo? Milta hai $0.333333...$ — 3's kabhi nahi ruke! Lekin ye hamesha repeat karte hain. Ise repeating decimal kehte hain. Ye tab hota hai jab bottom number mein aur numbers hon (jaise 3).

Yahan cool part ye hai: agar tum mujhe koi bhi aisa decimal do jo ya to ruke ya repeat kare, main use fraction mein convert kar sakta hoon! Lekin agar decimal bina repeat kiye hamesha chalta rahe (jaise $π$ ), to ye ek rational number NAHI hai — ise irrational kehte hain, aur ise ek simple fraction ke roop mein nahi likha ja sakta.

Isliye rational numbers khaas hain: inke decimals hamesha rules follow karte hain (rukna ya repeat karna). Irrational numbers rebels hain!

Connections

Prime factorization — terminating condition check karne ke liye zaruri
Euler's totient function — maximum repetend length determine karta hai
Real numbers — rationals, reals ka ek proper subset hain
Irrational numbers — woh numbers jinke decimals terminate ya repeat nahi karte
Dense sets — rationals, reals mein dense hain
Long division algorithm — wo computational process jo decimal patterns reveal karta hai
Modular arithmetic — explain karta hai ki repeating decimals mein remainders cycle kyun karte hain

Flashcards

Rational number kya hota hai? :: Ek aisa number jo $\frac{p}{q}$ ke roop mein express kiya ja sake jahan $p, q$ integers hain aur $q \neq = 0$ .

Ek rational number ka terminating decimal kab hoga?

Jab denominator (lowest terms mein) ka form

q = 2^{a} \cdot 5^{b}

ho — sirf prime factors 2 aur 5.

Ek rational number ka repeating decimal kab hoga?

Jab denominator (lowest terms mein) mein 2 ya 5 ke alawa kam se kam ek aur prime factor ho.

Repeating decimals repeat kyun karte hain?

q

se long division mein sirf

q

possible remainders hote hain. Aakhirkar ek remainder zaroor repeat hoga, jisse division pattern repeat ho jaata hai.

0. \overline{ab c}

ko fraction mein kaise convert karein?

Maano

x = 0. \overline{ab c}

1 0^{n}

se multiply karo (jahan

n

repetend length hai), original subtract karo,

x

ke liye solve karo. Result:

x = \frac{ab c}{1 0 ^{n} - 1}

\frac{1}{8}

decimal mein kya hai?

0.125

(terminate karta hai kyunki

8 = 2^{3}

)

\frac{1}{6}

decimal mein kya hai?

0.1 \overline{6}

(repeat karta hai kyunki

6 = 2 \cdot 3

mein prime 3 hai)

Sach ya Jhoot: Saare decimals rational numbers hain.

Jhoot. Sirf terminating ya repeating decimals rational hain. Non-repeating infinite decimals irrational hain.

0. \overline{9}

kya hai?

Exactly 1. Proof: Maano

x = 0. \overline{9}

, tab

10 x = 9. \overline{9}

, isliye

10 x - x = 9

, jisse

x = 1

milta hai.