Hello Students! ๐
Aaj hum ek bahut hi interesting aur important concept padhenge โ Irrational Numbers.
Is chapter ko samajhna zaroori hai kyunki rational numbers ke baad jab hum number line ko expand karte hain, toh irrational numbers hume ek complete number system samajhne me help karte hain.
๐ฅ Video Explanation
English Explanation
Hindi Explanation
๐ Infographic Summary
๐ Irrational Numbers Kya Hote Hain?
Imagine karo ek traveler jo chalta ja raha hai. Na toh uska raasta kabhi khatam hota hai (non-terminating) aur na hi woh kabhi same jagah wapas aata hai (non-repeating). Irrational numbers bilkul aise hi hote hain!
Irrational numbers woh numbers hote hain:
- โ Jise
p/qform (fraction form) me nahi likha ja sakta. - โ Jinki decimal expansion non-terminating (khatam nahi hoti) hoti hai.
- โ Aur non-repeating (koee pattern repeat nahi hota).
โญ Simple Definition (Easy Version):
๐ Aise numbers jo baar-baar repeat nahi hote,
๐ Jinka decimal kabhi khatam nahi hota,
๐ Aur jine fraction me convert nahi kar sakte,
Unhe Irrational Numbers kehte hain.
๐ Examples of Irrational Numbers
โ Famous irrational numbers:
โ2= 1.414213562โฆ (endless, no pattern)โ3= 1.732050807โฆโ5= 2.2360679โฆฯ(Pi) = 3.141592653โฆe= 2.718281828โฆ
โ Easy examples:
โ7,โ11,โ13,โ15- 0.101001000100001โฆ (non-repeating + infinite)
๐ซ Kaunse Square Roots Irrational Hote Hain?
Socho kuch numbers "perfect" hote hain, jaise 4 ya 9. Inka square root clean hota hai (2 aur 3). Lekin kuch numbers "ziddi" hote hain, jaise 2 ya 3. Inka square root kabhi seedha nahi nikalta, bas chalta rehta hai. Ye "ziddi" numbers hi Irrational hote hain.
Yeh rule yaad rakho:
- โ Agar kisi number ka perfect square hota hai โ uska square root rational hota hai.
- โ Agar number perfect square nahi hai โ uska square root irrational hota hai.
Examples:
| Number | Perfect Square? | โNumber | Type |
|---|---|---|---|
| 4 | Yes | โ4 = 2 | Rational |
| 9 | Yes | โ9 = 3 | Rational |
| 16 | Yes | 4 | Rational |
| 2 | No | โ2 = 1.414โฆ | Irrational |
| 3 | No | โ3 = 1.732โฆ | Irrational |
| 7 | No | โ7 = 2.645โฆ | Irrational |
๐ง Decimal Form ko Kaise Identify Kare?
If decimalโฆ
- โ terminates (khatam ho jaata hai) โ Rational
- โ repeats (pattern dikhai deta hai) โ Rational
- โ non-terminating + non-repeating โ Irrational
Examples:
| Decimal | Type | Reason |
|---|---|---|
| 2.5 | Rational | Terminates |
| 7.123123123โฆ | Rational | Repeating pattern (123) |
| 1.414213562โฆ | Irrational | No pattern, infinite |
| 3.14159265โฆ | Irrational | ฯ is non-repeating, non-ending |
๐ฏ Important Properties of Irrational Numbers
- โ 1. Two irrational numbers ke beech bhi infinite irrational numbers hote hain. (Example: โ2 aur โ3 ke beech me countless irrational numbers.)
- โ 2. Irrational + rational = mostly irrational. (E.g.
โ2 + 5= irrational) - โ 3. Irrational ร rational = irrational (mostly). (E.g.
3 ร โ3= irrational) - โ 4. (Irrational + irrational) kabhi rational ho sakta hai. (Example:
(โ2 + (โโ2)) = 0โ rational) - โ 5. Saare irrational numbers โ Real numbers ka part hain. (We will explain in next chapter)
๐งฉ Number Line Par Irrational Numbers
Irrational numbers number line par chupe hue khazane (hidden treasure) ki tarah hote hain. Woh wahan hote zaroor hain, par unhe dhoondhne ke liye humein geometry (jaise Pythagoras theorem) ki madad leni padti hai.
Irrational numbers number line me points ki tarah hote hain, par exact location banana thoda tough hota hai.
Example: โ2 ko Number Line par plot karna
Hum ek right triangle banate hain:
-
Step 1: Take a line from 0 to 1 (length = 1 unit).
0 ---- 1 -
Step 2: Construct a perpendicular of length 1 at point 1.
| | 1 unit | 0--+----1 1 unit -
Step 3: Join the point 0 to the top of the perpendicular line. The length of this new line (hypotenuse) is
โ2.
Yeh Pythagoras theorem se aata hai: Hypotenuseยฒ = Baseยฒ + Perpendicularยฒ
Hypotenuse = โ(1ยฒ + 1ยฒ) = โ2
Is hypotenuse ko compass se measure karke number line par 0 se mark karne se hume โ2 ka exact position mil jaata hai.
๐ Why โ2 is Irrational? (Easy Explanation)
โ2 = 1.414213562โฆ
Ye decimal:
- Khatam nahi hota (Non-terminating)
- Repeat nahi hota (Non-repeating)
- Fraction me convert nahi hota
Isliye โ2 irrational hai.
๐งฎ Common Irrational Numbers List
| Irrational Number | Description |
|---|---|
| โ2 | First known irrational number |
| โ3 | Triangle geometry me use hota hai |
| โ5 | Golden ratio se related |
| ฯ (Pi) | Circles ke circumference/area me use |
| e | Natural logarithm base |
โ๏ธ Practice Examples
โ Identify which numbers are irrational:
- 3.14159265โฆ
- โ16
- โ18
- 7.77777โฆ
- 0.1010010001โฆ
- โ1
Answers (for teacher use):
Irrational, Rational (โ16 = 4), Irrational, Rational, Irrational, Rational (โ1 = 1)
๐ Simple Real-Life Connections
Irrational numbers directly real life me use hote hain:
- โ ฯ โ Circle ka area, circumference
- โ โ โ Distance formula (Maps, GPS), Pythagoras theorem
- โ Golden ratio (ฮฆ) โ Nature, art, architecture
- โ e โ Growth/decay, banks, population models