Root 2 (√2) kya hai?

Square ko doguna karna

Lagbhag 800 BCE mein Baudhayana ne poochha ki diye gaye square se doguna bada square kaise banaayein. Bhuja doguni karna kaam nahi karta, kyunki usse area chaar guna ho jaata hai. Unka jawab (Shulbasutra, verse 1.9) hai naya square purane ke diagonal par banaana. Poorab-pashchim aur uttar-dakshin lines batati hain kyon: purana square 2 ek jaise right triangles ka hai aur naya 4 ka. Ye kadam dohraane par 2, 4 aur 8 triangles wale squares milte hain, har ek pichhle se doguna.

Aadha karna ulta chalta hai. Square kaagaz ke chaaron kone uske beech mein modo: silvatein bhujaon ke beech ke binduon ko jodti hain aur ek tircha square banaati hain jiska area theek aadha hota hai, kyunki mude hue chaaron tukde use poora dhak lete hain. Aadhi bhuja wale square ka area sirf chauthaai hota.

1-1 right triangle ka hypotenuse

Unit square PEAR ko uske diagonal ER se kaato to aisa right triangle milta hai jiski do chhoti bhujaayein 1 hain. Uske hypotenuse par bana square REST wahi doguna square hai, isliye uska area 2 × 1 = 2. Hypotenuse ki lambaai c ho to c² = 2, aur c = √2, wo sankhya jiska varg 2 hai. Kisi bhi lambaai a ki barabar bhujaon ke liye yahi tark c² = 2a² deta hai, yaani c = a√2. Barabar bhujaayein 12 hon to c = √288, 16 aur 17 ke beech; hypotenuse √72 ho to bhujaayein 6.

√2 kitna bada hai?

Varg √2 ko do sankhyaon ke beech pakad sakte hain. 1² = 1 aur 2² = 4, isliye √2, 1 aur 2 ke beech hai; inhe lower aur upper bound kehte hain. Phir 1.4² = 1.96 aur 1.5² = 2.25, to 1.4 < √2 < 1.5. Phir 1.41² = 1.9881 aur 1.42² = 2.0164, to 1.41 < √2 < 1.42. Phir 1.414² = 1.999396 aur 1.415² = 2.002225, to 1.414 < √2 < 1.415. Har kadam ek aur sahi ank jodta hai, aur ye kaam jitna chaahein utna chal sakta hai.

Kabhi khatm na hone wala decimal

Kya ank kabhi rukenge? Maan lo √2 terminating decimal hai. Uska aakhri ank non-zero hoga, aur aisi sankhya ka varg bhi decimal ke baad non-zero ank par khatm hota hai, isliye wo kabhi theek 2 ya 2.000 nahi ho sakta. Isliye √2 ka decimal hamesha chalta rehta hai: √2 = 1.41421356… Calculator bhi sirf iske shuruaati ank dikhata hai; poora decimal kabhi likha nahi ja sakta, chaahe kitne bhi ank likh lein.

Fraction bhi nahi

Kya √2 ginti ki sankhyaon ka fraction m/n ho sakta hai? Tab 2 = m²/n², yaani 2n² = m². Kisi varg ke prime factors mein har prime even baar aata hai. To m² mein prime 2 even baar hoga, jabki 2n² mein odd baar, jo asambhav hai. Ye saboot Euclid ne apni kitaab Elements mein lagbhag 300 BCE mein diya. √2 jaisi sankhyaon ke baare mein aage ki classes mein aur padhaaya jaayega. Aage theorem kya hai? aur Pythagorean triples kya hain? padho, ya Chapter 2 ke notes aur NCERT solutions dekho.