please solve the problem of Number System
Answers
Step-by-step explanation:
Given :-
The two numbers √2 and √3
To find:-
Find the three irrational numbers between them ?
How many irrational numbers between them ?
Solution:-
Given numbers are √2 and √3
We know that
a and b are two numbers then the irrational number between a and b is √(ab)
We have a = √2 and b=√3
First irrational number = √{√2×√3)
=> √√6 = (6)^1/4
Second irrational number between √2 and√√6
= √{√2×√√6}
= √(√{√2×√6})
=√√√(4×6)
=√√√24
=(24)^1/8
Third irrational number between√2 and 24^1/8
=> √{√2×(24)^1/8}
=> √{2^1/2×24^1/8}
=> √{2^4/8×24^1/8}
=> √(16×24)^1/8
=>√(432)^1/8
=>√√√√432
=> (432)^1/16
Method -2:-
√2 = 1.414....
√3 = 1.732...
The irrational numbers between them
1.521283..., 1.6102636383..., 1.7126363630...
If we continue like this we get infinite number of many irrational numbers between them
Answer:-
The three irrational numbers are 1.521283..., 1.6102636383..., 1.7126363630...
(or)
√√6 = (6)^1/4 ,√√√24=(24)^1/8 ,
√√√√432= (432)^1/16
There are infinitely number of many irrational numbers between them
Used formulae:-
- a and b are two numbers then the irrational number between a and b is √(ab)
- There are infinitely number of many irrational numbers between two numbers is called Densitive Property.
Answer:
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