Find an irrational number between 2root 3 and 3root 5

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Find an irrational number between 2root 3 and 3root 5

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Kylie 3 days 2021-10-11T15:53:26+00:00 1 Answer 0 views 0

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    2021-10-11T15:55:06+00:00

    Answer:

    the answer is

    Step-by-step explanation:

    So, let’s solve!!

    TO PROVE :- 3+2root 5 is irrational.

    PROOF :- Assume that the given real number is rational. This means that the number can be expressed in the form p/q where p and q belong to integers as well as are co-prime.

    So,

    3 + 2root5 = p/q

    Or,

    2root5 = p/q – 3 = (p -3q)/q Or,

    Root5 = (p-3q)/2q ……. (i)

    Now, (p-3q)/2q is a … (more)

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    Carlos Eugenio Thompson Pinzón

    , Triple IMO bronze medalist

    Answered March 5, 2019

    Rational numbers are a field and it is close to addition, subtraction, multiplication and division (except division by zero).

    So let’s suppose that 3+25–√ is rational.

    If 3+25–√ is rational, then (3+25–√)−3=25–√ is rational. And if 25–√ is rational, then 25–√÷2=5–√ would be rational.

    Now, if 5–√ is rational, then there would exist to integers p,q such as 5–√=pq. If q is negative, then 5–√=−p−q for a positive integer −q. And, among all possible values of pq=5–√ with positive q there is one pair with the less possible positive q.

    So let’s take … (more)

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    Girija Warrier

    , former Mathematics Teacher

    Answered February 26, 2019

    TO PROVE:( 3 + 2√5) is irrational

    For proving this , we can apply the theorem “ √5 is an irrational number”

    PROOF: Let (3 + √5 ) be a rational number.

    => 3+√5= p/q ( where, p & q are integers, ‘q’ not = 0 )

    => √5 = (p/q) -3

    => √5 = (p-3q)/q

    Here, in RHS , numerator is the difference of 2 integers, which always remains an integer. & denominator is also an integer, not = 0.

    => RHS is a rational number ( as, all the conditions for being a rational number, have been satisfied )

    But LHS is an irrational number. ( by theorem, √5 is an irrational number)

    => LHS not = RHS

    => our assumption (that 3+√5 is a ratio … (more)

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