prove that √5 is an irrational number
hey guys help me
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Question

prove that √5 is an irrational number
hey guys help me
Don’t spam​

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Alice 10 months 2021-07-07T16:16:40+00:00 2 Answers 0 views 0

Answers ( )

    0
    2021-07-07T16:18:27+00:00

    Answer:

    Let us assume the opposite, i.e., √5 is a rational number. Hence, √5 can be written as in the form ab where a and b(b≠0) are co-prime (no common factor other than 1 ). By theorem: if p is a prime number and p divides a2, then p divides a, where a in a positive number.

    0
    2021-07-07T16:18:32+00:00

    Answer:

    I HOPE THIS MAY HELP YOU ITS EASY❤❤❤

    Step-by-step explanation:

    Given: √5

    We need to prove that √5 is irrational

    Proof:

    Let us assume that √5 is a rational number.

    So it can be expressed in the form p/q where p,q are co-prime integers and q≠0

    ⇒ √5 = p/q

    On squaring both the sides we get,

    ⇒5 = p²/q²

    ⇒5q² = p² —————–(i)

    p²/5 = q²

    So 5 divides p

    p is a multiple of 5

    ⇒ p = 5m

    ⇒ p² = 25m² ————-(ii)

    From equations (i) and (ii), we get,

    5q² = 25m²

    ⇒ q² = 5m²

    ⇒ q² is a multiple of 5

    ⇒ q is a multiple of 5

    Hence, p,q have a common factor 5. This contradicts our assumption that they are co-primes. Therefore, p/q is not a rational number

    √5 is an irrational number.

    Hence proved

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