(HELP MEH) Why is √5 irrational? A.) because it is less than one
B.) because it has no opposite
C.) because all square roots are irrational
D.) because it cannot be expressed as a ratio of integers

  D.)  because it cannot be expressed as a ratio of integers

Step-by-step explanation:

The root of any integer that is not a perfect square is irrational. 5 is not a perfect square, so is irrational—it cannot be expressed as the ratio of integers.

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Proof

Suppose √5 = p/q, where p and q are mutually prime. Then p² = 5q².

If p is even, then q² must be even. We know that √2 is irrational, so the only way for q² to be even is for q to be even—contradicting our requirement on p and q.

If p is odd, then both p² and q² will be odd. We can say p = 2n+1, and q = 2m+1, so we have ...

  p² = 5q²

  (2n+1)² = 5(2m+1)²

  4n² +4n +1 = 20m² +20m +5

  4n² +4n = 4(4m² +4m +1)

  n(n+1) = (2m+1)²

The expression on the left will be even for any integer n; the expression on the right will be odd for any integer m. This equation cannot be satisfied for any integers m and n, so contradicting our assumption √5 = p/q.

We have shown using "proof by contradiction" that √5 cannot be the ratio of integers.

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answer

false

explanation

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