Rationals and irrationals Calculator

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Example

Created on 2024-06-20Asked by Michael Nelson (Solvelet student)

Determine whether the number

\sqrt{2}

is rational or irrational.

Solution

To determine whether

\sqrt{2}

is rational or irrational: 1. **Assume that

\sqrt{2}

is rational.**

\sqrt{2} = \frac{p}{q},

where

p

and

q

are coprime integers (i.e., their greatest common divisor is 1). 2. **Square both sides of the equation:**

2 = \frac{p^2}{q^2} \quad \Rightarrow \quad 2q^2 = p^2.

3. **Since

p^2

is even (because it equals

2q^2

p

must also be even.** Let

p = 2k

for some integer

k

. 4. **Substitute

p = 2k

into the equation:**

2q^2 = (2k)^2 \quad \Rightarrow \quad 2q^2 = 4k^2 \quad \Rightarrow \quad q^2 = 2k^2.

5. **Since

q^2

is even,

q

must also be even.** This leads to a contradiction because both

p

and

q

being even implies they have a common factor of 2, contradicting the assumption that they are coprime. Hence,

\sqrt{2}

is irrational. Solved on Solvelet with Basic AI Model

Some of the related questions asked by Penelope Lewis on Solvelet

1. Classify the number

\sqrt{2}

as rational or irrational.2. Determine whether the number

-\frac{7}{4}

is rational or irrational.

DefinitionAn rational number nonrepeating decimal expansion that does not have an end that cannot be written as a fraction of two whole numbers. For example: 21 is a rational number while π is an irrational number.

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