A student claims √2 is rational and writes √2 = a/b in lowest terms. Squaring gives 2 = a^2/b^2, so a^2 = 2b^2, meaning a^2 is even, hence a is even. If a = 2k, what does the equation become, and what contradiction follows?

Question

Accepted Answer

b^2 = 2k^2, so b is also even - contradicting lowest terms. With a = 2k: a^2 = 4k^2 = 2b^2, so b^2 = 2k^2, making b^2 even and b even. But then a and b share factor 2, contradicting 'lowest terms'. Hence √2 is irrational.

A student claims $2$ is rational and writes $2 = \frac{a}{b}$ in lowest terms.
Squaring gives $2 = \frac{a ^{2}}{b ^{2}}$ , so $a^{2} = 2 b^{2}$ , meaning $a^{2}$ is even, hence a is even.
If $a = 2 k$ , what does the equation become, and what contradiction follows?

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More Class 9 Rational and Irrational Numbers practice

A student claims 2​ is rational and writes 2​=ba​ in lowest terms.Squaring gives 2=b2a2​, so a2=2b2, meaning a2 is even, hence a is even.If a=2k, what does the equation become, and what contradiction follows?

Step-by-Step Solution

Master this chapter in 30 minutes a day.

More Class 9 Rational and Irrational Numbers practice

A student claims $2$ is rational and writes $2 = \frac{a}{b}$ in lowest terms.
Squaring gives $2 = \frac{a ^{2}}{b ^{2}}$ , so $a^{2} = 2 b^{2}$ , meaning $a^{2}$ is even, hence a is even.
If $a = 2 k$ , what does the equation become, and what contradiction follows?