Question
Show that $4\sqrt{2}$ is an irrational number.
✓
Answer
Let us assume that $4\sqrt{2}$ is a rational number .
So, we can find co-prime integers $‘a’$ and $‘b’ (b \neq 0)$ such that
$4 \sqrt{ } 2=\frac{a}{b}$
$\therefore \sqrt{ } 2=\frac{a}{4 b}$
Since, $a$ and $b$ are integers, $\frac{a}{4 b}$ is a rational number and so $\sqrt{ 2}$ is a rational number.
Alternate Proof:
Let us assume that $4\sqrt{2}$ is a rational number.
So, we can find co-prime integers $‘a’$ and $‘b’ (b \neq 0)$ such that
$ 4 \sqrt{2}=\frac{a}{b}$
$\therefore b(4 \sqrt{2})=a$
$\therefore 32 b^2=a^2$
$\therefore b^2=\frac{a^2}{32}$
Since, 32 divides $a^2$, so 32 divides 'a' as well.
So, we write $a=32 c$, where c is an integer.
$\therefore a ^2=(32 c )^2 \ldots$ [Squaring both the sides]
$\therefore 32 b^2=32 \times 32 c ^2 \ldots[$ [From(i) $]$
$\therefore b^2=32 c ^2$
$\therefore c^2=\frac{b^2}{32}$
Since, 32 divides $b ^2$, so 32 divides ' b '.
$\therefore 32$ divides both a and b .
a and b have at least 32 as a common factor.
But this contradicts the fact that a and b have no common factor other than 1.
$\therefore$ Our assumption that $4 \sqrt{ } 2$ is a rational number is wrong.
$\therefore 4 \sqrt{ } 2$ is an irrational number.
So, we can find co-prime integers $‘a’$ and $‘b’ (b \neq 0)$ such that
$4 \sqrt{ } 2=\frac{a}{b}$
$\therefore \sqrt{ } 2=\frac{a}{4 b}$
Since, $a$ and $b$ are integers, $\frac{a}{4 b}$ is a rational number and so $\sqrt{ 2}$ is a rational number.
Alternate Proof:
Let us assume that $4\sqrt{2}$ is a rational number.
So, we can find co-prime integers $‘a’$ and $‘b’ (b \neq 0)$ such that
$ 4 \sqrt{2}=\frac{a}{b}$
$\therefore b(4 \sqrt{2})=a$
$\therefore 32 b^2=a^2$
$\therefore b^2=\frac{a^2}{32}$
Since, 32 divides $a^2$, so 32 divides 'a' as well.
So, we write $a=32 c$, where c is an integer.
$\therefore a ^2=(32 c )^2 \ldots$ [Squaring both the sides]
$\therefore 32 b^2=32 \times 32 c ^2 \ldots[$ [From(i) $]$
$\therefore b^2=32 c ^2$
$\therefore c^2=\frac{b^2}{32}$
Since, 32 divides $b ^2$, so 32 divides ' b '.
$\therefore 32$ divides both a and b .
a and b have at least 32 as a common factor.
But this contradicts the fact that a and b have no common factor other than 1.
$\therefore$ Our assumption that $4 \sqrt{ } 2$ is a rational number is wrong.
$\therefore 4 \sqrt{ } 2$ is an irrational number.
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