Recall, $\pi$ is defined as the ratio of the circumference (say $c$ ) of a circle to its diameter
(say $d$ ). That is, $\pi=\frac{c}{d}$. This seems to contradict the fact that $\pi$ is irrational. How will you resolve this contradiction ?
Recall, $\pi$ is defined as the ratio of the circumference (say $c$ ) of a circle to its diameter
(say $d$ ). That is, $\pi=\frac{c}{d}$. This seems to contradict the fact that $\pi$ is irrational. How will you resolve this contradiction ?
When we measure the length of a line with a scale or with any other device, we only get an approximate rational value, i.e. $c$ and $d$ both are irrational.
$\therefore \frac{ c }{ d }$ is irrational and hence $\pi$ is irrational. Thus, there is no contradiction in saying that $\pi$ is irrational.
Similar Questions
Rationalise the denominators of the following :
$(i)$ $\frac{1}{\sqrt{7}}$
$(ii)$ $\frac{1}{\sqrt{7}-\sqrt{6}}$
$(iii)$ $\frac{1}{\sqrt{5}+\sqrt{2}}$
$(iv)$ $\frac{1}{\sqrt{7}-2}$
Rationalise the denominators of the following :
$(i)$ $\frac{1}{\sqrt{7}}$
$(ii)$ $\frac{1}{\sqrt{7}-\sqrt{6}}$
$(iii)$ $\frac{1}{\sqrt{5}+\sqrt{2}}$
$(iv)$ $\frac{1}{\sqrt{7}-2}$
You know that $\frac{1}{7}=0 . \overline{142857}$. Can you predict what the decimal expansions of $\frac{2 }{7},\, \frac{3}{7}$, $\frac{4}{7},\, \frac{5}{7}, \,\frac{6}{7}$ are, without actually doing the long division ? If so, how ?
You know that $\frac{1}{7}=0 . \overline{142857}$. Can you predict what the decimal expansions of $\frac{2 }{7},\, \frac{3}{7}$, $\frac{4}{7},\, \frac{5}{7}, \,\frac{6}{7}$ are, without actually doing the long division ? If so, how ?
Show that $3.142678$ is a rational number. In other words, express $3.142678$ in the form $\frac {p }{q }$, where $p$ and $q$ are integers and $q \ne 0$.
Show that $3.142678$ is a rational number. In other words, express $3.142678$ in the form $\frac {p }{q }$, where $p$ and $q$ are integers and $q \ne 0$.
Find three different irrational numbers between the rational numbers $\frac{5}{7}$ and $\frac{9}{11}$.
Find three different irrational numbers between the rational numbers $\frac{5}{7}$ and $\frac{9}{11}$.
Represent $ \sqrt{9.3}$ on the number line.
Represent $ \sqrt{9.3}$ on the number line.