Question

Alex says that there is no numbers between 1/3 and 1/4 . Why might he think that and what would you tell him? Is there a number that falls between these?

Answer 1

Answer:

Why might he think that and what would you tell him?

He thought perhaps that numbers were formed by rational numbers of the form $\frac{1}{x}$, where $x\in \mathbb{N}$. I would tell him that there are rational numbers $\frac{y}{z}$, such that $\frac{1}{x}\le \frac{y}{z} \le \frac{1}{x+1}$, where $x$, $y$, $z \in \mathbb{N}$.

Is there a number that falls between these?

$\frac{7}{24}$ is a rational number between $\frac{1}{3}$ and $\frac{1}{4}$.

Step-by-step explanation:

Why might he think that and what would you tell him?

He thought perhaps that numbers were formed by rational numbers of the form $\frac{1}{x}$, where $x\in \mathbb{N}$. I would tell him that there are rational numbers $\frac{y}{z}$, such that $\frac{1}{x}\le \frac{y}{z} \le \frac{1}{x+1}$, where $x$, $y$, $z \in \mathbb{N}$.

Is there a number that falls between these?

Indeed, the average number of $\frac{1}{3}$ and $\frac{1}{4}$, for instance. That is:

$\frac{y}{z} = \frac{\frac{1}{3}+\frac{1}{4} }{2}$

$\frac{y}{z} = \frac{\frac{7}{12} }{2}$

$\frac{y}{z} = \frac{7}{24}$

$\frac{7}{24}$ is a rational number between $\frac{1}{3}$ and $\frac{1}{4}$.