# 1) Find 100 irrational numbers between 0 and 1/100. 2) Find 50 rational numbers betwee 1 and 2. 3) Find 50 irrational numbers betwee 1 and 2.

1) Find 100 irrational numbers between 0 and $\frac{1}{100}.$
2) Find 50 rational numbers betwee 1 and 2.
3) Find 50 irrational numbers betwee 1 and 2.
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Cristiano Sears

1) We know 2 So, $0<\frac{e}{1000}<\frac{1}{100}ase<3<10$
So, $0<\frac{e}{{10}^{3}}<\frac{1}{100}$
Define , $0<\frac{e}{{10}^{103}}<\frac{e}{{10}^{102}}<\frac{e}{{10}^{101}}<\dots ..<\frac{e}{{10}^{5}}<\dots ..<\frac{e}{{10}^{3}}<\frac{1}{100}$
Hence $\frac{e}{{10}^{i}}$,
$3⇐i⇐103$ are the irrational numbers between $0\to \frac{1}{100}$.
2) Clearly we know that $0<\frac{1}{10}<1$ then $1<1+\frac{1}{10}<2$
Also, $0<\frac{1}{{10}^{2}}<1$, then $1<1+\frac{1}{{10}^{2}}<2$
So, $1+\frac{1}{10},1+\frac{1}{{10}^{2}},1+\frac{1}{{10}^{3}},\dots .,1+\frac{1}{{10}^{50}}$ are rational numbers between 1 and 2.
They are rational numbers because they are sum of rational number.
3) We know that $e,{e}^{2},{e}^{3},\dots .,{e}^{50}$ all are irrational numbers .
So, $1<1+\frac{1}{e}<2$
For $1\le i\le 50,1<1+\frac{1}{{e}^{i}}<2$
sum of one rational number and one irrational number is again a irrational number.
So , are 50 irrational numbers between 1 and 2.