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Question Number 141997 by jlewis last updated on 25/May/21

$$\mathrm{find}\mathrm{two}\mathrm{irrational}\mathrm{numbers}$$$$\mathrm{between}0.333....\mathrm{and}0.444...$$$$$$

Answered by MJS_new last updated on 25/May/21

$$.33312345678910111213...$$$$\frac{1}{\mathrm{e}}\approx .367879441...$$

Commented by MJS_new last updated on 25/May/21

$$.333000333300003333300000...$$

Commented by jlewis last updated on 25/May/21

$$\mathrm{wah}!\mathrm{i}\mathrm{have}\mathrm{not}\mathrm{gotten}\mathrm{how}\mathrm{u}$$$$\mathrm{arrived}\mathrm{to}\mathrm{this}$$

Commented by MJS_new last updated on 25/May/21

$$\mathrm{easy}:\mathrm{invent}\mathrm{an}\mathrm{infinite}\mathrm{non}-\mathrm{periodic}\mathrm{pattern}$$$$.01001000100001000001...$$$$.0123456789101112...$$$$.149162536496481100...$$$$.248163264128256512...$$$$.01123581321345589...$$

Commented by jlewis last updated on 25/May/21

$$\mathrm{i}\mathrm{get}\mathrm{it}\mathrm{now}.\mathrm{thanks}$$

Answered by MJS_new last updated on 25/May/21

$$\mathrm{you}\mathrm{could}\mathrm{also}\mathrm{do}\mathrm{this}:$$$$.333...=\frac{3}{9}$$$$.444...=\frac{4}{9}$$$$\sqrt[3]{{\left(\frac{3}{9}\right)}^2\times \frac{4}{9}}=\sqrt[3]{\frac{4}{81}}\approx .366880805...\notin \mathbb{Q}$$$$\sqrt{\frac{3}{9}\times \frac{4}{9}}=\frac{2\sqrt{3}}{9}\approx .384900179...\notin \mathbb{Q}$$$$\sqrt[4]{\frac{3}{9}\times {\left(\frac{4}{9}\right)}^3}=\frac{\sqrt[4]{192}}{9}\approx .413602159...\notin \mathbb{Q}$$

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