Question and Answers Forum

All Questions      Topic List

Number Theory Questions

Previous in All Question      Next in All Question      

Previous in Number Theory      Next in Number Theory      

Question Number 166911 by rexford last updated on 02/Mar/22

Answered by MathsFan last updated on 02/Mar/22

suppose (√2) is rational  (√2)=(p/q) ⇒ 2=(p^2 /q^2 ) ⇒  2q^2 =p^2 .....(1)   2 divides p   2 divides p^2   let  r=(p/2) ⇒  p=2r.......(2)  2q^2 =4r^2   ⇒   q^2 =2r^2   2 divides q  2 divides q^2   p and p have 2 as common factor  but contradict the fact that p and q  are coprime  hence  (√2) is an irrational number.

$$\boldsymbol{{suppose}}\:\sqrt{\mathrm{2}}\:\boldsymbol{{is}}\:\boldsymbol{{rational}} \\ $$$$\sqrt{\mathrm{2}}=\frac{\boldsymbol{{p}}}{\boldsymbol{{q}}}\:\Rightarrow\:\mathrm{2}=\frac{\boldsymbol{{p}}^{\mathrm{2}} }{\boldsymbol{{q}}^{\mathrm{2}} }\:\Rightarrow\:\:\mathrm{2}\boldsymbol{{q}}^{\mathrm{2}} =\boldsymbol{{p}}^{\mathrm{2}} .....\left(\mathrm{1}\right) \\ $$$$\:\mathrm{2}\:\boldsymbol{{divides}}\:\boldsymbol{{p}} \\ $$$$\:\mathrm{2}\:\boldsymbol{{divides}}\:\boldsymbol{{p}}^{\mathrm{2}} \\ $$$$\boldsymbol{{let}}\:\:\boldsymbol{{r}}=\frac{\boldsymbol{{p}}}{\mathrm{2}}\:\Rightarrow\:\:\boldsymbol{{p}}=\mathrm{2}\boldsymbol{{r}}.......\left(\mathrm{2}\right) \\ $$$$\mathrm{2}\boldsymbol{{q}}^{\mathrm{2}} =\mathrm{4}\boldsymbol{{r}}^{\mathrm{2}} \:\:\Rightarrow\:\:\:\boldsymbol{{q}}^{\mathrm{2}} =\mathrm{2}\boldsymbol{{r}}^{\mathrm{2}} \\ $$$$\mathrm{2}\:\boldsymbol{{divides}}\:\boldsymbol{{q}} \\ $$$$\mathrm{2}\:\boldsymbol{{divides}}\:\boldsymbol{{q}}^{\mathrm{2}} \\ $$$$\boldsymbol{{p}}\:\boldsymbol{{and}}\:\boldsymbol{{p}}\:\boldsymbol{{have}}\:\mathrm{2}\:\boldsymbol{{as}}\:\boldsymbol{{common}}\:\boldsymbol{{factor}} \\ $$$$\boldsymbol{{but}}\:\boldsymbol{{contradict}}\:\boldsymbol{{the}}\:\boldsymbol{{fact}}\:\boldsymbol{{that}}\:\boldsymbol{{p}}\:\boldsymbol{{and}}\:\boldsymbol{{q}} \\ $$$$\boldsymbol{{are}}\:\boldsymbol{{coprime}} \\ $$$$\boldsymbol{{hence}}\:\:\sqrt{\mathrm{2}}\:\boldsymbol{{is}}\:\boldsymbol{{an}}\:\boldsymbol{{irrational}}\:\boldsymbol{{number}}. \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com