Math, asked by alphyrose9688, 10 days ago

Show that root 2 is irrational number

Answers

Answered by Anonymous
3

ᴀɴSᴡᴇƦ :-

ʟᴇᴛ ᴜs ᴀssᴜᴍᴇ ᴛʜᴀᴛ √2 ɪs ᴀ ʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀ.

ᴛʜᴇʀᴇғᴏʀᴇ,

 \frac{ \sqrt{2} }{1}  =  \frac{a}{b}

ᴡʜᴇʀᴇ ᴀ ᴀɴᴅ ʙ ᴀʀᴇ ɪɴᴛᴇɢᴇʀs ᴀɴᴅ ʙ ɪs ɴᴏᴛ ᴇǫᴜᴀʟs ᴛᴏ 0.

ɴᴏᴡ,

 \sqrt{2 } b = a \\

Sǫᴜᴀʀɪɴɢ ʙᴏᴛʜ sɪᴅᴇs.

2 {b}^{2}  =  {a}^{2}  \\  \\  {b}^{2}  =  \frac{ {a}^{2} }{2}

ɪғ 2 ᴅɪᴠɪᴅᴇs

 {a}^{2}

ɪᴛ ᴀʟsᴏ ᴅɪᴠɪᴅᴇs ᴀ

ɴᴏᴡ,

a = 2c

Sǫᴜᴀʀɪɴɢ ʙᴏᴛʜ sɪᴅᴇs,

 {a}^{2}  = 4 {c}^{2} \\  \\ 2 {b}^{2}  = 4 {c}^{2}  \\  \\  {b}^{2}  = 2 {c }^{2} \\  \\  {c}^{2}  =  \frac{ {b}^{2} }{2}  \\  \\

ɪғ 2 ᴅɪᴠɪᴅᴇs

 {b}^{2}

2 ᴀʟsᴏ ᴅɪᴠɪᴅᴇs ʙ

ᴛʜᴇʀᴇғᴏʀᴇ, ᴛʜɪs ɪs ᴀ ᴄᴏɴᴛʀᴀᴅɪᴄᴛɪᴏɴ,

ʜᴇɴᴄᴇ,

√2 ɪs ᴀɴ ɪʀʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀ,

ʜᴏᴘᴇ ɪᴛ ʜᴇʟᴘs ʏᴏᴜ.

{\huge{\pink{\ddot{\smile}}}}

Answered by Anonymous
1

ᴀɴSᴡᴇƦ :-

ʟᴇᴛ ᴜs ᴀssᴜᴍᴇ ᴛʜᴀᴛ √2 ɪs ᴀ ʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀ.

ᴛʜᴇʀᴇғᴏʀᴇ,

\frac{ \sqrt{2} }{1} = \frac{a}{b}[tex]</p><p>1</p><p>2</p><p>	</p><p> </p><p>	</p><p> = </p><p>b</p><p>a</p><p>	</p><p> </p><p></p><p>ᴡʜᴇʀᴇ ᴀ ᴀɴᴅ ʙ ᴀʀᴇ ɪɴᴛᴇɢᴇʀs ᴀɴᴅ ʙ ɪs ɴᴏᴛ ᴇǫᴜᴀʟs ᴛᴏ 0.</p><p></p><p>ɴᴏᴡ,</p><p></p><p>[tex]\begin{lgathered}\sqrt{2 } b = a \\\end{lgathered}

2

b=a

Sǫᴜᴀʀɪɴɢ ʙᴏᴛʜ sɪᴅᴇs.

\begin{lgathered}2 {b}^{2} = {a}^{2} \\ \\ {b}^{2} = \frac{ {a}^{2} }{2}\end{lgathered}

ɪғ 2 ᴅɪᴠɪᴅᴇs

{a}^{2}a

2

ɪᴛ ᴀʟsᴏ ᴅɪᴠɪᴅᴇs ᴀ

ɴᴏᴡ,

a = 2ca=2c

Sǫᴜᴀʀɪɴɢ ʙᴏᴛʜ sɪᴅᴇs,

\begin{lgathered}{a}^{2} = 4 {c}^{2} \\ \\ 2 {b}^{2} = 4 {c}^{2} \\ \\ {b}^{2} = 2 {c }^{2} \\ \\ {c}^{2} = \frac{ {b}^{2} }{2} \\ \\\end{lgathered}

ɪғ 2 ᴅɪᴠɪᴅᴇs

{b}^{2}b

2

2 ᴀʟsᴏ ᴅɪᴠɪᴅᴇs ʙ

ᴛʜᴇʀᴇғᴏʀᴇ, ᴛʜɪs ɪs ᴀ ᴄᴏɴᴛʀᴀᴅɪᴄᴛɪᴏɴ,

ʜᴇɴᴄᴇ,

√2 ɪs ᴀɴ ɪʀʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀ,

ʜᴏᴘᴇ ɪᴛ ʜᴇʟᴘs ʏᴏᴜ.

{\huge{\pink{\ddot{\smile}}}}

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