Question

prove that √5 is irrational​

Answer 1

Answer:

ʜᴏᴘᴇ ɪᴛ ʜᴇʟᴘs ʏᴏᴜ ғʀɪᴇɴᴅ

Step-by-step explanation:

ʟᴇᴛ ᴜs ᴀssᴜᴍᴇ ᴛʜᴀᴛ √ɪs ʀᴀᴛɪᴏɴᴀʟ. sᴏ, ᴡᴇ ᴄᴀɴ ғɪɴᴅ ɪɴᴛᴇɢᴇʀs ᴘ ᴀɴᴅ ǫ (≠ ) sᴜᴄʜ ᴛʜᴀᴛ √ = .

sᴜᴘᴘᴏsᴇ ᴘ ᴀɴᴅ ǫ ʜᴀᴠᴇ ᴀ ᴄᴏᴍᴍᴏɴ ғᴀᴄᴛᴏʀ ᴏᴛʜᴇʀ ᴛʜᴀɴ .

ᴛʜᴇɴ, ᴡᴇ ᴅɪᴠɪᴅᴇ ʙʏ ᴛʜᴇ ᴄᴏᴍᴍᴏɴ ғᴀᴄᴛᴏʀ ᴛᴏ ɢᴇᴛ √ = , ᴡʜᴇʀᴇ ᴀ ᴀɴᴅ ʙ ᴀʀᴇ ᴄᴏᴘʀɪᴍᴇ.

sᴏ, ʙ√ = ᴀ.

sǫᴜᴀʀɪɴɢ ᴏɴ ʙᴏᴛʜ sɪᴅᴇs, ᴡᴇ ɢᴇᴛ

ʙ = ᴀ

ᴛʜᴇʀᴇғᴏʀᴇ, ᴅɪᴠɪᴅᴇs ᴀ.

ɴᴏᴡ, ʙʏ ᴛʜᴇᴏʀᴇᴍ ᴡʜɪᴄʜ sᴛᴀᴛᴇs ᴛʜᴀᴛ ʟᴇᴛ ᴘ ʙᴇ ᴀ ᴘʀɪᴍᴇ ɴᴜᴍʙᴇʀ. ɪғ ᴘ ᴅɪᴠɪᴅᴇs ᴀ , ᴛʜᴇɴ ᴘ ᴅɪᴠɪᴅᴇs ᴀ, ᴡʜᴇʀᴇ ᴀ ɪs ᴀ ᴘᴏsɪᴛɪᴠᴇ ɪɴᴛᴇɢᴇʀ,

ᴅɪᴠɪᴅᴇs ᴀ.

sᴏ, ᴡᴇ ᴄᴀɴ ᴡʀɪᴛᴇ ᴀ = ᴄ ғᴏʀ sᴏᴍᴇ ɪɴᴛᴇɢᴇʀ ᴄ

sᴜʙsᴛɪᴛᴜᴛɪɴɢ ғᴏʀ ᴀ, ᴡᴇ ɢᴇᴛ ʙ = ᴄ ,ɪ.ᴇ. ʙ = ᴄ .

ᴛʜɪs ᴍᴇᴀɴs ᴛʜᴀᴛ ᴅɪᴠɪᴅᴇs ʙ, ᴀɴᴅ sᴏ ᴅɪᴠɪᴅᴇs ʙ (ᴀɢᴀɪɴ ᴜsɪɴɢ ᴛʜᴇ ᴀʙᴏᴠᴇ ᴛʜᴇᴏʀᴇᴍ ᴡɪᴛʜ ᴘ = ). ᴛʜᴇʀᴇғᴏʀᴇ, ᴀ ᴀɴᴅ ʙ ʜᴀᴠᴇ ᴀᴛ ʟᴇᴀsᴛ ᴀs ᴀ ᴄᴏᴍᴍᴏɴ ғᴀᴄᴛᴏʀ.

ʙᴜᴛ ᴛʜɪs ᴄᴏɴᴛʀᴀᴅɪᴄᴛs ᴛʜᴇ ғᴀᴄᴛ ᴛʜᴀᴛ ᴀ ᴀɴᴅ ʙ ʜᴀᴠᴇ ɴᴏ ᴄᴏᴍᴍᴏɴ ғᴀᴄᴛᴏʀs ᴏᴛʜᴇʀ ᴛʜᴀɴ .

ᴛʜɪs ᴄᴏɴᴛʀᴀᴅɪᴄᴛɪᴏɴ ʜᴀs ᴀʀɪsᴇɴ ʙᴇᴄᴀᴜsᴇ ᴏғ ᴏᴜʀ ɪɴᴄᴏʀʀᴇᴄᴛ ᴀssᴜᴍᴘᴛɪᴏɴ ᴛʜᴀᴛ ɪs ʀᴀᴛɪᴏɴᴀʟ.

sᴏ, ᴡᴇ ᴄᴏɴᴄʟᴜᴅᴇ ᴛʜᴀᴛ √ ɪs ɪʀʀᴀᴛɪᴏɴᴀʟ.

Answer 2

Step-by-step explanation:

Let

5

be a rational number.

then it must be in form of

q

p

where, q



=0 ( p and q are co-prime)

5

=

q

p

5

×q=p

Suaring on both sides,

5q

2

=p

2

--------------(1)

p

2

is divisible by 5.

So, p is divisible by 5.

p=5c

Suaring on both sides,

p

2

=25c

2

--------------(2)

Put p

2

in eqn.(1)

5q

2

=25(c)

2

q

2

=5c

2

So, q is divisible by 5.

.

Thus p and q have a common factor of 5.

So, there is a contradiction as per our assumption.

We have assumed p and q are co-prime but here they a common factor of 5.

The above statement contradicts our assumption.

Therefore,

5

is an irrational number.