What is the difference between associative and commutative property?
Answers
commutatative property concerns the order of certain mathematical operations. In commitative property elements in equation are rearranged. In associative property elements are merely regrouped.
♕︎ǫᴜᴇsᴛɪᴏɴ♕︎
ᴡʜᴀᴛ ɪs ᴛʜᴇ ᴅɪғғᴇʀᴇɴᴄᴇ ʙᴇᴛᴡᴇᴇɴ ᴀssᴏᴄɪᴀᴛɪᴠᴇ ᴀɴᴅ ᴄᴏᴍᴍᴜᴛᴀᴛɪᴠᴇ ᴘʀᴏᴘᴇʀᴛʏ?
♕︎ᴀɴsᴡᴇʀ♕︎
ғᴏʀ ᴀᴅᴅɪᴛɪᴏɴ, ᴛʜᴇ ᴏʀᴅᴇʀ ᴏғ ᴛʜᴇ ɴᴜᴍʙᴇʀs ᴅᴏᴇs ɴᴏᴛ ᴍᴀᴛᴛᴇʀ. sᴏ, ᴇᴠᴇɴ ɪғ ʏᴏᴜ ʀᴇᴠᴇʀsᴇ ᴛʜᴇ ᴏʀᴅᴇʀ ᴏғ ᴛʜᴇ ɴᴜᴍʙᴇʀs ᴀɴᴅ ᴀᴅᴅ ᴛʜᴇᴍ, ᴛʜᴇ ᴇɴᴅ ʀᴇsᴜʟᴛ ʀᴇᴍᴀɪɴs ᴛʜᴇ sᴀᴍᴇ. ᴛʜɪs ᴘʀᴏᴘᴇʀᴛʏ,ᴡʜᴇʀᴇ ᴛʜᴇ ʀᴇsᴜʟᴛ ᴅᴏᴇs ɴᴏᴛ ᴅᴇᴘᴇɴᴅ ᴏɴ ᴛʜᴇ ᴏʀᴅᴇʀ ɪɴ ᴡʜɪᴄʜ ᴛʜᴇ ɴᴜᴍʙᴇʀs ᴀʀᴇ ᴛᴀᴋᴇɴ ғᴏʀ ᴀɴ ᴀʀɪᴛʜᴍᴇᴛɪᴄ ᴏᴘᴇʀᴀᴛɪᴏɴ, ɪs ᴄᴀʟʟᴇᴅ ᴄᴏᴍᴍᴜᴛᴀᴛɪᴠᴇ ᴘʀᴏᴘᴇʀᴛʏ.
ᴇxᴀᴍᴘʟᴇ:- 1 + 2 = 2 + 1
ᴀssᴏᴄɪᴀᴛɪᴠᴇ ᴘʀᴏᴘᴇʀᴛʏ ᴏғ ᴀᴅᴅɪᴛɪᴏɴ sᴀʏs ᴛʜᴀᴛ ғᴏʀ ᴀɴʏ ᴛʜʀᴇᴇ ʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀs ᴀ/ʙ, ᴄ/ᴅ, ᴀɴᴅ ᴇ/ғ, ᴛʜᴇ ᴀssᴏᴄɪᴀᴛɪᴠᴇ ᴘʀᴏᴘᴇʀᴛʏ ᴏғ ᴀᴅᴅɪᴛɪᴏɴ sᴛᴀᴛᴇs ᴛʜᴀᴛ:-
[ ᴀ/ʙ + ᴄ/ᴅ] + ᴇ/ғ = ᴀ/ʙ + [ ᴄ/ᴅ + ᴇ/ғ ]
ᴀssᴏᴄɪᴀᴛɪᴠᴇ ᴘʀᴏᴘᴇʀᴛʏ ᴏғ ᴍᴜʟᴛɪᴘʟɪᴄᴀᴛɪᴏɴ sᴀʏs ᴛʜᴀᴛ ғᴏʀ ᴀɴʏ ᴛʜʀᴇᴇ ʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀs, ᴀ/ʙ, ᴄ/ᴅ, ᴀɴᴅ ᴇ/ғ, ᴛʜᴇ ᴀssᴏᴄɪᴀᴛɪᴠᴇ ᴘʀᴏᴘᴇʀᴛʏ ᴏғ ᴍᴜʟᴛɪᴘʟɪᴄᴀᴛɪᴏɴ sᴛᴀᴛᴇs ᴛʜᴀᴛ :-
[ ᴀ/ʙ × ᴄ/ᴅ] × ᴇ/ғ = ᴀ/ʙ × [ ᴄ/ᴅ × ᴇ/ғ]
♕︎ᴀᴅᴅɪᴛɪᴏɴᴀʟ ɪɴғᴏʀᴍᴀᴛɪᴏɴ♕︎
- ᴛʜᴇ ᴄᴏᴍᴍᴜᴛᴀᴛɪᴠᴇ ᴘʀᴏᴘᴇʀᴛʏ ᴏғ ᴀᴅᴅɪᴛɪᴏɴ ғᴏʀ ʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀs sᴛᴀᴛᴇs ᴛʜᴀᴛ ғᴏʀ ᴀɴʏ ᴛᴡᴏ ʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀs, ᴀ/ʙ ᴀɴᴅ ᴄ/ᴅ, ᴀ/ʙ + ᴄ/ᴅ = ᴄ/ᴅ + ᴀ/ʙ
- ғᴏʀ sᴜʙᴛʀᴀᴄᴛɪᴏɴ ʜᴏᴡᴇᴠᴇʀ, ᴏʀᴅᴇʀ ᴏғ ᴛʜᴇ ɴᴜᴍʙᴇʀs ᴍᴀᴛᴛᴇʀ. ᴄʜᴀɴɢɪɴɢ ᴛʜᴇ ᴏʀᴅᴇʀ ɢɪᴠᴇs ᴀ ᴅɪғғᴇʀᴇɴᴛ ᴀɴsᴡᴇʀ. ᴛʜᴀᴛ's ᴡʜʏ ᴜɴᴅᴇʀ sᴜʙᴛʀᴀᴄᴛɪᴏɴ, ᴛʜᴇ ʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀs ᴀʀᴇ ɴᴏᴛ ᴄᴏᴍᴍᴜᴛᴀᴛɪᴠᴇ.
- sᴏ, ғᴏʀ ᴀɴʏ ᴛᴡᴏ ᴅɪsᴛɪɴᴄᴛ ʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀs ᴀ/ʙ ᴀɴᴅ ᴄ/ᴅ, ᴀ/ʙ - ᴄ/ᴅ = ᴄ/ᴅ - ᴀ/ʙ
- ʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀs sᴀᴛɪsғʏ ᴄᴏᴍᴍᴜᴛᴀᴛɪᴠᴇ ᴘʀᴏᴘᴇʀᴛʏ ᴜɴᴅᴇʀ ᴀᴅᴅɪᴛɪᴏɴ ᴀɴᴅ ᴍᴜʟᴛɪᴘʟɪᴄᴀᴛɪᴏɴ.
- ғᴀᴄᴛ :- ᴛʜᴇ ᴡᴏʀᴅ "ʀᴀᴛɪᴏɴᴀʟ" ʜᴀs ɪᴛs ʀᴏᴏᴛs ɪɴ ᴛʜᴇ ᴡᴏʀᴅ "ʀᴀᴛɪᴏ" ɪɴ ᴍᴀᴛʜᴇᴍᴀᴛɪᴄs. ᴇᴠᴇʀʏ ʀᴀᴛɪᴏɴᴀʟ ɴᴜᴍʙᴇʀ ɪs ᴀᴄᴛᴜᴀʟʟʏ ᴊᴜsᴛ ᴀ ʀᴀᴛɪᴏ ᴏғ ᴛᴡᴏ ɪɴᴛᴇɢᴇʀs.
- ғᴀᴄᴛ :- ᴍᴀɴʏ ᴀʟɢᴇʙʀɪᴄ ɪᴅᴇɴᴛɪᴛɪᴇs [ ғᴏʀ ᴇxᴀᴍᴘʟᴇ, ( ᴀ+ʙ)^2 = ᴀ^2 + 2ᴀʙ + ʙ^2] ᴀʀᴇ ᴀᴘᴘʟɪᴄᴀᴛɪᴏɴs ᴏғ ᴛʜᴇ ᴅɪsᴛʀɪʙᴜᴛɪᴠᴇ ᴘʀᴏᴘᴇʀᴛʏ ᴏғ ɴᴜᴍʙᴇʀs ᴏᴠᴇʀ ᴀᴅᴅɪᴛɪᴏɴ ᴀɴᴅ sᴜʙᴛʀᴀᴄᴛɪᴏɴ.