What is the difference: 剪 or 理, if I need to tell hiardress about hiarcut?
I still can't understand what is the difference between these two words "cut": 剪 and 理? When I go to the hairdressing, which is word should I tell to hiardress?

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你可以跟他们说：我想理发或者我想剪一个新的发型或者我想剪头发

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Both of them they are right. But the point is the first exactly means u wanna cut ur hair. The second one means u wanna fix ur hair. Both of them actually have the same meaning lmao.

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Simply put
1.when it comes to cutting hair，these two words share the same meaning.
2.剪informal/理formal 
But，nowadays in China，it makes no different between 剪发 and 理发.So choose whatever u like.
3.related words u have to keep in mind
理发店 / 造型师 /发型 /染发 /烫发 /洗剪吹

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剪头 is colloquial and should be only used in spoken language while 理发 is much more formal. The barber's in Chinese is 理发店 rather than 剪头店. The verb 剪 means to cut sth. into two pieces (but 剪头 does not mean to cut your head into two pieces, lol). However, the verb 理 means cut (hair) only when it's in the fixed word 理发. It has various of meanings in other combinations.
If you want to tell your barber you want a haircut, you can say 我想剪个头 or 我想理个发, both are ok.

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In Taiwan they mean the same.
But usually we use "理"頭髮 on man, while "剪"頭髮 on both man and woman.
EX : 去把頭髮理一理 (Go cut your hair)
.
I think you can simply say 我想剪頭髮 (I wanna cut my hair)
or 我想換髮型 (I wanna change my hair-style)