Difference between 忘 and 忘记
Please read Result complement  

Also 了 :  'aspect marker' indicating completed action 

'忘' (forget) as a verb cannot be used without a result complement or aspect marker 

Example:


  '忘掉' 
  
  掉 is the result complement of 忘. 
  
  It describes the result of forget is the object is 'forgot away' (from one's mind)
  
  You cannot remove the result complement '掉' from '(忘掉)自己的手電號碼' and write '(忘)自己的手電號碼'


~


  '忘了' 
  
  了 is a aspect marker that indicates 'completion of an action'
  
  You cannot remove the aspect marker '了' from '(忘了)自己的手電號碼' and write '(忘)自己的手電號碼'


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'忘' = 'forget'
'记' = 'remember/ recollect'
'忘记' (forget to remember) = 'to forget' 


'忘记' is a compound word verb

It can be used with or without a result complement or aspect marker

Example:

忘记自己的手電號碼 = forget one's own cell-phone number

忘记(掉)自己的手電號碼 = forget one's own cell-phone number (away)

(忘记了)自己的手電號碼 = (forgotten) one's own cell-phone number 

Terms like '忘本'，'忘情' that directly apply the verb 忘 before a noun, are in effect, compound words  as in "做人忘本" (being ungrateful), 忘情演出 (passionately perform).  

The other cases of 忘 directly apply before a noun are 1. 'in idiomatic form',  like "見利忘義" 2. 'with an adverb',  like 難忘, 不忘, 未忘, and they are all literary terms