THE TERM RAHMAH- Lexicography
The term ‘Rahmah’ denotes tenderness and urge to show kindness to others including connotations of love, compassion, benevolence and generosity. Therefore the terms Ar-Rahmaan and Ar-Rahiim denote two aspects of the same root word ‘Rahmah’, which has different connotations and derivatives also,e.g:

‘Rahimun’:Mother's womb which bears, nurtures, nourishes, protects the child.
‘Rahimatun’:That gift or bounty which makes up any obvious or hidden deficiencies or meets such requirements.It has to be free of cost, price, wages etc – It is Gracious Bestowment from Allah. (Ta'leemul Quran, M.A.Khan)

‘Rahimun’: Relationship. (John Penrice)
F. Steingass in the Arabic English Dictionary, has given following meanings also:
Rahm : Commiseration
Rahiim: Sincere in friendship.

"There is a statement by Ibn Jarir that indicates that there is a consensus on this meaning". Since both these terms denote very intense and multi-layered meanings and nuances of the origin from the Mercy, there are a number of explanations of each.

"And My (Allah's) Mercy embraces all things".

There are dual and plural for the word Rahiim but not for the word Rahmaan. "The signification of the word `Rahiim’, on the contrary, does not contain anything which it should be impossible to find in a created being ..So, the word `Rahim' may justifiably be employed in the case of humanbeing.." (Page 59, Maariful Quran, M.M.M.Shafi). For details, also see Pages 61-66 of Lughatul Quran, Muhammad Abdur-Rashid Nomani, and for derivatives of the root word Rahm as used in the Quran, see ‘Al-Muajamul Mufahris’ - Muhammad Fawad Abdul Baqi.

The term Rahmaan is used at 53 places in the Quran. (Lughatul Quran, Muhammad Abdur-Rashid Nomani). There are different opinions whether the term Rahmaan is an Arabic or Hebrew word. Those who agree that it is an Arabic word, believe that it is derived from ‘Rahmah’. (The detailed discussion will follow). The others argue to the contrary that if it was an Arabic term and was derived from an Arabic root the Arabs would not have shown refusal with ignorance about this term when it was revealed to them, particularly when they were not ignorant and opposed to the belief in the Mercy of God, e.g:

"And they were those, who disbelieve at the mention of Ar-Rahmaan" 21-36

"And when it is said to them:
"Prostrate yourselves to The Most Gracious (Rahmaan)!
They say: `And what is the Most Gracious (Ar-Rahmaan)?
Shall we fall down in prostration to that which you command us? '
And it increases in them only aversion." 25-60

Anyway, it is a fact that before the revelation of the Quran the term `Rahmaan' for Allah was not common in Arabs. Only Christians, Jews and followers of some other religions used this term e.g. in Najad, etc. However they were familiar with the term Allah as is evident from verses: 29-61 and 63. It was because of this, that when Ali (May Allah be pleased with him) wrote "Bismillaahir-Rahmaanir-Rahiim" as the head line of the Treaty of Hudaibia, the Quraish objected to the term `Rahmaan.’

Ar-Rahmaan has more meanings that pertain to Mercy than Ar-Rahiim. Ar-Rahiim gives perennial expression and effect to ‘Rahmah’ covering everything in the universes every moment. The mention of the two aspects together in such a way that they magnify each other, emphasizes the all-embracing and ever available umbrella of Rahmah.

From above explanations it would be clear that it is not possible to translate the Attribute (in fact all the Attributes) of Allah in one word or one phrase in any language. In fact this is true for translation of whole of the Quran that one word or word for word translation does not convey the entire sense. Anyhow some of the translations of "Ar-Rahmaanir-Rahiim", in fact highlighting different aspects of the Main Fountain of Mercy, are given below:

"Excessively Compassionate and Extremely Merciful"
-(Tafsir-e-Usmani, Muhammad Ashfaque Ahmed)
"The All-Merciful, The Very Merciful"
-(Ma'ariful Quran, M.M.M.Shafi.)
"The Most Gracious, The Most Merciful"
-(The Noble Quran, M.M. Khan and Al-Hilali)
"The Most Affectionate, The Merciful"
-(Kanzul Iman, Shah Faridul Haque )
"Most Benevolent, Ever Merciful" - (Ahmed Ali.)
"The Mercy-Giving, The Merciful" - (T.B. Irving)
Some other scholars have translated this as
"The Ever All-Merciful, The Ever All-Beneficent"

Inorder to understand the true sense conveyed by these two terms, `Ar-Rahmaan and Ar-Rahiim', and difference from synonyms, it may be useful to study their comparison and contradistinction , e.g:

"Rauufun" :One who feels for the pain of others,
"Latiifun" :One who takes care in small details,
"Hanaanun" :One who is affectionate like mother,
(Pages 828-830, Mutaradifatul Quran, Abdur-Rahman Kailani).