[Ummu Sufyaan]

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When combined 'Ar Rahmaan' and 'Ar Raheem' mean that the Rahmah (mercy) of Allaah is expansive wide-spreading and it reaches [all] the creation, this is what some [Scholars] alluded to by saying: 'Ar Rahmaan' is a universal mercy [to all the creation] and 'Ar Raheem' is a mercy that is specific to the believers.

The mercy of Allaah upon the disbelievers is specific to the life of this world only, so it is as if there is really no mercy upon them because in the after life when they ask Allaah to take them out of the Hellfire seeking to get close to Allaah through His Lordship and by confessing upon themselves saying:
{Our Lord! Take us out of it [Hellfire] If ever we return (to evil), then indeed we are Dhalimoon (polytheists, oppressors, unjust, and wrong doers)} [Al Mu-minoon: 107]

They will not find mercy, rather they will find justice, Allaah ‘Azza wa Jall will say to them: {Remain in it in humiliation! And speak not to Me!} [Al Mu-minoon: 108]

Source

 

[Ummu Sufyaan]

*bump:

Just a bit more on this:
Ar-Rahmaan: (the Most Merciful) is one of the names which are particular to Allaah, the Mighty the Majestic, and cannot be applied to other than Him. Ar-Rahmaan means the one who has as His attribute very great and extensive Mercy.

Ar-Raheem: (The Bestower of Mercy) is a name which is applied to to Allaah, the Mighty, the Majestic, and the term may be restrictively applied others also. It's meaning is the great who is merciful to others. Ar-Rahmaan is the one who possess very great and extensive mercy, and ar-Rahmeem is the one who bestows that Mercy upon others. (i.e He has mercy upon them). So when these two names of Allah come together, then what is meant by ar-Raheem is the one who has mercy upon whomever He wishes from His servants. As Allaah, the Most High, says:

بعذب من يشاء ويرحم من يشاء واليه تقلبون​

"He justly punishes whomever He wills (i.e the disobedient) and shows mercy to whomever He wills, and to Him you will be returned." (Surat al-Ankaboot, 21)


Source: Sharh Usool ath-Thalaath, by Sheikh ibn Uthaymeen, rahimahullah...

 

[Mazhara]

Ar-Rehman is the PERSONAL NAME of Allah,

Ar-Raheem is the attribute of Ar-Rehman.

 

[Yanal]

A difference of wording and letters make such a difference for the names of All mighty Allah. I remember in grade 6 I attempted to meomorize the 99 words but I gave up thinking it's hard.