Under the commentary of verse 4:13, a brief portion explains what the divisions constitute:
It may incidentally be noted here that heirs are divided into two main groups:
Dhawu’l-Fara’id (sharer) i.e. persons to whom Islam allots a fixed and definite share (1/2, ¼, 1/8, 2/3, 1/3, or 1/6) in the estate of a deceased person.
‘Asabat (residuary) i.e. the heirs who receive the residue after the Dhawu’l Fara’id have received their allotted shares.
It is, however, not necessary that a member of the first group should always remain a Dhu’l Farida (sharer). In certain cases, he may be both a Dhu’l Farida (sharer) and an ‘Asaba (residuary) and in other cases he may be simply an ‘Asaba (residuary).
The Dhawu’l-Fara’id (sharers) are 12 in number, four males and eight females.
The males are:
Half-brother on the maternal side.
The females are:
Half-sister on paternal side
Half-sister on maternal side
Grandmother whether paternal or maternal.
The ‘Asabat (residuaries) are of four kinds:
Ascendants of the deceased i.e. father, paternal grandfather, etc.
Descendants of the deceased i.e. son, son’s son, etc.
Descendants of the father of the deceased i.e. full brother, brother’s son, etc.
Descendants of the grandfather i.e. full paternal uncle, paternal uncle’s son, etc.
[Larger Edition of the Commentary of the Qur’an, Volume 2, pp. 504-505, under verse 4:13]
The objections raised are specifically against the Sharers (Dhawu’l Fara’id), which constitutes the fractional portions. Anti-Islamists claim that these don’t add up to a whole number, but they are not meant to necessarily round up to a whole number. There are residue amounts left over which are considered as well, thus no matter what the case may be, regardless of whether there are fractional portions left over or not, the divisions happen according to the amounts which are left, debts and strangers are paid off first, then the remaining portion is divided among family members, some inheriting at one stage of division, then others get remaining portions of the division. If anyone wishes to consult this matter further they are invited to read the book “Algebra of Mohammed Ben Musa” edited and translated by Frederic Rosen, 1831, specifically the latter portion of the book starting from the chapter entitled “On Legacies”. To give some background, this book was written by the famous Al-Khawarizmi, who invented the modern algebra by utilizing it for the shares of inheritance. Thus, the problem with those trying to grasp the numbers in Islamic inheritance “not adding up” is due to a lack of mathematics on their part and their ignorance of algebra. Algebra was named because of Al-Khawarizmi (Al-Jabr), so anyone who wishes to learn how inheritances were computed should look to the father of modern algebra before they raise further objections.>Summary
The allegations against the Quran that its maths is wrong when dividing inheritance is due to lack of knowledge of how inheritance works and lack of knowledge of maths on the part of those who raise this objection. A simple reading on how the system of inheritance in Islam works and what’s the maths behind it would alleviate any doubts on this subject.