Islamic Inheritance jurisprudence is a field of Islamic jurisprudence (Arabic: ) that deals with inheritance, a topic that is prominently dealt with in the Qur'an. It is often called M?r?th, and its branch of Islamic law is technically known as ?ulm al-fari? (Arabic: ?, "the science of the ordained quotas"). All Muslims are to follow and implement the rules of Islamic inheritance.
The Qur'an introduced a number of different rights and restrictions on matters of inheritance, including what were at that time general improvements to the treatment of women and family life. The Qur'an also presented efforts to fix the laws of inheritance, and thus forming a complete legal system. This development was in contrast to pre-Islamic societies where rules of inheritance varied considerably.  They do, however, also differ from ongoing secular egalitarian improvements since that time, up to, though principally in, the modern era.
Furthermore, the Qur'an introduced additional heirs that were not entitled inheritance in pre-Islamic times, mentioning nine relatives specifically of which six were female and three were male. The laws of inheritance in the Qur'an also included other male relatives, like the husband and half-brothers from the mother's side, which were excluded from inheritance in old customs. The heirs mentioned in the Qur'an are the mother, father, husband, wife, daughter, uterine brother, full sister, uterine sister, and consanguine sister.
In general, the Qur'an improved the status of women by identifying their share of inheritance in clear terms. It also completely forbade the practice of inheriting widows.[4:19]Orientalist Joseph Schacht states that "this is not meant as a regular legal ordinance, but is part of the Qur'anic endeavor to improve the position of women." The Qur'an does not explicitly mention the shares of male relatives, such as the decedent's son, but provides the rule that the son's share must be twice that of the daughter's. Muslim theologians explain this aspect of inheritance by looking at Islamic law in its entirety, which bestows the responsibility and accountability on men to provide safety, protection and sustenance to women.[Qur'an 4:34] One explanation of why a daughter is entitled to only half that of the son is that Islam decrees that women, upon marriage are entitled to a "dowry" from the husband (in addition to any provision by her parents). It is thereafter the husband's obligation to care for and maintain his wife and the "dowry" is, therefore, essentially an advance of inheritance rights from her husband's estate which returns to his possession after the formalities over.
In addition to the above changes, the Qur'an grants testamentary powers to Muslims in disposing their property.[Qur'an, 2:180-182, 2:240, 4:33, 5:106-107] In their will, called wa?eyya, Muslims are allowed to give out a maximum of one third of their property. Muslims are also encouraged to give money to the orphans and poor if they are present during the division of property.
The Qur'an contains only three verses [4:11, 4:12 and 4:176] which give specific details of inheritance and shares, in addition to few verses dealing with testamentary power. It has also been reported in Hadith that Muhammad allotted great importance to the laws of inheritance and ordered his followers to learn and teach them. Muslim jurists used these verses as a starting point to expound the laws of inheritance even further using Hadith, as well as methods of juristic reasoning, like Qiyas. In later periods, large volumes of work have been written on the subject.
This amalgamation of old agnatic customs and Islamic law led to a number of problems and controversies that Muslim jurists have solved in different ways. Through the use of deductive reasoning (Qiyas), Muslim jurists added three additional heirs: the paternal grandfather, maternal grandmother, and agnatic granddaughter. These heirs, if entitled to inherit, are given their fixed shares and the remaining estate is inherited by the residuaries (?a?aba). In some cases, they have also upheld the rule of men having twice the share of women in circumstances not readily mentioned in the Qur'an, and tried to deal with complex cases in a variety of different contexts. This led to some minor differences between jurisprudence schools of the Sunni maddhabs. Also, the laws of inheritance for Twelver Shia, despite being based on the same principles, differ in a number of features due to the rejection of certain accounts of Hadith and based on their understanding of certain events in early Islam. On the other hand, the system of inheritance of the Kharajite Ibadis and Zaidis closely resemble that of the Sunni system. In modern Muslim countries, usually a mixture of different schools of jurisprudence (including Shia) is in effect, in addition to a number of important reforms to the traditional system. The main achievements of such modern systems was the codification of inheritance laws.
Inheritance is considered as an integral part of Shariah Law. Muslims inherit from one another as stated in the Qur'an.[Qur'an 4:7] Hence, there is a legal share for relatives of the decedent in his estate and property. The major rules of inheritance are detailed in Qur'an, Hadith and Fiqh.
When a Muslim dies there are four duties which need to be performed. They are:
Therefore, it is necessary to determine the relatives of the deceased who are entitled to inherit, and their shares.
These laws take greater prominence in Islam because of the restriction placed on the testator (a person who makes a will). Islamic law places two restrictions on the testator:
Heirs referred to as primary heirs are always entitled to a share of the inheritance, they are never totally excluded. These primary heirs consist of the spouse relict, both parents, the son and the daughter. All remaining heirs can be totally excluded by the presence of other heirs. But under certain circumstances, other heirs can also inherit as residuaries, namely the father, paternal grandfather, daughter, agnatic granddaughter, full sister, consanguine sister and mother. Those who inherit are usually confined to three groups:
In Islamic law, only relatives with a legitimate blood relationship to the deceased are entitled to inherit. Thus, illegitimate children and adopted children have no shares in inheritance. In general, a full brother will exclude a half-brother who shares a common father ("consanguine brother), but not a half-brother who shares a common mother ("uterine brother"). In cases where a deceased man leaves a pregnant woman, the unborn child's share will be reserved. Also a woman during the time of waiting (?iddat) after divorce is considered a wife of the deceased for purposes of inheritance.
There are even further rules of exclusion and inclusion of different relatives. The only "practical situations" which may cause disqualification are differences of religion and homicide. But schools of Islamic jurisprudence differed whether a Muslim can inherit from a non-Muslim or not. All the jurists agree that intentional or unjustifiable killing would exclude a person from inheritance.
In Islam, women are entitled the right of inheritance. In general circumstances, though not all, Islam allots women half the share of inheritance available to men who have the same degree of relation to the decedent. For example, where the decedent has both male and female children, a son's share is double that of a daughter's. Additionally, the sister of a childless man inherits half of his property upon his death, while a brother of a childless woman inherits all of her property. However, this principle is not universally applicable, and there are other circumstances where women might receive equal shares to men. For example, the share of the mother and father of a decedent who leaves children behind. Also the share of a uterine brother is equal to the share of a uterine sister, as do the shares of their descendants.
Sometimes, women get double the share as that of men; for example, if there are only parents and husband, husband will receive half, father gets 1/6 and mother gets 2/6. This is according to Ibne Abbas's interpretation of verses 11, 12 of Surah An-Nisa. [Quran 4:11,12] Also the Qur'an does not discriminate between men and women in cases of kalalah relation.Kalalah describes a person who leaves behind neither parents nor children; it also means all the relatives of a deceased except his parents and children, and it also denotes the relationships which are not through [the deceased's] parents or children. Islamic scholars hold that the original reasons for these differences are the responsibilities that are allotted to spouses. A husband in Islam must use his inheritance to support his family while a wife has no support obligations. Additionally, Arab society traditionally practiced the custom of bride price or dower rather than dowry; i.e., the man paid a gift to his wife or her family upon marriage, rather than the opposite, placing a financial burden on men where none existed on women. This custom was continued but changed materially by Islam. The divine injunction stipulated that the dowry (mahr) is due to the wife only not her family. It can also be deferred thereby reducing the burden if the husband is unable to afford the requested dowry at the time of the marriage. The wife can defer it till a stipulated date or it can become a debt on the estate when the husband dies. And give their dowries willingly to women (as an obligation), but if they, of their own accord, remit a portion of the dowry, you may enjoy it with pleasure. 
The Islamic law of inheritance served as an impetus behind the development of algebra (derived from the Arabic al-jabr) by Muhammad ibn M?s? al-Khw?rizm? and other medieval Islamic mathematicians. Al-Khw?rizm?'s Hisab al-jabr w'al-muqabala, the foundational text of algebra, devoted its third and longest chapter to solving problems related to Islamic inheritance using algebra. He formulated the rules of inheritance as linear equations, hence his knowledge of quadratic equations was not required.
Al-Hass?r, a mathematician from the Maghreb (North Africa) specializing in Islamic inheritance jurisprudence during the 12th century, developed the modern symbolic mathematical notation for fractions, where the numerator and denominator are separated by a horizontal bar. The "dust ciphers he used are also nearly identical to the digits used in the current Western Arabic numerals. These same digits and fractional notation appear soon after in the work of Fibonacci in the 13th century.
In the 15th century, Ab? al-Hasan ibn Al? al-Qalas?d?, a specialist in Islamic inheritance jurisprudence, used a mathematical notation for algebra which took "the first steps toward the introduction of algebraic symbolism." He represented mathematical symbols using characters from the Arabic alphabet.
Fibonacci uses ibn al-Y?samin's fraction notations to the full in the Liber abbaci [ed. Boncompagni 1857], writing composite fractions from right to left and mixed numbers with the fraction to the left - all in agreement with Arabic customs. Further, he often illustrates non-algebraic calculations in rectangular marginal frames suggesting a lawha.
At this point it would be well to make a few remarks about Fibonacci's fractions. The first thing to note is the format, 1/2 4, which means four and a half. The format is unique to Andalusia and the Maghrib and reflects the Arabic method of writing from right to left, something Fibonacci most probably learned as a student in a Moslem school in Bougie.