Middle Eastern Maths

For a number we take for granted as an essential in modern day mathematics, 0 hasn’t always been a part of maths, especially European maths. Today it has two main purposes in mathematics. One of these is to be a place holderinnumberstomakesurethatotherdigitsfallinthe right place to have the right value. For example, without the 0 in 206, you would get 26, which is a number a lot smaller. The other, more abstract use of 0 is when it is referringtotheconceptoftherebeingnothing.

Thefirstoftheseideascanbedatedbacktotheancient Babylonians(inmodern-dayIraq).Theyhadusedaplacevalue number system and had been doing so since at least 1700 BC, although they used cuneiform numbers, and they used a base-60 system rather than a base-10 one. Additionally, it wasn’t until around 400BC that they found it necessary to use placeholders in the numeral system, and these took numerous forms, such as 2 wedges or 3 hooks. However, it was Greek mathematicianslikePtolemywhostartedtointroducethe current notation as they used a 0 as a placeholder in his treatise Almagest, along with the ancient Babylonian numerals. In fact, even the Mayans had a place-value numeralsystem(althoughagainusingabase-20system). In comparison, few intellectuals in Europe knew of this innovation, and everyone else clung on to old Roman numerals, which were very inconvenient for larger arithmeticoperations,includingmultiplication.

For the use of 0 as standalone number of his own, Leonardo Da Pisa, more famously known as Fibonacci, is often credited. He is most well-known for the Fibonacci sequence of numbers, and helped to introduce 0 and our modern numeral systems to the Europeans with his book Liber Abbacci. However, the use of 0 in this way can be traced back far before him and all the way to the Indian sub-continent. In the early 800s during the reign of the caliph Al-Ma’mun (caliphs were the political and religious leaders of the Islamic empire, considering themselves to be successors of the prophet Mohammed), he founded the House of Wisdom in Baghdad, a great library and intellectual centre, which contained translations of works from Greece to India. Therefore, by the time Fibonacci arrived in the Middle East in his 20s in the late 1100s, scholars in the Middle East had already known about the decimal system and Hindu-Arabic numerals for centuries before, mostly from the works of Indian mathematicians like Brahmagupta, who is from what historians know, the first person to try and extend the numeral system beyond positivenumbersallthewayisthe600s.

Initially, however, there were some troubles with how 0 should be used in arithmetic calculations, especially with division with Brahmagupta initially writing that 0 divided by is 0, which is not the case. Mahavira wrote 200 years later that a number remains unchanged when divided by 0, while Bhaskara wrote another 300 years later that a number divided by 0 equals infinity- and both claims were incorrect. Despite this though, these mathematicians had made great process in introducing 0 negative numbers to our modern numeral system.

Liber Abbaci, which first appeared in 1202, was primarily based on the work of the 9th century mathematician Al-Khwarizmi, who had presented the first way to systematically solve quadratic equations. For this reason, he is known as the father of algebra, a word which itself is derived from the Arabic word al-jabr, meaning “the restoration of broken parts”. His work also introduced the decimal system, where we use 10 digits with their valued determined by the position, from India to the Middle East. This system required a place holder for positions that didn’t have a value, and 0 fulfilled this role as a place holder. Furthermore, Al-Samawal, another Islamic mathematician in the later 12th century, was describing how “if we subtract a positive number from zero the same negative number remains… if we subtract a negative number from zero the same positive number remains.”

Fibonacci never went as far as those mathematicians in the Middle East or India before him, as he never treated 0 as a number in the same way as 1, 2, 3 or 4, referring to it as a symbol. In fact, even after this, it took centuries for the use of 0 to become widespread and standard in European mathematics, with even mathematicians in the 1500s, such as Cardan, omitting 0. It took centuries of work in India and the Middle East before Fibonacci even introduced it to Europe to lay the foundations of how mathematics, which included 0, worked. Furthermore, the work of the House of Wisdom should not be understated. Although little remains from the academy which was destroyed back in the 13th century, what does remain shows evidence of plenty of work to translate works into Arabic and some historians consider it close to a true academy. This is a great contrast to Europe, which following the collapse of the Western Roman Empire, was relatively intellectual behind the global curve, contributing to the name sometimes given to the centuries in the latter half of the first millennium, the Dark Ages. For something so fundamental as 0 in our modern-day mathematical system, it isn’t as old as you might have expected, and the story of its development take place across the world over hundreds of years.