The Arabic numerals (0, 1, 2, 3, 4, 5, 6, 7, 8, 9…) that we still use today were introduced to Europe at some point around the 12th century. They were introduced via Arab culture, hence the name Arabic Numerals (although they actually originated in India), and were introduced by Italian Mathematician Leonardo of Pisa.
(Side Note: Leonardo of Pisa is more commonly known as Fibonacci, who discovered Fibonacci’s sequence – the sequence where each item is the sum of the two previous items, so 0, 1, 1, 2, 3, 5, 8 etc.. Initially this sequence seemed inconsequential, but over time we’ve realised it has applications in lots of areas of Mathematics including nature, cryptography and statistics.)
However, the history of zero is much more complex than the other Arabic numerals. To understand it’s history, we need to understand how it developed as a concept. When you think of one hundred written down, or one thousand, or ten thousand, the image in your mind is likely a digit followed by several zeros, and we can see that adding or taking away a zero from the end of the number would change the amount drastically. In this case, zero acts as a placeholder, representing no digits in a particular column. We call this column numbering system positional notation. Still used today, positional notation is a system of expressing numbers where digits are arranged in succession, with each position having a place value – e.g. 1s, 10s and 100s etc.
There is some dispute about the oldest evidence of the placeholder zero, but it’s widely believed to come from the Sumerians roughly 5,000 years ago. Found scratched into the wall of a cave, this placeholder consisted of two slanted wedges, used to denote the lack of a number in its place. The Sumerians were the first to develop a numbering system to keep a record of goods and cattle using positional notation.
This placeholder became more commonplace as positional notation become prevalent in Babylonian society. The symbol developed as the concept passed through different cultures, with the Babylonians displaying zero with two slanted wedges, the Mayans with an eye-like character, the Chinese with the open circle we now use for zero, and the Hindus with a dot, but the placeholder concept remained constant and became part of the Arabic Numerals that we still use today.
The other part of the history of zero is the number zero as its own concept, instead of as a placeholder. Formally defined as the average of 1 and – 1, the number zero is closely linked to the philosophical idea of nothingness. Marcus du Sautoy, a University of Oxford Mathematics Professor, is quoted as saying “[T]he creation of zero as a number in its own right, which evolved from the placeholder dot symbol found in the Bakhshali manuscript, was one of the greatest breakthroughs in the history of mathematics.” Some philosopher’s think of nothingness as whatever’s left when you take away the entire universe, and all the laws governing it. This idea ties into different religions, with emptiness being a key idea in Mahayana Buddhism, and being referenced in the King James Bible (Genesis 1:2) with ‘The Earth was without form, and void.’
Perhaps equally interesting to the development of the concept of zero are the reasons the concept of zero took such a long time to develop. As the idea developed, it was met with a lot of resistance and suspicion, with multiple cultures linking it to dark magic. The Catholic Church branded zero as Satanic, because if God was in everything then the idea of nothingness was godless. When zero was brought to Europe in the 12th century with the other Arabic Numerals, it wasn’t received warmly, with the Italians in particular very suspicious of changing their ancestors way of numbering. In fact, in 1259 a law was passed that banned accountants from using zero or any new Arabic numerals, and in 1299 all Arabic Numerals were banned in Florence.
Interestingly, zero bypassed both the Ancient Greeks and the Romans numbering systems completely. The Ancient Greeks were aware of the concept of zero but did not consider it to be a number, with Aristotle dismissing it since it couldn’t be used in division. Similarly, zero did not have its own Roman Numeral. In 525, Dionysius Exiguus was known to use Nulla alongside Roman Numerals, and N, the initial of Nulla, was used in 725 in a table of Roman Numerals. The reason the Romans didn’t have a Roman Numeral for zero was because the Romans never used their numerals for arithmetic, hence avoiding the need for a placeholder zero. Instead, addition and subtraction were completed with an abacus or counting frame.
Without zero, the Romans had to rely on Roman Numerals, a decimal system that was developed as early as 800 BCE. Roman Numerals consisted of letters of the alphabet, combined in different orders to represent different numbers. Some of the letters used were:
- I to represent 1
- V to represent 5
- X to represent 10
- L to represent 50
- C to represent 100
- D to represent 500
- M to represent 1,000
They didn’t write more than three of the same letter in a row, so while 1 would be written as I, 2 written as II and 3 written as III, 4 would be written as IV (i.e. 5 – 1) as they couldn’t write it as IIII. Although this numbering system served its purpose, being effectively used for trade and record-keeping, if we look at a longer number such as three thousand seven hundred and thirty, the Roman Numeral MMMDCCXXX is far more convoluted than the positional notation number 3730, which is reliant on the zero placeholder.
Although today zero is more commonly accepted and used, it still has some interesting properties. As a number that can be divided by two with no remainder, zero is classified as an even number, but zero is neither positive nor negative. (Weirdly, this means we can say that + 0 = – 0). In fact, in most cultures, zero was identified as a concept before the concept of negative numbers existed. Since zero can be divided by anything, zero is said to have an infinite number of factors.
Zero has many modern day uses in areas including Mathematics, Physics and Computer Science. In Mathematics, there are four main elementary rules to consider when working with zero, as follows.
- Addition: x + 0 = 0 + x = 0
- That is to say that zero is an identity element or neutral element with respect to addition, because adding zero to any number has no effect.
- Subtraction: x – 0 = x and 0 – x = – x.
- Multiplication: x * 0 = 0 * x = 0
- Division: 0 / x = 0 for any non-zero x. However, x / 0 is undefined.
- Powers: x 0 = 1 and 0 x = 0.
In Physics, zero is the lowest possible state for many physical quantities. For example, the Kelvin scale is commonly used in Physics to measure temperature, with zero being used to represent absolute zero, the coldest possible temperature. Similarly, the zero-point energy of any physical system is the ground state of the system, and is the lowest possible energy the physical system could possess i.e. no energy.
In Computer Science, binary is a numbering system that is the primary language of computers. Binary consists solely of the numbers one and zero, where zero represents off while one represents on. For a computer or logical system, this could mean zero = ‘electricity is not allowed to flow’ while one = ‘electricity is allowed to flow’.
The history of the number zero is just one of countless fascinating areas of Mathematics and Mathematical history that aren’t normally explored in schools. Tutoring can be used to explore academic ideas more thoroughly and to learn about subjects in more depth. This can enrich a student’s education, and engage them while inspiring genuine interest in academic areas of study.
When in a tutoring session, the student has the benefit of one to one learning with a tutor who’s passionate about their subject, and this creates the perfect environment for a student to explore areas that interest them and to explore areas not normally covered by the national curriculum. This can provide further explanation to help your child academically, and creates a session tailored specifically to your child’s areas of understanding and interest. The tutors at We Make Academics have hundreds of hours of maths tutoring experience, so feel free to get in touch!