What is mathematics? - Session 2

The Romans, like any other great civilization, needed a way to count their goods, keep track of their victories and build their empire. Instead of using the numbers we know today, they used a unique system of letters known as Roman numerals. Roman numerals were created over two thousand years ago. The Romans developed this system to help them with everyday tasks. This system was simple for them to use because it was based on the Latin alphabet, which they were already familiar with.

In the Roman numeral system, there are seven basic symbols. Each symbol represents a different value.I stands for the number one. V stands for the number five. X stands for the number ten. L stands for the number fifty. C stands for the number one hundred. D stands for the number five hundred. M stands for the number one thousand.

When a smaller numeral comes after a larger one, you add their values together. For example, VI is five plus one, which equals six.When a smaller numeral comes before a larger one, you subtract the smaller value from the larger one. For example, IV is five minus one, which is equal to four.

You can repeat a symbol up to three times to add value. Do not use the same symbol more than three times in a row. For example, instead of writing IIII to represent four, write IV. The numeral II is simply one plus one, which is equal to two. The numeral III is simply one plus one plus one, which is equal to three. The numeral XIII is ten plus one plus one plus one, which is equal to thirteen. Can you tell the value of the numeral VIII?.

The numeral IX is ten minus one, which is equal to nine. The numeral XXIV is ten plus ten plus five minus one, which is equal to twenty four. Here, instead of adding I to X, it is subtracted from V. This is because after I, there is a larger numeral which is V. So we shall first subtract it from V and then add the final value to previous value. Can you tell the value for the numeral XIX?.

For larger numbers, the same rules apply. Here are some examples. The numeral XC is hundred minus ten, which is equal to ninety.The numeral CM is thousand minus hundred, which is equal to nine hundred. What is the value of the numeral XD?.

Zero, a concept so integral to modern mathematics and daily life, was not always a part of the numerical systems. Isn’t that surprising? Its invention revolutionized mathematics. It provided a foundation for complex calculations and the development of various scientific fields. Understanding the history of zero gives us insight into how human thought has evolved over centuries.

In ancient times, many civilizations developed numerical systems for trade, astronomy, and record-keeping. These early systems, such as those used by the Egyptians, Babylonians and Romans did not have a concept of zero as a number. They had symbols to represent quantities. They lacked a way to denote the absence of a quantity.

Egyptians used a system based on hieroglyphs with separate symbols for each power of ten. They did not have any symbol for zero. Babylonians used the sexagesimal system and had a placeholder for zero. It was not a true zero as we understand it today.

Romans used Roman numerals, which lacked any symbol for zero. The concept of zero as both a placeholder and a number, was independently developed in several regions. That is, many people around the world thought it was important, on their own. The most significant development of zero occurred in ancient India around the fifth century AD. Indian mathematicians were among the first to treat zero as a number with its own value and symbol.

Aryabhata was one of the earliest to use a place value system, which is a critical feature of the modern decimal system. This system uses positions to represent the value of digits, making calculations more straightforward and efficient. He did not have a symbol for zero. His work implied the concept of place value, which required the use of zero to denote empty places in numbers.

While Aryabhata's work laid the foundation, it was Brahmagupta who formalized the concept of zero as a number. Brahmagupta explicitly defined zero and provided rules for arithmetic operations involving zero. He treated zero as a number in its own right, distinct from a mere placeholder. Brahmagupta described operations such as addition, subtraction and multiplication involving zero.

For instance, adding zero to a number results in leaving the number unchanged. Subtracting zero from a number results in leaving the number unchanged. Multiplying any number by zero gives zero. The Indians used a dot or small circle to represent zero. This symbol evolved into the zero we use today. The Sanskrit word for zero was śūnya which means empty or void.

Islamic scholars translated and expanded upon Indian mathematical works. Persian mathematician Al-Khwarizmi and Arab mathematician Al-Kindi played crucial roles in disseminating the concept of zero. Al-Khwarizmi's works introduced the Hindu-Arabic numeral system, including zero, to the Islamic world. By the twelfth century, the concept of zero reached Europe through translations of Arabic texts. Italian mathematician Fibonacci helped popularize the Hindu Arabic numeral system in Europe with his book Liber Abaci.

Modern mathematical innovations have revolutionized problem solving across disciplines. Calculus analyzes change in physics and economics. Probability theory and statistics predict outcomes and analyze data in various fields. Number-theory supports cryptography and computer algorithms. Set-theory and abstract algebra organize data and study structures, vital in computer science and physics. Category-theory explains fundamental aspects of mathematics and the world. These advancements continue to improve engineering, social sciences, and beyond, shaping our technological world.