You may have used zero in simple arithmetic, but have you ever stopped to think about its history and why we cannot divide by it? Zero is one of the simplest and yet enigmatic numbers in mathematics. It represents nothing, but its discovery revolutionised mathematical thinking and calculation.
Explore the history of zero, its applications, and the mathematical paradoxes that arise when we attempt to use it in certain calculations.
Where does zero come from?
Zero as a concept is a new development in the broad history of mathematics. The ancient civilizations such as the Babylonians, Egyptians, and Romans had no digit specifically to represent zero. They utilised placeholders to indicate that there was no value, such as creating a gap within their number system. The Babylonians, for instance, used a special symbol to divide up numbers in their positional notation system but used no standalone sign for zero.
It was in ancient India that zero came to be established as an independent number. Indian mathematician Brahmagupta (c. 598–668 CE) formally defined zero and provided rules for arithmetic operations with zero in his seminal book Brahmasphutasiddhanta. He gave zero the definition of being the product of subtracting a number from itself and laid the foundation for advancements in algebra to follow. His efforts also set forth rules such as how zero reacts with negative numbers and how it performs under multiplication.
The employment of zero spread to the Islamic world through scholars such as Al-Khwarizmi and Al-Samawal, who further built upon its usage in calculations. Al-Khwarizmi, also called the father of algebra, contributed to the standardization of the use of zero in equations, leading to the development of algebraic expressions. By the 12th century, Fibonacci brought zero to Europe via the Hindu-Arabic numbering system. Zero then became an integral aspect of mathematical notation and calculation, eventually leading to making modern mathematics and science possible.
Using Zero
Zero is a cornerstone of modern mathematics and computing. It plays a crucial role in various branches of mathematics, including algebra, calculus, and number theory. Here are some key areas where zero is indispensable:
- Place Value System: Zero is essential in the decimal place value system, allowing us to differentiate between numbers such as 40 and 400.
- Algebra: Zero is an identity element of addition (a + 0 = a) and is responsible for the formulation of solutions to equations.
- Calculus: The concept of calculus limits and derivatives relies on arriving at values arbitrarily close to zero.
- Computing: Binary coding, which is the basis for modern computing, is based around zero and one (0s and 1s) so that digital technology can function.
- Physics: Most physical laws and formulas employ zero as a reference, for example, absolute zero in thermodynamics.
- Probability and Statistics: Zero is crucial in the context of defining probability, where a probability of zero indicates an event is impossible.
- Finance and Economics: Zero is important in financial computations, interest rates, and balance sheets in that they enable firms to monitor gains and losses effectively.
Zero is especially important in modern electronics and digital technology. In binary code, which underlies all modern computing systems, everything is represented in terms of only two digits: 0 and 1. This binary system is the foundation of digital logic circuits, microprocessors, and memory storage.
Logic gates, the cornerstone of digital electronics, operate on binary states, with zero often standing for the “off” state and one for the “on” state. Without zero, we could never efficiently process information, store data in memory, or execute complex calculations in computers today. From simple calculators to advanced artificial intelligence, the power of zero makes it all possible.
The problem with dividing by zero
While it is useful, zero also comes with its issues, particularly when dividing. Division by zero is undefined in elementary arithmetic. Why is that the case?
“Black holes are where God divided by zero” – Albert Einstein
Take an easy division question: 10 divided by 2 is 5 because 5 times 2 is 10. But if you attempt to divide 10 by zero, then you would have to come up with a number that, when multiplied by zero, is 10. Because any number multiplied by zero is zero, there is no answer. It causes mathematical inconsistencies and is the reason why division by zero is undefined.
Mathematicians have tried to expand our knowledge of division by zero. For instance, in calculus, limits may get arbitrarily close to zero but never divide by it. In abstract algebra and some extensions of mathematics, mathematicians have tried to redefine division by zero but with the end result being inconsistency. In some mathematical theories, such as the extended complex plane, zero can be treated differently, but it is used more as a mathematical curiosity than as an applied tool.
Zero is not just a mathematical concept; it also has a powerful presence in physics and nature. The absolute zero, for example, is the lowest temperature, at which molecular motion is brought to a halt. In black hole physics, singularities include dividing by zero, leading to undefined values in general relativity.
In addition, zero is significant in measuring potential energy, electrical charge equilibrium, and quantum mechanics. In quantum physics, zero-point energy refers to the lowest possible energy a quantum mechanical system can have, demonstrating that there is still activity at zero.
Zero is infinite
While zero represents emptiness, it has an infinite number of possibilities somehow contained within it. In set theory, the empty set (no elements) is an axiomatic framework employed to define numbers. In physics, black holes make our understanding of zero problematic because division by zero is possible in equations for singularities. In calculus, the limit as one approaches zero is key in defining instantaneous rates of change.
Zero also raises philosophical questions. If zero represents nothing, how can it be something? Philosophers and mathematicians have long debated the concept of nothingness, with some arguing that zero is more than just a number – it is a concept that reshapes our understanding of reality. The very nature of zero forces us to reconsider the idea of existence and non-existence.
A paradox and a necessity
Zero is a powerful number that denies its seemingly rudimentary looks. It is the backbone of modern numeral systems, central to scientific advancements, and yet the source of paradoxes in mathematics. From its humble inception in ancient India to its necessity in computing, electronics, and physics, zero remains to be one of the most fascinating numbers in mathematics.
The next time you see zero on a calculator, stop and consider how much of a foundational impact this ‘nothing’ has had in shaping all of mathematics.