The first number that was truly "invented" by the human mind is the number zero, 0. Many mathematicians would regard zero as the most important number in mathematics, since a number system without a notation for zero is extremely impractical for dealing with very small or very large numbers. Just imagine how to represent one million or one thousandth as a decimal number without having a symbol for the concept of zero.
The development of a place-value system by the Babylonians in ancient Mesopotamia in the second millennium BCE was one of the first historical highpoints of mathematics. In a place-value system, the exact position of a digit within a number is relevant for the value it represents. The Babylonians had no concept or notation for zero, which caused difficulties for many-digit numbers and resulted in severe notational problems. Since their numeral system was a sexagesimal system (base-60), a Babylonian three-digit number D2D1D0 would represent the number D2 · 602 + D1 · 601 + D0 · 600 in the decimal system, but the notational problems that arise when there is no zero digit are essentially the same for all place-value systems. To demonstrate these notational problems, we will consider familiar decimal numbers rather than the Babylonian sexagesimal numbers. If there is no digit 0, how can we, for example, distinguish the number 2018 from 218? The Babylonians indicated the absence of a certain positional value by a space between sexagesimal numerals. This would correspond to writing "2 18" for the number 2018. Clearly, this practice becomes quite problematic when two or more consecutive positions need to be left out, as in the number 1001. Moreover, using spaces for empty positions does not resolve all ambiguities, since we can hardly add a space at the end of a number. Indeed, the Babylonians used identical representations for 1, 60, and 602, which would correspond to writing just "1" for the numbers 1, 10, 100, or any other power of 10. The number that was actually intended always had to be inferred from the context. Around 300 BCE, the Babylonians introduced a placeholder symbol for empty positions, making the notation less ambiguous. Yet this symbol was never used at the end of a number, so the written numbers 1, 60, and 602 still could not be distinguished from one another without knowing the context.
Babylonian clay tablets dating from the second millennium BCE already cover calculations with fractions, algebra, quadratic and cubic equations, and the Pythagorean theorem-more than a thousand years before Pythagoras was born. It might appear strange to us that one of the most mathematically advanced civilizations of their time never developed a concept of zero, especially when we think about the associated notational difficulties the Babylonians had to deal with.
Who invented the zero, then? It is documented that the Greco-Egyptian mathematician and philosopher Claudius Ptolemy (ca. 100–ca. 170) used a symbol for zero within a sexagesimal numeral system in his astronomical writings. However, the symbol usually played the role of a placeholder, and it was only used in the fractional part of a number, that is, the part corresponding to minutes and seconds. The modern concept of zero as an integral number was actually developed in India, together with the decimal place-value system. The oldest clear evidence for a thorough understanding of the role of zero was found in the text Brāhmasphuṭasiddhānta, which was the main work of the Indian mathematician and astronomer Brahmagupta (ca. 598–ca. 665). Some older writings indicate that the number zero was used in India as early as the fifth century CE. In the ninth century, the concept of zero was transmitted to the Islamic culture and later brought to Europe by the famous Italian mathematician Fibonacci (ca. 1170–1250), who in his book Liber Abaci (1202) used the term zephyrum as a translation of the Arabic ṣifr, meaning "empty." The word zephyrum turned into zefiro in Italian and was then contracted to zero in Venetian.
The introduction of zero into the decimal system dramatically simplified calculations with large numbers, thereby making mathematics, in general, a much more practicable and powerful tool. It surely was the most significant step in the development of a number system, and it laid the basis for many great advances in commerce, navigation, astronomy, physics, and engineering. Inventing a notion of zero, and raising this symbol for "nothing" to a number on its own right, requires a considerable mental abstraction that should be ranked among the greatest achievements of the human mind in the ancient world.