Whether the set of integers Z includes 0 is a foundational question in mathematics. The integers are the numbers used for counting, and comprise the positive natural numbers (1, 2, 3, …), their negatives (-1, -2, -3, …), and zero. So the question arises – is zero itself an integer? This article will examine the definition and properties of the integer set Z, look at historical and mathematical arguments for and against including zero, and provide a definitive answer supported by modern mathematical conventions.
Definition and Properties of Z
The set of integers Z is defined formally as:
Z = {…, -3, -2, -1, 0, 1, 2, 3, …}
The key properties of Z are:
- Closed under addition: The sum of any two integers is an integer.
- Closed under multiplication: The product of any two integers is an integer.
- Additive identity: 0 is the additive identity, meaning a + 0 = 0 + a = a for any integer a.
- Additive inverses: Every integer a has an additive inverse -a, such that a + (-a) = (-a) + a = 0.
These closure and identity properties are required for Z to form a mathematical ring. The inclusion of 0 in the definition and its role as the additive identity are key to these properties holding true.
Historical Perspectives on Zero as an Integer
The idea of zero as a number representing nothingness and acting as an additive identity developed gradually over thousands of years. Some key historical perspectives include:
- Early civilizations like the Babylonians, Mayans, and Indians used a placeholder symbol to represent the absence of a value in their positional notation number systems. This evolved into the modern zero digit.
- The ancient Greeks debated extensively about whether nothing could be considered a number. Pythagoras and his followers viewed 1 as the origin of numbers and did not accept zero.
- In the 7th century, the Indian mathematician Brahmagupta formally defined zero in his treatise Brahmasphutasiddhanta, including key properties like 0 + a = a for any number a.
- Persian and Arab mathematicians expanded on Brahmagupta’s work, developing the modern Hindu-Arabic decimal system which spread to Europe in the Middle Ages.
- European mathematicians were initially distrustful of the concept of zero, viewing it as occult and referring to it as “the devil’s invention.” But by the 17th century, zero was universally accepted in Europe.
So while zero was controversial historically, it was eventually accepted as both a number and the additive identity element in almost all mathematical traditions.
Arguments For and Against Zero in Z
Looking at the set Z = {…, -3, -2, -1, 0, 1, 2, 3, …}, there are reasonable mathematical arguments on both sides of whether zero should be considered an integer:
Arguments for including zero in Z:
- Zero is needed for the closure property under addition. If zero was not in Z, sums like 1 + (-1) would not be integers.
- Zero serves as the additive identity needed for Z to be a ring.
- Zero is necessary to have additive inverses and fulfill a – a = 0 for all integers a.
- Zero acts as a separator between the positive and negative integers.
- In the standard ordering of integers with
Arguments against including zero in Z:
- Historically, zero was viewed as different in nature than the “counting numbers” 1, 2, 3, …. It represents an absence, not a concrete quantity.
- In some sense, zero lies outside the positive and negative integers rather than between them.
- When extending Z to the integers modulo n, for prime n the cosets are {1, 2, …, n-1}. Zero is not included.
- Under the Peano axioms for natural numbers, zero is not the successor of any natural number.
- In set theory, specifying zero as an element of Z is redundant since it can be derived as the additive identity.
So there are reasonable perspectives on both sides of this debate. Including zero in Z does provide key mathematical properties, but zero also has unique properties setting it apart from the other integers.
Modern Conventions on Zero and Z
While historical opinions were mixed, modern mathematicians almost universally agree that zero should be included in the set of integers Z. The key points supporting this consensus are:
- Mathematical convenience – Defining integers as Z = {…, -3, -2, -1, 0, 1, 2, 3, …} streamlines proofs about their properties.
- Zero is essential for the ring properties of addition/multiplication and the well-ordering of Z.
- All modern set-theoretic, algebraic, and number system approaches explicitly specify zero as an integer.
- Zero has clear analogues to the other integers when extending Z to new algebraic structures like rational numbers.
- Applied mathematics requires zero to represent quantities like zero velocity or a zero change in a variable.
While zero has unique properties, it fits most cleanly within the integer framework compared to excluding it or creating a special “zeroth” number system separate from Z. This provides advantages in both pure and applied mathematics.
Conclusion
In summary, while the status of zero was ambiguous historically, the modern mathematical consensus is clear – zero is included in the set of integers Z. Key advantages of defining Z = {…, -3, -2, -1, 0, 1, 2, 3, …} include:
- Preserving ring properties like closure under addition/multiplication.
- Retaining zero as the additive identity element.
- Keeping a well-ordered set with less than properties and additive inverses.
- Aligning with algebraic, set theoretic, and applied perspectives on zero as an integer.
So in the end, despite some unique properties, zero has become universally accepted as a full-fledged integer. The set Z would simply not function as cleanly without including zero among the counting numbers, their negatives, and zero itself anchoring the integers between their positive and negative values. Both pure and applied math require zero to be part of Z.