No, infinity and plus infinity are not strictly the same, although the distinction depends on the context. While "infinity" often refers to positive infinity, it can also represent negative infinity or other forms of infinity depending on the number system or mathematical context.
Understanding Infinity
Infinity (symbol: ∞) represents a quantity or concept that is without any bound or end. It is not a real number but rather a concept used to describe limits and behaviors in mathematics.
Plus Infinity (+∞)
Plus infinity (+∞) specifically denotes the positive unbounded quantity. It is used when a quantity is increasing without limit in the positive direction.
Key Differences and Context
The apparent similarity between "infinity" and "+infinity" stems from common usage. Here's a breakdown:
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Real Numbers: In the context of the real number line extended with a single infinity, 'infinity' typically means positive infinity. So, in many informal discussions, "infinity" is used synonymously with "+infinity".
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Extended Real Numbers: The extended real number line includes both positive infinity (+∞) and negative infinity (-∞). In this context, +∞ is distinct from -∞ and the term "infinity" by itself might be ambiguous or require qualification, potentially referring to either +∞ or the absolute value of arbitrarily large numbers.
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Complex Numbers: In the complex plane, "infinity" often refers to a single point at infinity, approached by going infinitely far in any direction. There isn't a notion of "+∞" or "-∞" in the same way as with real numbers.
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Ordinal Numbers and Cardinal Numbers: In set theory, there are different sizes of infinity (different cardinalities). For example, the cardinality of the set of natural numbers (ℵ₀) and the cardinality of the set of real numbers (c) are both infinite but distinct. The concept of "plus" doesn't directly apply to cardinal numbers in the same additive sense.
Why the Distinction Matters
The distinction between different types of infinity (positive, negative, complex, cardinal) is crucial for mathematical rigor and avoiding contradictions. For example:
- The limit of
1/xasxapproaches 0 from the positive side is +∞, while the limit asxapproaches 0 from the negative side is -∞. - In calculus, improper integrals may converge to a finite value, diverge to +∞ or -∞, or simply diverge without approaching any specific value.
- When dealing with transfinite numbers in set theory, addition must be carefully defined and has different properties than addition of finite numbers.
Example: Adding Infinity
The statement "Infinity plus infinity is still infinity," as quoted in the references, is a simplified way of thinking about limits and cardinalities. Specifically:
- If you have an infinite set and add another infinite set to it, the resulting set is still infinite (assuming both sets are countably infinite).
For example, if you have an infinite number of odd numbers and an infinite number of even numbers, combining them yields an infinite number of integers. Similarly, ℵ₀ + ℵ₀ = ℵ₀. However, as noted above, addition of transfinite numbers needs to be defined carefully.
Conclusion
While the term "infinity" is often used informally to mean positive infinity (+∞), particularly in basic calculus, it's important to recognize that infinity is a broader concept with different interpretations in different mathematical contexts. The distinction between +∞, -∞, and other forms of infinity is essential for rigorous mathematical reasoning.
