The question "What are the sum of the positive?" is grammatically incomplete and open to multiple interpretations. To provide a comprehensive and exact answer, we will rephrase it to "What is the Sum of All Positive Numbers?", considering both finite and infinite sets of positive numbers.
In conventional mathematics, the sum of an infinite series of positive numbers, such as all positive integers, generally diverges to positive infinity (∞).
Understanding the Sum of Positive Numbers
The concept of "the sum of the positive" depends heavily on whether you are referring to a finite collection of numbers or an infinite sequence.
1. Sum of a Finite Set of Positive Numbers
When you add a finite number of positive values, the sum will always be a positive and finite number. This is a fundamental property of positive numbers: if every number in a set is greater than zero, their total sum must also be greater than zero.
For example:
- The sum of 1, 5, and 10 is 16.
- The sum of 0.5, 1.2, and 3.7 is 5.4.
As stated by mathematical principles, if all the numbers you are adding are positive, then the sum of those numbers will be positive. If you are adding a finite number of positive numbers, the sum will be a finite positive number.
2. Sum of an Infinite Set of Positive Numbers
This is where the concept becomes more nuanced.
a. Sum of All Positive Integers (1 + 2 + 3 + 4 + ...)
When considering the sum of all positive integers (1, 2, 3, 4, and so on, extending infinitely), the standard mathematical interpretation is that the sum diverges to positive infinity (∞). This means that as you add more and more positive integers, the sum grows without bound and never reaches a finite value.
This is classified as a divergent series because its partial sums do not approach a finite limit.
b. Advanced Mathematical Interpretations (Ramanujan Summation)
While the conventional sum of positive integers diverges to infinity, in certain advanced mathematical contexts, such as Ramanujan summation or Zeta function regularization, a finite value can be assigned to this series. The most famous result is that the sum of all positive integers is assigned the value -1/12.
It is crucial to understand that this result is not the standard sum in everyday arithmetic or calculus. It arises from specific methods of assigning values to divergent series, which are used in theoretical physics (e.g., string theory) and advanced number theory, but do not imply that adding positive numbers in a conventional sense results in a negative value.
c. Sum of All Positive Real Numbers
The set of all positive real numbers (any number greater than 0, including fractions, decimals, irrational numbers) is uncountably infinite. Attempting to sum all positive real numbers also results in a sum that diverges to positive infinity (∞). There is no finite sum for such an infinitely dense set.
Conclusion
In summary, the "exact answer" to the sum of positive numbers depends on the context:
- For any finite collection of positive numbers, the sum is always a finite positive number.
- For an infinite series of positive numbers (like all positive integers or all positive real numbers), the sum conventionally diverges to positive infinity (∞). While special mathematical methods can assign a finite value like -1/12 in specific contexts, this is not the general or conventional sum.
