Different styles of writing for different situations
The question is, "What is one plus one?"
Here is how to answer in a math class:
Two
Here is how to answer in a science class:
One plus one is two.
Here is how to answer in an English class:
The sum "one plus one" is generally widely accepted to have a definite value, depending, of course, on the number system being used. The ring most commonly used for arithmetic is the real numbers (actually, the real numbers are generally considered as a field because doing so introduces additional useful rules). For the sake of simplicity, however, we will discuss only the integers, following the rule of thumb not to add extra terms which are not needed. In fact, since addition is the only operation in question, the integers shall be treated as a group rather than as a ring. The integers are quite similar to a cyclic group. There is a single element which can be repeatedly added to itself to generate half of the non-zero elements in the group; taking the inverses of these elements gives the remaining non-zero elements. Zero itself is assumed to be part of the group, since there must be an identity element. The near-cyclic structure of the integers means that each of the non-zero elements is equal to the repeating sum of either one or the inverse of one (zero itself is the empty sum, the sum with no terms). Elements which are repeated sums of one are named "positive," while those which are repeated sums of the inverse of one are named "negative." The nth positive element would be the element which is the repeating sum of n ones. When considering the integers as a ring, this also happens to be equal to the product of n and one; since one is the multiplicative identity element, this product is n. Since it would be counterintuitive to have addition function differently when a set is taken as the base of a ring and when the same set is taken as the base of a group, this rule, that the repeating sum of n ones is equal to n, should be preserved in this current inquiry. Since there are two ones in the sum in question, this sum would, according to the above rule, be equal to two. Extending the result to larger groups, or rings, is easy. The process shall start with the rational numbers. One only need note that the group of rational numbers has identical additive structure to the integers, at least with the elements that concern us; it may even be homomorphic to the group of integers, but that is not the subject of this discussion. The only change made to move from the group of integers to the group of real numbers is to add elements, so addition between existing elements is not altered. Drawing on the same rule as before, since the similar structure allows it, the repeating sum of two ones is still equal to two. This process will yield the same result when extended to the rational numbers, or even the complex numbers. It becomes only slightly more complicated when applied to finite groups over some subset of the integers. Since the additive structure is still preserved, one plus one is still two, with a single exception case. In the "trivial" group, which has only two elements, one and zero, the sum cannot be two. Two is not in the group, but additive groups, such as this one, must still be closed under addition. At the very least, convention calls for this sum to be considered equal to zero. A short investigation would show that if it were considered to be equal to one, the only other possibility, one would have no inverse in the trivial group. This too is a violation of one of the group axioms, which declares that all elements in a group must have an inverse in that group. Since zero plus one would not be equal to zero, and one plus one would not be equal to zero, one would have no inverse, which is not allowable. Thus, in the singular case of the group of two elements, commonly called the "trivial group," one plus one is equal to zero, while in all other commonly encountered cases, it is equal to two.
Here is how to answer in a math class:
Two
Here is how to answer in a science class:
One plus one is two.
Here is how to answer in an English class:
The sum "one plus one" is generally widely accepted to have a definite value, depending, of course, on the number system being used. The ring most commonly used for arithmetic is the real numbers (actually, the real numbers are generally considered as a field because doing so introduces additional useful rules). For the sake of simplicity, however, we will discuss only the integers, following the rule of thumb not to add extra terms which are not needed. In fact, since addition is the only operation in question, the integers shall be treated as a group rather than as a ring. The integers are quite similar to a cyclic group. There is a single element which can be repeatedly added to itself to generate half of the non-zero elements in the group; taking the inverses of these elements gives the remaining non-zero elements. Zero itself is assumed to be part of the group, since there must be an identity element. The near-cyclic structure of the integers means that each of the non-zero elements is equal to the repeating sum of either one or the inverse of one (zero itself is the empty sum, the sum with no terms). Elements which are repeated sums of one are named "positive," while those which are repeated sums of the inverse of one are named "negative." The nth positive element would be the element which is the repeating sum of n ones. When considering the integers as a ring, this also happens to be equal to the product of n and one; since one is the multiplicative identity element, this product is n. Since it would be counterintuitive to have addition function differently when a set is taken as the base of a ring and when the same set is taken as the base of a group, this rule, that the repeating sum of n ones is equal to n, should be preserved in this current inquiry. Since there are two ones in the sum in question, this sum would, according to the above rule, be equal to two. Extending the result to larger groups, or rings, is easy. The process shall start with the rational numbers. One only need note that the group of rational numbers has identical additive structure to the integers, at least with the elements that concern us; it may even be homomorphic to the group of integers, but that is not the subject of this discussion. The only change made to move from the group of integers to the group of real numbers is to add elements, so addition between existing elements is not altered. Drawing on the same rule as before, since the similar structure allows it, the repeating sum of two ones is still equal to two. This process will yield the same result when extended to the rational numbers, or even the complex numbers. It becomes only slightly more complicated when applied to finite groups over some subset of the integers. Since the additive structure is still preserved, one plus one is still two, with a single exception case. In the "trivial" group, which has only two elements, one and zero, the sum cannot be two. Two is not in the group, but additive groups, such as this one, must still be closed under addition. At the very least, convention calls for this sum to be considered equal to zero. A short investigation would show that if it were considered to be equal to one, the only other possibility, one would have no inverse in the trivial group. This too is a violation of one of the group axioms, which declares that all elements in a group must have an inverse in that group. Since zero plus one would not be equal to zero, and one plus one would not be equal to zero, one would have no inverse, which is not allowable. Thus, in the singular case of the group of two elements, commonly called the "trivial group," one plus one is equal to zero, while in all other commonly encountered cases, it is equal to two.