Exploring Integer Pairs That Sum to 4

When considering the sum of two integers totaling 4, it is surprising to note the vast number of possible pairs that meet this criterion. This exploration begins by understanding the fundamental properties of integer pairs that add up to a specific sum, in this case, 4. By diving into examples and deriving a general formula, we will demonstrate that there are infinitely many pairs of integers that can sum to 4.

Examples of Integer Pairs

Let's start by listing some examples of integer pairs that sum to 4:

Notice a pattern? Each pair has a first integer, and the second integer is simply the difference between 4 and the first integer. For instance, if the first integer is -2, then the second integer must be 6 to ensure the sum is 4.

General Formula and Proof

Mathematically, if one integer is represented as x, the other integer can be expressed as 4 - x. This relationship means that for any integer value of x, there is a corresponding integer 4 - x that satisfies the equation x 4 - x 4.

To prove that this relationship holds for all integers, consider choosing any integer n. The integer m where m 4 - n will indeed satisfy the condition n m 4. Let's verify this for a few values:

Example Verification

Let n 3. Then we have m 4 - n 4 - 3 1. So, the pair (3, 1) does indeed sum to 4.

Similarly, if n -25, then m 4 - n 4 - (-25) 29. Therefore, the pair (-25, 29) also satisfies the condition.

Conclusion

In summary, the sum of two integers being 4 means that there are infinitely many pairs of integers that can satisfy this condition. For any integer n, the integer m 4 - n ensures that n m 4. This exploration into the integer pairs provides insight into the properties and relationships between these pairs, highlighting the infinite possibilities and the elegance of basic algebra.