How does the well-ordering principle guarantee the existence of a least positive integer linear combination for integers a and b not both zero?
Conditions
- Integers a and b are not both zero.
- The set consists of positive values of the form au+bv where u and v are integers.
Reasoning, step by step
- Form the set of all positive values generated by the linear combination au+bv.
- Verify that this set is nonempty because a and b are not both zero.
- Apply the well-ordering principle, which states that every nonempty set of positive integers has a least element.
- Conclude that there exists a smallest positive value d in this set, which is itself an integer linear combination.
Example
The script states: 'For integers a,b not both zero, positive values au+bv form a nonempty set. The well-ordering principle gives a smallest value d, itself an integer linear combination.'
Common misconceptions
- Believing that the set of linear combinations might be empty if a and b are large.
- Thinking that the well-ordering principle applies to sets of negative integers or real numbers.
- Assuming that the least positive combination is always equal to a or b.
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