Write 4–6 sentences explaining why it is important to have precise definitions in mathematics. Choose five terms, some from geometry and some from other areas of mathematics, and give a precise definition of each. Then change each of those definitions so that it still describes the term but not precisely enough to avoid ambiguity. Formulate counterexamples, with sketches when feasible, that show how things could satisfy the less precise definitions but not match the intended terms.
Having precise definitions in mathematics is crucial because it ensures clear communication and understanding among mathematicians and prevents ambiguity or misinterpretation. Without precise definitions, mathematical concepts can become vague, leading to confusion and erroneous conclusions.
Here are five terms, including some from geometry and others from different areas of mathematics, with both precise definitions and modified imprecise definitions:
1. Triangle (Geometry):
Precise Definition: A polygon with three sides and three angles.
Imprecise Definition: A shape with three sides and three angles, but the angles may or may not add up to 180 degrees.
Counterexample: A shape with three unequal sides and three angles, where the angles add up to less than 180 degrees. Here, the modified definition allows for non-Euclidean triangles like those found on the surface of a sphere.
2. Polynomial (Algebra):
Precise Definition: An expression consisting of variables, coefficients, and mathematical operations such as addition, subtraction, multiplication, and exponentiation.
Imprecise Definition: An expression consisting of variables, coefficients, and mathematical operations, but it may also include roots, logarithms, or trigonometric functions.
Counterexample: A function that includes an irrational root or a logarithmic term. By introducing additional functions beyond the basic operations, the modified definition blurs the boundaries of polynomials.
3. Prime Number (Number Theory):
Precise Definition: A positive integer greater than 1 that has no positive divisors other than 1 and itself.
Imprecise Definition: A positive integer greater than 1 that has no immediate divisors other than 1 and itself, but it may have other divisors at a higher level.
Counterexample: A number that is divisible by a prime number other than 1 and itself at a higher level. For example, 6 is divisible by 2 (which is a prime number), although it doesn't appear as an immediate divisor in the modified definition.
4. Function (Calculus):
Precise Definition: A relation that assigns each element of one set (the domain) to a unique element of another set (the codomain) in such a way that each element of the domain corresponds to exactly one element of the codomain.
Imprecise Definition: A relation that assigns elements of one set to elements of another set, but it may not be one-to-one or onto.
Counterexample: A relation where an element in the domain corresponds to multiple elements in the codomain or vice versa, violating the requirement of uniqueness. The modified definition allows for functions that are not well-defined, leading to potential inconsistencies in mathematical reasoning.
5. Limit (Analysis):
Precise Definition: A fundamental concept in calculus that describes the behavior of a function as it approaches a particular value or point.
Imprecise Definition: An approximate description of how a function behaves around a specific value or point.
Counterexample: A function that diverges or oscillates instead of approaching a particular value, as described by the precise definition. The modified definition allows for more general behaviors that do not align with the intended meaning of a limit.