# The least upper bound property and the irrationals.

• RediJedeye
In summary, the conversation discusses whether irrational numbers have the least upper bound property. It is determined that the set of irrationals does not satisfy this property, as there is always a rational number that serves as a least upper bound. The concept of limit points and completeness is also mentioned in relation to the LUB property.

#### RediJedeye

Hi

Does anybody know if the irrational numbers have the least upper bound property?

RediJedeye said:
Hi

Does anybody know if the irrational numbers have the least upper bound property?

The set of irrationals less than zero is nonempty (it contains -pi, for example) and is bounded above (by pi, for example) yet has no least upper bound. So the irrationals do not satisfy the LUB property.

Of course zero is a LUB for that set in the reals, but 0 is not irrational. That's the beauty of the LUB concept. It encapsulates the intuition of there being "no holes" in a given set.

Cool that makes perfect sense thanks for the help.

To generalize SteveL's answer, for any rational q,

consider the intervals (-oo,q) . Notice that the LUB of a subset of real numbers is

a limit point of that set S . So if a subset S of R does not contain all its limit points you

can constructuct a subset of S that does not contain its LUB-- so that closed subsets

contain their LUB's. Think of the relation with completeness of a set...

I can confirm that the irrational numbers do have the least upper bound property. This property states that for any non-empty subset of real numbers, there exists a least upper bound (or supremum) that is also a real number. This means that for any set of irrational numbers, there will always be a smallest number that is greater than or equal to all the other numbers in the set. This is one of the defining characteristics of the real numbers, and it applies to both rational and irrational numbers. So yes, the irrational numbers do have the least upper bound property.

## 1. What is the least upper bound property?

The least upper bound property is a mathematical concept that states that any non-empty set of real numbers that is bounded above must have a least upper bound (also known as a supremum). This means that there is a smallest number in the set that is greater than or equal to all other numbers in the set.

## 2. How is the least upper bound property related to the irrationals?

The least upper bound property is closely related to the irrationals because the set of irrational numbers is unbounded. This means that for any irrational number, there is always a larger irrational number. The least upper bound property helps to define the real numbers as a continuous and complete number system, which includes both rational and irrational numbers.

## 3. What is an example of a set of numbers that does not have the least upper bound property?

An example of a set of numbers that does not have the least upper bound property is the set of rational numbers. This is because there are gaps in the rational number line, where there is no rational number that can serve as the least upper bound for the set.

## 4. How is the least upper bound property used in mathematical proofs?

The least upper bound property is often used in mathematical proofs to show the existence of a supremum for a given set of numbers. It is also used to prove the completeness of the real numbers as a number system, which is an important concept in advanced mathematical analysis.

## 5. Can the least upper bound property be generalized to other number systems?

Yes, the least upper bound property can be extended to other number systems, such as complex numbers and p-adic numbers. However, the specific definition of the least upper bound may differ depending on the number system being considered.