# What is the definition of greater than or less than in terms of real numbers?

• mitcho
In summary: The least-upper-bound property states that for any two sequences a and b there is a rational number c such that a<c<b.In summary, the definition of 'greater than (>)' and 'less than (<)' in the real number system is that they are defined as the equivalence relation between a set of greater cardinality and a set of lesser cardinality. The construction of the rational numbers from ZFC set theory is done by starting with the natural numbers N, inducing an ordering on Z, constructing Q, and then constructing R. The definition of 'greater than (>)' and 'less than (<)' in the real number system is based on the equivalence relation between two sets of greater cardinality.
mitcho
A thought just struck me today after watching a lecture on the construction of the rational numbers. What is the definition of 'greater than (>)' and 'less than (<)' in the real number system. The only way I can think to describe it is to make some reference to the Euclidean distance between the number and zero and see which distance is greater or smaller but of course that is just using the term in the definition itself. I also thought about the number of steps it takes to construct the number from ZFC set theory. Again though, you have to have some concept of greater than or less than to determine which took more steps.
Any help would be appreciated
Thanks

mitcho said:
A thought just struck me today after watching a lecture on the construction of the rational numbers. What is the definition of 'greater than (>)' and 'less than (<)' in the real number system. The only way I can think to describe it is to make some reference to the Euclidean distance between the number and zero and see which distance is greater or smaller but of course that is just using the term in the definition itself. I also thought about the number of steps it takes to construct the number from ZFC set theory. Again though, you have to have some concept of greater than or less than to determine which took more steps.
Any help would be appreciated
Thanks

Think in terms of a set of greater cardinality having all sets of lesser cardinality as proper subsets.

One popular construction of the real numbers is to start with the natural numbers N with their natural ordering defined by 0 < n for all natural numbers n.

Then you continue to construct the integers Z = N x N / ~ where ~ is the equivalence relation such that (a,b) ~ (c,d) if a+c = d+b, where n = [(n,0)], and -n = [(0,n)] for positive n. We induce an ordering on Z by [(a,b)] < [(c,d)] if a+d > c+b. Note that this gives the natural order on Z we are used to.

Then we construct Q = Z x (N-{0}) / ~, where ~ is defined by (a,b) ~ (c,d) if ad = bc, and [(a,b)] < [(c,d)] if ad < bc, where < is the order of Z.

Finally, we construct R by looking at the cauchy-sequences of Q^N (i.e. sequences or rational numbers that are cauchy). A sequence (q_n) of rational numbers is cauchy if for every rational number e, there is a natural number N such that |q_n-q_m| < e for all n,m >= N. Let this set of cauchy-sequences be C.

We define R = C /~ where (q_n) = (p_n) if the sequence (q_n -p_n) converge to 0. The order of R induced is defined as [(q_n)] < [(p_n)] if there is a rational number e such that there exists a natural number N such that p_n-q_n >= e for all n >= N. It will require a proof of that this in fact is a well-defined ordering, but when you do that it will be the natural ordering of R we are used to.

From this definition of R we can prove all the known axioms of R we need, most importantly the least-upper-bound property.

## 1. What is the meaning of "greater than" in terms of real numbers?

The phrase "greater than" refers to a comparison between two real numbers, where the first number is larger than the second number.

## 2. How is "greater than" represented in mathematical notation?

In mathematical notation, the symbol ">" is used to represent "greater than". For example, 5 > 3 means "5 is greater than 3".

## 3. What is the definition of "less than" in terms of real numbers?

The term "less than" describes a comparison between two real numbers, where the first number is smaller than the second number.

## 4. How is "less than" expressed mathematically?

In mathematical notation, the symbol "<" is used to represent "less than". For instance, 2 < 6 means "2 is less than 6".

## 5. Can a number be both "greater than" and "less than" another number at the same time?

No, a number cannot be both "greater than" and "less than" another number at the same time. A number is either greater than, less than, or equal to another number.

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