Question

The lines below outline the argument that zero is not a positive number. Read each line, then determine the property of the real numbers which will complete line 4 of the argument below:

1. Zero is not a positive number.
2. Choose x as the smallest number.
3. The ________ property tells us that _________.
4. Therefore, between x and 0 there will be a third number, smaller than x.
5. Call this new number x and return to step 2.
6. We never find a smallest number and we never arrive at 0.

Answer 1

The real number property that completes line 4 of the argument about zero not being a positive number is the Density Property, which indicates that between any two real numbers, another number can always be found.

Step-by-step explanation:

The question asks us to identify the property of real numbers that would complete line 4 of the given argument regarding why zero is not a positive number. The appropriate property to use here is the Density Property (also known as the Archimedean Property) of the real numbers. This property states that for any two real numbers, there is another real number that lies between them. Applying this property, line 3 should read:

3. The Density Property tells us that for any two real numbers x and y, with x < y, there exists a third number z, such that x < z < y.

4. Therefore, between x and 0 there will be a third number, smaller than x.

Following this logic iteratively, we conclude that we can infinitely find numbers between zero and any positive number, and hence we never truly find a smallest positive number.