Zero And The Numberline

petrushkagoogol · February 2, 2017

What is the significance of the following :

0 can be expressed as the average of 2 infinite series from 1 to + infinity and -1 to - infinity.

Any other integer (positive or negative) is expressed as the average of it's 2 adjacent numbers eg) 0 and 2 average gives +1

Is this unique property advantageous or should we think of 0 as the average of +0 and -0 (which become a necessary requirement thereby)? :sherlock:

OceanBreeze · February 3, 2017

What is the significance of the following :

0 can be expressed as the average of 2 infinite series from 1 to + infinity and -1 to - infinity.

No, I don’t believe it can be so expressed because ∞ - ∞ is not equal to zero, but in fact is left undefined.

The reason is simple: start by assuming ∞ - ∞ = 0, and since (∞ + ∞) = ∞, then (∞ + ∞) - ∞ = 0, and from our starting assumption that ∞ - ∞ = 0, we arrive at ∞ + 0 = 0, or simply ∞ = 0. Now you can see why ∞ - ∞ is undefined!

Any other integer (positive or negative) is expressed as the average of it's 2 adjacent numbers eg) 0 and 2 average gives +1

The same is true for 0. The two adjacent numbers are +1 and -1 which average to zero. You can extend that as far as you like on the real number line, but not to infinity because infinity is not a real number.

Is this unique property advantageous or should we think of 0 as the average of +0 and -0 (which become a necessary requirement thereby)?

Zero is certainly unique, but not for the reason you have given. For example, there is no other number the same distance from the origin as zero, because zero is the origin! Zero is unique in other ways, but I am not a numerologist so I better just leave it at that.