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All numbers are interesting. The proof is by contradiction.
Assume we have found an uninteresting number. If it is not the first uninteresting number, we can find earlier uninteresting numbers until we have found the first one. But then this number becomes interesting because it is the first uninteresting number which is a characteristic no other number has.
Thus we have arrived at a contradiction and there can be no uninteresting numbers.

A number is said to be "interesting" if it has some property which is not shared with at least one other number.

Consider the integer 0. 0 has the property that, for any number k, k+0=k. This property is not shared with any other one dimensional number. So 0 is interesting. Let x be any positive number. Then -x has the property of being negative, and it does not share this property with x. Thus x and -x are interesting numbers. But, x is arbitrary, so any positive number and its additive inverse are interesting. So all real numbers are interesting.

This "proof" can be extended to the complex numbers. All complex numbers have an imaginary component to them, i.e. a number z is complex iff it can be expressed as a+bi for real numbers a and b. Because of the two dimensional nature of the complex numbers, it makes no sense to say "z1<z2", i.e. the complex numbers are not ordered. But, the real numbers are ordered, so each complex number has a property not shared with any real number (or it is a real number), and thus all numbers are interesting.

Induction does not work here, except for the integers, since there is no smallest number greater than 0 in any number set except the integers.

An alternate proof

Let x be a mathematical object of some kind (a set, a function, a matrix, etc.). Consider the property 'is equal to x'. By definition of equals, x=x, but at the same time, x is the only mathematical object of any kind that equals x. So x is interesting because it has a property not shared with any other mathematical object. (This proof provided by pfft.)

There are many interesting numbers, according to the Penguin Book of Curious and Interesting Numbers (David Wells, 1986, 0-14-008029-5), the smallest interesting number is...