To show that negative exponents are indeed the representation of fractions in powers of the radix let z be any radix and n be any exponent. For simplicity assume that both z and n are integers but the proof is equally valid if they are real numbers.

In this proof the caret symbol, ^, means "raised to the power of".

Now by definition z^n is greater than z^(n-1), which in turn is greater than z^(n-2) for some integer n. This sequence can continue until z^(n-n)=z^0=1 is reached. The sequence that is established by the monotonically decreasing powers of z is such that the magnitude of every value of a power is less than the magnitude of the previous value.

Now, when the power reaches zero, there are no more positive integers. In order for the sequence to continue any further, exponents must be less than zero or negative. Also, the value of the power which follows z^(n-n)=z^0=1 is z^(n-n-1)=z^(-1) and must be less than its predecessor and therefore must be a fraction. Also, it must be a power of z to belong to the sequence and continue the pattern established with the positive exponents. The only number that satifies both of these criteria is the fraction 1/z^1. Likewise z^(-2) must be written as the fraction 1/z^2 to be a member of the sequence of decreasing powers of z.

The negative exponents, therefore, generate fractions as a consequence of the monotonic nature of the powers of a radix and the pattern established in the sequence generated by monotonically decreasing exponents of that radix.