How do you Show that the set of powers of ten is countably infinite?

Answer 1

By:

1. Establishing a one-to-one relationship between the powers of 10 and the natural numbers which is a countably infinite set; AND
2. Restricting yourself to the rational powers of 10.

If you are talking about integer powers of 10, then condition 2 above is automatically enforced as all integers are rational numbers.

The easiest way is:

1. to take the logarithm to base 10 of the power of 10;
2. the result is a unique integer number and the integer numbers is a countable infinite set.

10⁰ → 0

10⁻¹ → -1

10¹ → 1

10⁻² → -2

10² → 2

etc.

Similarly, if the power is a rational number, the result of taking the logarithm to base 10 is the set of rational numbers which is also countably infinite:

The complete set of rationals can be generated by considering the numbers created by taking the integer pair (i, j) where i + j = n are all possible combinations of i ≥ 0, j ≥ 0 as n = 1, 2, 3, ..., and listing the rational numbers in the order -i/j (if i > 0) and i/j. This list includes all possible numerators and denominators, and can be placed in a one-to-one relationship with the counting numbers.

If the power is an irrational number (eg √2, e, π), then the powers of 10 are not an infinitely countable set as taking the logarithm results in a set of Irrational Numbers and the set of irrational numbers is not countable:

Assume the irrational numbers are countable, then they can all be listed in order.

Take the number by adding 1 to the first digit of the first number, 1 to the second digit of the second number, 1 to the third digit of the third number and so on. (If the original digit was a 9 it becomes a 0).

This creates a new number which is not on the list as it is different from the first, second, third, etc numbers.

Thus the assumption that the list is complete is false

Thus the cannot all be listed in order and so cannot be put in a one-to-one relation with the counting numbers.

Thus the irrational numbers is not an infinitely countable set.

Answer 2

By establishing a one-to-one correspondence with the set of natural numbers - or to the set of integers, which is also countably infinite.For example, for the positive powers of ten, you set the correspondence:

0 <--> 10^0

1 <--> 10^1

2 <--> 10^2

etc.

Answer 3

You cannot because it is not true. It is true only if you consider the set of integer powers of ten - which is NOT the same as the set of powers of ten. Even a minute spent thing of log10 will tell you that the set of powers of ten has a cardinality equal to the continuum.

Consider the mapping 10x to x where x is in Z.

This is a one-to-one mapping and so the cardinality of the two sets is the same. That is, the set of powers of ten is countably infinite.