The square root of 2 also has a non-repeating and non-terminating decimal expansion, and the square root of 2 shows up all the time in basic geometry (e.g. diagonal of a square with side length 1, comes up in 45-45-90 triangles, etc.). The following also have non-repeating and non-terminating decimal expansions:

0.101001000100001000001...

0.12345678910111213141516...

I mention these because people sometimes think "non-reapeating and non-terminating" (an irrational number) automatically means no pattern (or at least, no easily detectable pattern) in the decimal expansion, but that's not true. It is true, however, that all the standard irrational numbers one comes into contact with have decimal expansions that have no easily detectable pattern. In fact, no one knows whether the decimal expansion for the square root of 2 (or pi, or the square root of any non-perfect square, or the cube root of 47, etc.) contains each of the 10 digits infinitely many times (or anything at all about the limiting proportion for any of the digits).

The square root of 2 also has a non-repeating and non-terminating decimal expansion, and the square root of 2 shows up all the time in basic geometry (e.g. diagonal of a square with side length 1, comes up in 45-45-90 triangles, etc.). The following also have non-repeating and non-terminating decimal expansions:

ReplyDelete0.101001000100001000001...

0.12345678910111213141516...

I mention these because people sometimes think "non-reapeating and non-terminating" (an irrational number) automatically means no pattern (or at least, no easily detectable pattern) in the decimal expansion, but that's not true. It is true, however, that all the standard irrational numbers one comes into contact with have decimal expansions that have no easily detectable pattern. In fact, no one knows whether the decimal expansion for the square root of 2 (or pi, or the square root of any non-perfect square, or the cube root of 47, etc.) contains each of the 10 digits infinitely many times (or anything at all about the limiting proportion for any of the digits).