My three favorite numbers, written in binary (base-two) notation are 3=0.0010010

Question

My three favorite numbers, written in binary (base-two) notation are
3=0.00100100001111110110...two
e 2=0.10110111111000010101...two
1/7 =0.00100100100100100100...two ,

where for each we have shown the first 22 binary digits. Of course each of these expansions is nonterminating so go on indefinitely.

Prove or disprove: two of these numbers have the same digit in infinitely many of the “binary” places.

(For example, the first two number have the same digit in positions 2, 3, 5, 6, 11, 16, 17, and 18 after the “binary point.” Must this list of positions of like digits be infinitely long for some pair of the numbers?)

Explanation / Answer

"Two of these numbers have the same digit in infinitely many of the “binary” places".

This statement is true.

Proof: Let us assume this statement is false i.e two of these numbers have the same digit in finitely many of the binary places. Let the last place where the same digit appears be k.

Now, from k+1 th place onwards, the digits are different i.e one number has digit 0 and the other has 1.

Then, if we add the two numbers, we get 1111...... from the k+1th place onwards which is recurring and therefore the sum is a rational number.

But two of the given numbers are transcadental. The sum of a transcadental number and any other number is transcadental and cannot be rational.

Therefore we arrive at a contradiction and our assumption is false.

Hence the given statement is true.

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