continuum in number theory

Number of real numbers {continuum} is greater than number of natural numbers.

number types

Natural numbers are counting numbers: 0, 1, 2, 3, ... Counting numbers can be binary numbers. Integers are natural numbers plus their negatives: ... -3, -2, -1, 0, 1, 2, 3, ... Zero is its own negative. Rational numbers are integer fractions, such as -3/-1, -3/1, 3/-1, 3/1, -1/-3, -1/3, 1/-3, 1/3, and 0/integer, but not integer/0, because it has no definition. Rational numbers are repeated decimal numbers. Irrational numbers are non-repeating decimal numbers. Infinite operation series, making digit series, can represent irrational numbers {countable irrational number}. The number pi and all rational-number roots are countable irrational numbers {countably infinite}. However, most irrational numbers are not countable. Real numbers are rational numbers plus irrational numbers.

countable infinity

Numbers of natural numbers, integers, rational numbers, and countable irrational numbers are equal, because they can be put in one-to-one correspondence with counting numbers. See Figure 1.

Fraction has integer numerator and denominator. Countable irrational numbers are countable, because you can count operations and digits. Therefore, natural, rational, or countable irrational numbers have one-to-one correspondence with counting numbers.

continuum

Try to align natural numbers with non-countable irrational numbers in one-to-one correspondence by making an array, with natural numbers on one axis and non-countable irrational numbers on other axis. See Figure 2.

Mark diagonal, to take diagonal slash. See Figure 3.

Change the mark at first-row-and-column first position, at second-row-and-column second position, and so on. See Figure 4.

The new sequence is non-countable irrational number, because it randomly comes from the non-countable irrational-number array. However, it is not the same as any row or column sequence, because it differs from first row and column at first number, differs from second row and column at second number, and so on. If natural numbers and non-countable irrational numbers have one-to-one correspondence, the list must contain all possible non-countable irrational numbers. Therefore, natural numbers and non-countable irrational numbers have no one-to-one correspondence, and number of non-countable irrational numbers, and so number of real numbers, is greater than number of natural numbers.