Repeating decimals and the real numbers

A terminating decimal is a fraction over a power of 10: 0.35 = 35/100 = 7/20.

A repeating decimal: subtract so the endless tail cancels. If the repeat starts at once, call the decimal x and multiply by 10ⁿ, where n is how many digits repeat:

• x = 0.666…: 10x = 6.666…, subtract to get 9x = 6, so x = 2/3;
• x = 0.4545…: 100x = 45.4545…, so 99x = 45 and x = 5/11.

Some digits before the repeat? First move them in front of the point (multiply by 10ᵐ), then multiply by 10ⁿ more for the repeat, and subtract:

• x = 0.1666…: 10x = 1.666…, 100x = 16.666…, so 90x = 15 and x = 1/6;
• x = 2.35777…: 100x = 235.777…, 1000x = 2357.777…, so 900x = 2122 and x = 1061/450.

A curious number. 1/7 = 0.142857142857…. Its block 142857 is a cyclic number: multiply it by 1 to 6 and you get the same digits, rotated (285714, 428571, 571428, 714285, 857142).

Irrational numbers have decimals that never end and never repeat: √2 = 1.41421356…, π = 3.14159265…. So:

• terminates or repeats → rational;
• neither → irrational.

0.999… = 1 exactly. With x = 0.999…, 10x = 9.999…, so 9x = 9 and x = 1. Every terminating decimal has a twin ending in repeating 9s.

The real numbers. The rationals and irrationals together make the real numbers, ℝ: ℕ inside ℤ, inside ℚ, with the irrationals filling the gaps, all on one unbroken line. Even that is not the end: no real number squares to –1, so mathematicians created the imaginary numbers, with i × i = –1.