Mathematics

Mathematics

Problem

Show that the recurring decimal 0.833‾0.83\overline{3} is a rational number. Express it in the form pq\frac{p}{q} where pp and qq are integers and q≠0q\neq0.
Intuitive explanation

The key insight is that any repeating decimal can be converted to a fraction by using algebra to "cancel out" the repeating part. Since only the 33 repeats, multiply by powers of 1010 strategically—once to shift past the non-repeating digit, and again to align the repeating blocks so you can subtract and eliminate the infinite tail. This transforms an infinite decimal into a finite equation you can solve.

Intuitive explanation

The key insight is that any repeating decimal can be converted to a fraction by using algebra to "cancel out" the repeating part. Since only the 33 repeats, multiply by powers of 1010 strategically—once to shift past the non-repeating digit, and again to align the repeating blocks so you can subtract and eliminate the infinite tail. This transforms an infinite decimal into a finite equation you can solve.

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