Conceptual Bottlenecks Math Example 3

Follow the full solution, then compare it with the other examples linked below.

Example 3

easy
Students often confuse 0.999…0.999\ldots with something 'just below 1'. Show that 0.999…=10.999\ldots = 1 exactly.

Solution

  1. 1
    Let x=0.999…x = 0.999\ldots. Multiply by 10: 10x=9.999…10x = 9.999\ldots.
  2. 2
    Subtract: 10x−x=9.999…−0.999…=910x - x = 9.999\ldots - 0.999\ldots = 9.
  3. 3
    So 9x=99x = 9, giving x=1x = 1.

Answer

0.999…=10.999\ldots = 1
The conceptual bottleneck here is understanding that an infinite decimal is a limit, not an approximation. 0.999…0.999\ldots represents the limit of the sequence 0.9,0.99,0.999,…0.9, 0.99, 0.999, \ldots, which equals 1 exactly.

About Conceptual Bottlenecks

Specific concepts or ideas whose misunderstanding blocks progress across a wide range of related mathematical topics.

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Example 4 medium

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