dyeing, dying

The words dyeing, dying sound the same but have different meanings and spellings. Why do dyeing, dying sound the same even though they are completely different words?

The answer is simple: dyeing, dying are homophones of the English language.

dyeing
  1. :: verb

    Present participle of dye.

dying
  1. :: verb

    Present participle of die1.

  2. :: adjective

    About to die: dying patients.

  3. :: adjective

    Drawing to an end; declining: in the dying hours of the legislative session.

  4. :: adjective

    Done or uttered just before death: a dying request.

Definitions from Wiktionary, Creative Commons Attribution/Share-Alike License, from The American Heritage® Dictionary of the English Language, 4th Edition and Wordnik.

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About Homophones

Homophones (literally "same sound") are usually defined as words that share the same pronunciation, regardless of how they are spelled.

If they are spelled the same then they are also homographs (and homonyms); if they are spelled differently then they are also heterographs (literally "different writing").