Proving 0.999... = 1
Proving 0.999... = 1 is a Grade 7 math skill in Reveal Math Accelerated, Unit 13: Irrational Numbers, Exponents, and Scientific Notation, where students use the properties of infinite repeating decimals and fraction equivalences to rigorously demonstrate that the repeating decimal 0.999... is exactly equal to 1. This result challenges intuition and deepens understanding of the real number system.
Key Concepts
Property The repeating decimal $0.\overline{9}$ (or $0.999...$) is mathematically equal to exactly 1.
$$0.\overline{9} = 1$$.
Examples Example 1: Algebraic Proof Let $x = 0.999...$ Multiply both sides by 10: $10x = 9.999...$ Subtract the original equation: $$10x x = 9.999... 0.999...$$ $$9x = 9$$ $$x = 1$$.
Common Questions
Does 0.999... actually equal 1?
Yes. 0.999... (where 9 repeats forever) is exactly equal to 1, not just approximately. This can be shown by writing x = 0.999..., multiplying both sides by 10 to get 10x = 9.999..., then subtracting to get 9x = 9, so x = 1.
What is an algebraic proof that 0.999... = 1?
Let x = 0.999.... Multiply both sides by 10: 10x = 9.999.... Subtract the first equation from the second: 9x = 9. Dividing both sides by 9 gives x = 1, so 0.999... = 1.
What fraction is equal to 0.999...?
0.999... = 1/1 = 1. Alternatively, since 1/3 = 0.333..., multiplying both sides by 3 gives 1 = 0.999..., showing the equivalence through fraction reasoning.
What is Reveal Math Accelerated Unit 13 about?
Unit 13 covers Irrational Numbers, Exponents, and Scientific Notation, including classifying real numbers, repeating and terminating decimals, exponent rules, and scientific notation.