Mathematics
Grade 10
15 min
Convert decimals to fractions
Convert decimals to fractions
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1
Introduction & Learning Objectives
Learning Objectives
Differentiate between terminating, repeating, and non-repeating decimals based on their fractional representations.
Formulate a logical, step-by-step algorithm for converting any terminating decimal into its simplest fractional form.
Construct an algebraic proof to convert any purely repeating decimal (e.g., 0.333...) into a rational number.
Develop and apply an algebraic method to convert mixed repeating decimals (e.g., 0.12333...) into fractions.
Justify why non-repeating, non-terminating decimals (irrational numbers) cannot be expressed as a fraction of two integers.
Apply the concept of infinite geometric series to validate the conversion of repeating decimals to fractions.
Is 0.999... really and truly equal to 1? 🤔 Let's use the power of algebr...
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Key Concepts & Vocabulary
TermDefinitionExample
Rational NumberAny number that can be expressed as a quotient or fraction p/q of two integers, where p is the numerator and q is a non-zero denominator. All terminating and repeating decimals are rational.7/3, -5 (as -5/1), and 0.25 (as 1/4) are all rational numbers.
Terminating DecimalA decimal number that has a finite number of digits after the decimal point. It can be written as a fraction with a denominator that is a power of 10.0.875 is a terminating decimal.
Repeating DecimalA decimal number that has a digit or a block of digits that repeat infinitely. The repeating block is denoted with a bar (vinculum) over the digits.0.555... is written as 0.overline{5}. 2.1343434... is written as 2.1overline{34}.
Period (of a repeating decimal)The digit or block of digits t...
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Core Formulas
Terminating Decimal Conversion Algorithm
d = \frac{n}{10^k}
For a terminating decimal `d`, write the digits of the decimal `n` as the numerator. The denominator is 10 raised to the power of `k`, where `k` is the number of digits after the decimal point. Always simplify the resulting fraction.
Purely Repeating Decimal Conversion (Algebraic Method)
Given x = 0.\overline{p}, then 10^k x - x = p. Solve for x.
Let `x` be the decimal. Let `k` be the length of the period (the repeating block). Multiply `x` by `10^k`. Subtract the original equation (`x`) from the new one to eliminate the repeating part, then solve for `x`.
Mixed Repeating Decimal Conversion (Algebraic Method)
Given x = 0.n\overline{p}, then 10^{k+j} x - 10^k x = \text{integer}. Solve for x.
Let `k` be the nu...
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Challenging
Convert the complex mixed repeating decimal 1.23overline{45} into a single, simplified fraction (improper fraction).
A.12222/9900
B.12333/9990
C.679/550
D.679/55
Challenging
Which statement provides the most rigorous logical justification for why a non-repeating, non-terminating decimal (an irrational number) cannot be expressed as a fraction of two integers?
A.The number of digits is infinite, so the denominator would have to be infinitely large.
B.Calculators cannot display the full number, so a fraction cannot be found.
C.The division of any two integers (p/q) will always result in a decimal that either terminates or eventually enters a repeating cycle of digits.
D.Irrational numbers were defined by the ancient Greeks as numbers that are not fractions, so they cannot be fractions.
Challenging
For the decimal x = 0.overline{7}, demonstrate that the algebraic method (10x - x) and the infinite geometric series formula (S = a / (1-r)) produce the same fractional result.
A.Both methods correctly result in 7/9.
B.The algebraic method gives 7/9, but the series method gives 7/10.
C.Both methods are approximations and result in slightly different fractions.
D.The algebraic method gives 7/10, and the series method gives 7/9.
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