Mathematics
Grade 10
15 min
Convert fractions to decimals
Convert fractions to decimals
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1
Introduction & Learning Objectives
Learning Objectives
Apply the long division algorithm to convert any rational number into its decimal representation.
Logically deduce whether a fraction will result in a terminating or repeating decimal by analyzing the prime factors of its denominator.
Justify why a fraction p/q, in simplest form, terminates if and only if the prime factorization of q contains only powers of 2 and 5.
Represent repeating decimals accurately using bar notation (vinculum).
Analyze the structure of the long division algorithm to prove why the decimal representation of any rational number must either terminate or repeat.
Apply the conversion of fractions to decimals in the context of trigonometric ratio approximations and geometric problems.
If a computer can only store a finite number of decim...
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Key Concepts & Vocabulary
TermDefinitionExample
Rational NumberA number that can be expressed as a quotient or fraction p/q of two integers, where p is the numerator and q is the non-zero denominator.The number 7/8 is a rational number.
Terminating DecimalA decimal representation that has a finite number of digits after the decimal point.3/4 = 0.75. The decimal ends after the digit 5.
Repeating DecimalA decimal representation that has a digit or a sequence of digits that repeats infinitely.2/11 = 0.181818... The sequence '18' repeats forever.
Vinculum (Bar Notation)A horizontal line placed over the repeating digits (the repetend) in a repeating decimal to indicate the repeating block.For 2/11 = 0.181818..., we write 0.overline{18}.
Prime FactorizationThe process of expressing a composite number as a uniq...
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Core Formulas
The Division Algorithm
p/q = p ÷ q
The fundamental rule for converting a fraction to a decimal is to divide the numerator (p) by the denominator (q). This algorithm's step-by-step process reveals the decimal representation.
Terminating Decimal Test
A simplified fraction p/q terminates if and only if its denominator q can be written in the form q = 2^n * 5^m, where n and m are non-negative integers.
This is a logical test to predict a terminating decimal without performing division. It works because our decimal system is base-10 (2 * 5), so any denominator with only prime factors of 2 and 5 can be multiplied to become a power of 10.
Repeating Decimal Test
A simplified fraction p/q repeats if the prime factorization of its denominator q contains any prime factor oth...
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Challenging
Which statement is the most critical component in a logical proof that a simplified fraction p/q results in a terminating decimal *if and only if* the prime factorization of q contains only powers of 2 and 5?
A.The long division algorithm must be used to show the remainder becomes zero.
B.The ability to create an equivalent fraction p'/q' where q' is a power of 10 (10^k) is the key link.
C.The numerator p must be smaller than the denominator q for the proof to hold.
D.The fundamental theorem of arithmetic guarantees a unique prime factorization for q.
Challenging
When converting the fraction p/q to a decimal, the division algorithm generates a sequence of remainders. For the specific fraction 1/7, what is the logical reason that the length of the repeating block of digits (the repetend) MUST be less than 7?
A.Because there are only 6 possible non-zero remainders (1, 2, 3, 4, 5, 6) when dividing by 7, a remainder must repeat within 6 steps.
B.Because 7 is a prime number, and the length of the repetend is always one less than the prime.
C.Because the numerator is 1, which forces the cycle to be short.
D.Because the first remainder is 1, and it must repeat before reaching a remainder of 7.
Challenging
The radius of a circle is given as (5/16) cm. The area is calculated using A = πr². A student claims that because π is irrational, the decimal representation of the exact area must be non-repeating and non-terminating. However, what is the logical nature of the decimal representation of the coefficient of π, which is r²?
A.r² = 25/32, which is a repeating decimal because 32 is not a power of 10.
B.r² = 25/256, which is a repeating decimal because 256 is a large number.
C.r² = 25/256, which is a terminating decimal because its denominator (256 = 2^8) only has 2 as a prime factor.
D.r² = 10/32, which simplifies to 5/16, a terminating decimal.
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